NP-Completeness is Illuminated

Another literary reference to a hard combinatorial problem. Jonathan Safran Foer describes plans for a wedding reception in his disjointed novel Everything is Illimuniated.

The hardwood floors were covered in white canvas, and tables were set in a line stretching from the master bedroom to the kitchen, each feathered with precisely positioned name cards, whose placement had been agonized over for weeks. (Avra cannot sit next to Zosha, but should be near Yoske and Libby, but not if it means seating Libby near Anshel, or Anshel near Avra, or Avra anywhere near the centerpieces, because he's terribly allergic and will die. And by all means keep the Uprighthers and Slouchers on opposite sides of the table.)

The Beauty of the Magic Number

Baseball has a large number of mathematical nuggets but since my childhood I have always liked the simplicity of the magic number. In a division race, the magic number is the minimum number of wins by the first place team and number of losses of the second place team to guarantee the first place team wins the division.

Let's do an example. As I write this the New York Yankees have 94 wins, the Boston Red Sox have 60 losses. The easiest way to compute the magic number comes from working backwards from the definition. There are 162 games in a season so the Yankees magic number is 162+1-(94+60) = 9. Any combination of nine Yankees wins and Red Sox losses and the Yankees wins the American League East. The "+1" comes from the fact that in a tie the Yankees would still need to win a one-game playoff to win the division.

What can the magic number teach us about complexity? Consider the RIOT Baseball Project at Berkeley. Not satisfied with the magic number, the project computes the First Place Clinch Number as the "Number of additional games, if won, guarantees a first-place finish." To compute this number one has to look not only at the current standings but the schedule of remaining games between the teams.

My main issue of the clinch number relates to complexity. Not only is it more complicated to compute; to update the clinch number after a game sometimes requires recomputing the number from scratch. The magic number has a simple update function counting down like a rocket launch. Yankees win the magic number drops by one. Red Sox lose the magic number drops by one. If the Yankees beat the Red Sox, both events happen so the magic number drops by two. And once the magic number hits zero you pop the champagne. That's the beauty of the magic number.

Quantum in the North; Random in the South

A couple of interesting workshops happening this week both studying information, not the usual classical notions of information but quantum and random information.

At Banff in the Canadian Rockies we have the BIRS Workshop on Quantum Computation and Information Theory. This workshop will examine the properties of quantum information and tie it together with people working on quantum algorithms and complexity.

Meanwhile in Córdoba, Argentina on the edge of the Sierra Chica mountain is the Conference on Logic, Computability and Randomness 2004. Much of this conference focuses on random sets, not from a probability distribution but single sets that "look" random. Rater surprisingly, whether we define random sets by passing statistical tests, foiling betting strategies or using Kolmogorov complexity, these notions often lead to equivalent definitions of random.

While circumstances keep me in Chicago I say to the participants of both meetings: Enjoy the mountains and save some theorems for me.

NSF News

Arden Bement, who has been serving as the acting director of the National Science Foundation since Rita Colwell stepped down in February, has been nominated for the permanent director position.

The US Senate continues to work on the appropriation bills to set funding levels for next years US fiscal year that starts October 1. The CRA has a summary of the various bills affecting research funding in computer science (see also this note from AIP). Most worrisome (as I've mentioned before) is the funding for the VA-HUD bill that covers the NSF. The House committee would cut the budget by 2%. The CRA has an Advocacy Alert, still time to contact your senators.

Email's Curse of Success

Computer Scientists have communicated via email since the mid-1980's. Back then email worked quite well: You would send a message and usually get a quick response. We avoided telephone tag, trying to reach each other by phone when we needed to talk to each other quickly. Research ideas spread quickly; the world became smaller. Even within a department communication became paperless as one can get a message out to everyone far quicker electronically.

So what went wrong? We still use email today as the primary source of communication among computer scientists. But send a message today and I've learned to wait on average a couple of days to expect a response, if I get one at all.

Spam is the obvious culprit. Spam does clog our inboxes and even worse many of us don't carefully go through our spam folders and some legitimate mail gets unread. Spam has also made some computer scientists reluctant to share their email addresses online. But spam is not the only issue.

Email has become the communication of choice in the rest of the world as well. Besides messages from other scientists, I get email about my daughter's soccer team, announcements of upcoming concerts, warnings from the local police departments, a morning summary of the New York Times, financial information, utility bills and much more. All legitimate and usually useful email but it takes longer to work through it and slows down the time to respond to other scientists. Not to mention the many other web distractions such as news and weblogs (So stop reading this blog and respond to my emails. You know who you are.)

I can't rely on older technologies; since computer scientists expect email they check even less often their phone messages and postal mail. I can't rely on newer technology; computer scientists are surprisingly slow in adapting to new tools (like mail attachments) and it'll be years before instant messaging becomes common in the scientific community.

Oddly enough in our highly connected society it becomes harder and harder to get someones attention. So what am I doing? Slowly collecting the cell phone numbers of other computer scientists. Want mine? Send me an email.

Is the AP Test to Blame for Shifting CS Enrollments?

It is no secret that undergraduate enrollments in computer science have been dropping over the past five years. Students follow the money and, with the perception of a weaker job market in computer-oriented careers, less students are willing to study computer science.

Why does computer science follow the job market so closely? We don't see such swings in physics or history but such swings are common in engineering disciplines. Are undergraduate viewing computer science more as engineering than science? And why?

One theory I recently heard puts the blame on the Advanced Placement (AP) Computer Science Exam given to high school students. The reasoning goes as follows: The AP exam has a strong emphasis on the Java programming language and so high school teachers, teaching to the exam, focus most of their course on syntax and coding of Java. This gives the impression to students that computer science = programming.

I don't agree with this assessment. I looked at some sample CS AP tests. The tests, particularly the AB exam, requires some knowledge of data structures and simple algorithms. Nothing deep but enough that students should realize that computer science is more than just programming.

There was a surge of interest in computer science when I started college in the early 1980's (with the advent of personal computers) before an AP test in Computer Science even existed. Also I've heard of declines in enrollments outside the US where they don't use the AP tests.

But in the end we shouldn't be that worried about shifting enrollments. Advances in computer technology have helped drive computer science from a virtually non-existent discipline forty years ago to one that many universities now consider one of their most important departments. Better to have enrollments that swing up and down with the state of the computer industry than one that stagnates at the low end.

Favorite Theorems: List Decoding

August Edition

In coding theory, one typically maps a string to a code such that with some small amount of error in the code one can still recover the original string. What if the amount of error is too much to give a unique decoding? In the 1950s Peter Elias suggested the idea of list decoding, coming up with a short list of possibilities one of which is correct.

Madhu Sudan showed that list decoding can be achieved in scenarios where one cannot do unique decoding.

Decoding of Reed-Solomon Codes Beyond the Error-Correction Bound by Madhu Sudan

In this paper Sudan gives a polynomial-time list-decoding algorithm that can deal with errors in the code beyond what regular codes can handle. Later Guruswami and Sudan give a polynomial-time algorithm that handles what is believed to be the best possible amount of error.

List-decodable codes have had applications to many areas including pseudo-random generators, extractors, hard-core predicates, probabilistically-checkable proofs and a neat result by Sivakumar on the implications of SAT being membership-comparable.

We've seen many other important papers in coding theory from computer scientists over the last decade. Besides the work on list decoding I should also mention Spielman's breakthrough result showing linear time encodable and decodable codes building on the initial work of using expander graphs for codes developed by Sipser and Spielman.

For much more on coding see the surveys by Sudan on List Decoding and Guruswami on Error-correcting codes and Expander Graphs.

Which Affiliation?

Dieter van Melkebeek and I had a mild disagreement over an issue in writing a paper and he suggested I discuss it on my weblog. So here goes.

We're working on a journal version of a paper where we did the research back when we were both in New Jersey (Dieter is now on the Wisconsin faculty). Dieter listed our current institutions on the paper. I think that we should list our institutional affiliation when we performed the research, or more precisely the place paying the salary at that time. I don't feel that strongly about the issue so I am letting Dieter have his way, especially since he did the vast majority of the writing. I did ask that we have footnotes explaining our affiliations at the time of the research. (As a side note, footnotes should also contain other support information such as grants, where one was physically located when the research was done if different and contact information, though these days one needs only know how to spell my name correctly when using Google).

So I'll open this up to my readers. Which affiliation should one use? Or are we just arguing over an issue that nobody really cares about?

Just Because I'm a Scientist Doesn't Mean I Know Anything

My six-year old daughter playing with her friend on the computer ran into some problems. So she found me in the house and said, "Daddy, I know you are a computer scientist so you know everything about computers and we need you to fix our game."

I've learned to expect this attitude from adults. Often when I talk to a non-academic about being a computer scientist I get responses like, "Should I get Windows or a Mac?", "I was thinking of setting up a wireless network." or "What do you think of that Google IPO?". Of course this is on top of them thinking I have a cushy academic job teaching three hours a week with summers off.

For my daughter I just went ahead and fixed her problem. I know when she becomes a teenager she'll run circles around me on the computer and will say "When I was a kid I thought you knew everything about computers. What do you computer scientists do anyway?" Then we can talk.

The Electoral College

As everyone knows from the 2000 election, the United States does not use a majority rule to choose the president, rather they use a more complicated system known as the Electoral College. With some calls for the abolishment of the Electoral College, let's take a look at the College from a computer science point of view and see the rather clever device our founding fathers have created.

In short the Electoral College works as follows: 538 electors are allocated to states as the sum of the senators (2 for each state) and representatives (proportional to population). In most states, each voter picks a single candidate and the candidate that wins the most votes receives all of the electoral votes for that state. The candidate winning the majority of the electoral votes becomes president. More details here.

In computer science terms (assuming two candidates), we have a weighted majority of majorites or a depth-2 neural net. It has some properties that you would want:

• Monotonicity: If a candidate wins the election and more people vote for him, he will still win.
• Fairness: Barring a tie, if all the votes were switched the other candidate would win.

The College does lack symmetry, a permutation of the voters could lead to a different result. Only the simple majority function has symmetry, monotonicity and fairness. But symmetry is not necessary for an election scheme.

The United States is just that, a collection of fifty states each with their own laws, cultures and economies, united under some common principles. A simple majority would have the large populations centers overwhelm the rest of the country in choosing our leader. A majority of majorities would give some states far more power than their size would dictate. So a compromise was formed, a weighted majority of majorities to give small states some but not too much influence. The fact that this process does not always agree with majority is not a bug but a feature that preserves the balance between small and big states, rural and urban America. It also keeps balance between states of the same size, an lopsided vote in California would not overwhelm a closer vote in New York.

Some things I would change in the Electoral College: Electors should be required to vote for the candidate they represent; for each state we should have a ranked voting method instead of plurality takes all; the tie-breaking rules should be changed, now they give too much power to the small states.

The winner of the World Series in baseball is not the team that scores the most runs but the team that wins the most games, a majority of majorities and most people feel it gives a better indication of the better team. Why shouldn't elections deserve a system at least as sophisticated?

Is Encryption Doomed?

Considerable Slashdot discussion about a Technology Review article Is Encryption Doomed? subtitled Our entire information society rests on a fragile foundation that mathematicians are racing to dismantle.

That fragile foundation is the P versus NP question. Much of current cryptography is based on the NP problem Factoring which would have an efficient algorithm if P were equal to NP. If P = NP than no other method of public-key cryptography would be possible either. However, as I have argued before, the loss in public-key crypto would be greatly offset by the gain in efficiency of nearly every other aspect of our lives. Encryption's doom would be society's gain.

Factoring is not believed to be NP-complete and we could have the worst of all possible worlds with no cryptography and no new algorithms for problems we care about, one of five possibilities explored by Russell Impagliazzo.

Racing to dismantle is a bit of an overstatement. First of all we have nothing to worry about. As Juris Hartmanis once said, "We all know that P ≠ NP, we just don't know how to prove it." Remember too that all mathematics is based on unprovable assumptions about consistencies of logical theories. Doesn't seem to seriously bother anyone.

The article quotes Adleman as saying "From my perspective, we are no nearer to solving the problem now that we were when bell-bottom pants were cool." Adleman is an optimist; we do not even currently have a good approach to showing P ≠ NP. We are much further away from solving the problem than ever before.

Theory in the Coca-Cola Capital of Iowa

  

Today I gave the talk at the Atlantic Theory Seminar, not Atlantic as in ocean but Atlantic, Iowa (population 7,257). Located halfway between the theory groups of Iowa State and Nebraska, the Cass County branch of the Iowa Western Community College hosts a theory seminar every couple of months. Pretty impressive that these theory groups, faculty, postdocs and students, drive two hours each just to join for a seminar and talk theory with each other. Quite an enjoyable day in rural America.

Sorry about the quality of the picture above. We didn't have a digital camera so I took the picture with my cell phone.

Computationally Simple One-Way Functions

An upcoming FOCS paper is drawing a lot of interest at TTI and Chicago, Cryptography in NC⁰ by Benny Applebaum, Yuval Ishai and Eyal Kushilevitz.

Don't let "NC⁰" scare you, it simply means that every output bit depends on a constant number of input bits. In this paper the authors show that under reasonable assumptions one can have one-way functions and pseudo-random generators where each output bit depends on only four input bits. Quite surprising that such simple functions can be so hard to invert.

Their proof uses a clever but not overly complicated reduction using randomized polynomials from a larger class (⊕L/poly) to NC⁰ and so if one-way function and PRGs exist in the larger class than slightly weaker versions exist in NC⁰ respectively. The larger class contains many functions conjectured to be one-way or a PRG, including those based on factoring.

I remember Mike Sipser mentioning in a complexity class I took back in the mid-80's that "We don't even know how to show there are no one-way functions in NC⁰." Now we know why.

What happened to the future?

Disney World has two areas originally designed to give a glimpse of "the future", Tommorowland in Magic Kingdom and Future World at EPCOT, which itself once stood for Experimental Prototype Community of Tomorrow. These areas give somewhat a nostalgic view of the future from the time of my childhood but not of the future from today.

Not only has society's view of the future changed but even the view about the future has changed. We live in an era of massive changes in technology particularly in access to communication and information. In the "Spaceship Earth" ride that shows the history and future of communications, the future shows two kids seeing and playing games with each other through video monitors, a task easily accomplished today with webcam-equipped networked PCs. But the ride missed many aspects of computers and the web. Who would have thought when I was a kid that I would now be writing this post on a commuter train that soon will be read around the world.

On the other hand we also dreamed of flying cars and space planes but the technology of transportation has not significantly changed since I was born and I don't expect major changes in the next forty years. So the future has come but not quite as we expected.

Technology has made the world a more homogeneous place. When at the Norway pavilion at EPCOT, an American visitor asked a Disney worker from Norway about what the country was like to visit. "The towns are like mid-size American cities" was the reply. I guess it is a small world after all.

ε and the Olympics

Back from Disney World and ready to tackle the new academic year. Thanks to Adam Klivans for his guest posts. Interesting to see that COLT glorifies the open problems. In complexity we prefer to see them closed.

Varsha Dani emailed a note about an article describing an Amazon tribe with no concept of numbers. I guess they don't send anyone to the olympics where numbers seem king.

Watching gymnastics I am always amazed how winning seems to depend on sticking the landing. After a rather complicated and lenghty routine why should whether your feet move at the end be the difference between Gold and Bronze? The answer lies in the ε; the best gymnasts can uniformly hit the major elements of their routines but consistently sticking the landing is much more difficult and that what makes or breaks the result.

We see the ε issue in many scenarios, most notably the 2000 US presidential election where confusing ballots in one county created a huge controversy. The budget of the NEC Research Institute is miniscule compared to the revenue of NEC corporate. A couple of years ago NEC eeked out a profit about the same size of the cost of the Institute and all of a sudden the NEC Research Institute looked very expensive to NEC. In another example, consider the relatively large power a small party can have in a coalition in a parlimentary system.

The best way to avoid the ε problem is by decisive victories. Solidly beat your opponent in an election or have enough parlimentary seats so you don't need small partners. Make enough of a profit so that a cheap research lab stays that way. One only gets the ε problem when one has a statistical tie and then the small things take on great importance.

In gymnastics one would want a system where a greatly superior gymnast can score high enough that a small jump on the landing won't make a difference. But the current system doesn't allow that; with a top score of 10 the world's best gymnasts all get well over nine, making it impossible to put enough of a gap between a the best and the rest.

"Conclusions and Open Problems" (by Adam Klivans)

Unless Lance has been involved in some unfortunate roller coster accident, this is most likely my last post. It remains to be seen whether I will be allowed to keep my posting privileges here on the board (place your bets on "No"). What I've taken away from this experience, more than anything else, is that posting every day is a tremendous time sink. Lance, kudos to you for the almost daily missive.

At the end of most technical talks is the obligatory "Conclusions and Open Problems" slide, usually the least thought out moment of the presentation, which consists of a brief summary of the talk and a list of the most obvious (difficult) open problems. For the last few years, COLT has glorified the open problems section and allocates about an hour of time for a presentation of open problems. The open problems themselves must be submitted months beforehand and are refereed (how rigorously is anyone's guess); accepted problems appear in the proceedings. A list of this year's open problems can be found on the COLT 2004 program schedule -- the session was held on Friday evening.

Are there any other computer science conferences where open problems are refereed and given their own slot in the program? It always seems to work out well at COLT, and while I am aware of open problems being presented at rump sessions at various conferences, I don't know of other venues which require advance submission of problems.

With that, gentle reader, I return you to Lance's wise embrace. If I ever get to guest blog again, I will reveal what's hidden in Lance's "Private" directory here on fortnow.com. Some "lost" theorems apparently-- here's an aborted post entitled "Soon I will be famous: a simple proof that NP = coNP" -- and here's another unfinished note with some not so nice things to say about Algorithms. And what's this? A love letter to Jessica Simpson? Oh if only we had more time.

Constant Depth Circuits, Fourier Transform, and Learnability (by Adam Klivans)

First, thanks to Jeff Erickson for the lone comment on my previous post. For a second I was worried that I would have to post about Quantum Learning to bait Scott Aaronson into commenting. Fortunately, we're past that. I also realize now that being in Chicago I'm just a two hour drive away from you Jeff at UIUC. Maybe you, Ernie, and I should get some 3-D hotcakes at the local Waffle House-- on me.

Many top notch researchers come to visit the Toyota Technological Insitute here in Chicago. This fall alone Lenore Blum, Bruno Codenotti, Prahladh Harsha, and Jaikumar Radhakrishnan will be in residence along with TTI's regular faculty.

Yishay Mansour visited in August for about two weeks, and I had the chance to ask him about his work along with N. Linial and N. Nisan which pioneered the use of discrete Fourier analysis in computational learning theory. The paper Constant Depth Circuits, Fourier Transform, and Learnability gives a quasi-polynomial time learning algorithm for constant depth circuits with respect to the uniform distribution. More importantly, it pointed out a connection between the Fourier concentration of a Boolean function and its learnability.

If f is a Boolean function, we could imagine writing f as a multilinear polynomial in n variables mapping {-1,1}^n to {-1,1}. Every Boolean function can be written as such a polynomial whose total degree is at most n. Each coefficient of this polynomial measures the correlation of f with that monomial. These are the Fourier coefficients of f. Linial, Mansour, and Nisan showed that if the sum of the squares of the coefficients of monomials of degree d and larger is small, then there is an n^{O(d)} time algorithm for learning f with respect to the uniform distribution-- roughly speaking it is sufficient to estimate the coefficients of only the low degree monomials and output this polynomial.

What does any of this have to do with constant depth circuits? They also proved that for any circuit of depth d, the sum of the squares of the coefficients on the terms of degree (log n)^d and higher decays rapidly. From the above paragraph this gives us a quasi-polynomial time algorithm. For a rough intuition as to why this is true, consider one implication of Hastad's Switching Lemma, namely that parity cannot even be approximated by constant depth circuits. Thus every coefficient of a sufficiently large monomial (each monomial is a parity function) must be small.

As it turns out, according to Yishay, the work stemmed from an effort to prove circuit lower bounds, rather than a plan to develop new learning algorithms. The authors were inspired by Kahn, Kalai, and Linial's work on the influence of variables on Boolean functions and thought that discrete Fourier analysis might be the right tool for studying circuits. They happened to be correct-- in an unexpected way.

Banff Revisited (by Adam Klivans)

Hey, you're back! And I'm really starting to feel comfortable at this posting gig. Thanks for all of the comments I got about my previous post-- all zero of them.

A few weeks ago Lance reported on the events from a complexity workshop he was attending at the Banff International Research Station, an institute similar to the International Space Station except that it's in Canada rather than outer space.

What Lance didn't mention was that at the exact same time, some important computer science conferences were taking place just down the street at the Banff Park Lodge. I was participating in the 17th annual Conference on Learning Theory (COLT) which was co-located with the International Conference on Machine Learning (ICML) and the Conference on Uncertainty in Artificial Intelligence (UAI).

Back when COLT stood for Computational Learning Theory (circa 1988-2002), the conference was known for focusing on the computational complexity of machine learning. Nowadays the conference still operates from a theoretical perspective, but, as the Conference on Learning Theory, the program covers everything from game theory and economics to kernel methods in addition to traditional PAC style results.

A PAC style result that appeared in COLT 2004 which may be of interest to the readers of this web log is Polynomial-Time Prediction Strategy with Almost Optimal Mistake Probability by Nader Bshouty. His paper solves a problem in the online learning setting. In this setting, an unknown function f is chosen from some fixed concept class and at time t a learner is presented with an input x chosen according to some distribution D.

The goal of the learner is to run in time polynomial in log t (and all of the other relevant parameters) and predict the value of f(x) with mistake probability O(1/t) (Haussler, Littlestone, and Warmuth proved that this mistake probability is optimal). Bshouty shows that if the concept class is PAC learnable, then there exists an online learning algorithm which runs in time polynomial in log t and achieves mistake probability O(log t/ t). His algorithm has an exponential improvement in running time as previous solutions ran in time polynomial in t. The algorithm works by iteratively creating a branching program based on hypotheses considered at previous time steps. A boosting-type procedure dictates which branch to take for any new input.

In the spirit of being controversial (Lance asked me to be controversial), I could discuss the pros and cons of the change from Computational Learning Theory to Conference on Learning Theory (as a concrete example of differences in the community, about half the participants at the COLT business meeting wanted to see COLT co-located with STOC in 2006-- University of Washington folks make your move-- and the others wanted to co-locate with ICML). I'll leave that, however, for my faithful readers to debate. I will point out that it's hard to argue with the increase in attendance at COLT over the last few years.

Favorite Theorems: The Harmonic Sieve (by Adam Klivans)

What does Lance Fortnow do after getting a paper accepted to FOCS? He goes to Disneyworld for a week, of course. While Lance blasts pasts celestial satellites on Space Mountain, I will humbly try to replace the irreplaceable.

Considering the topic of yesterday's post, if there were a list of favorite theorems in computational learning theory, a field which makes only brief appearances here on the weblog despite its many connections to computational complexity, Jeff Jackson's algorithm for learning DNF formulas would certainly be on it.

A DNF formula is a Boolean formula written as an OR of ANDs (e.g. x_1 and x_2 OR x_3 and x_5). The size of a DNF formula is equal to the number of terms or ANDs. The problem of PAC learning an unknown polynomial-size (in n, the number of variables) DNF formula with respect to an arbitrary distribution remains one of the most notorious open problems in the field (for background on PAC learning see this post or Kearns and Vazirani's excellent book An Introduction to Computational Learning Theory).

In fact, even if we restrict the underlying distribution on examples to be the uniform distribution, the fastest algorithm for learning DNF formulas runs in quasi-polynomial time (a result due to K. Verbeurgt-- the main idea being that only terms of logarithimic length have a chance at being satisfied, so longer terms can be ignored).

If, however, we allow the learner to make queries to the unknown DNF formula, i.e. if the learner can choose any input x and ask for the value of the DNF formula evaluated on x, then the learner can succeed in polynomial-time.

The solution, due to Jeff Jackson in 1994, shows how to learn polynomial-size DNF formulas with respect to the uniform distribution in polynomial-time (again assuming the learner has query access to the unknown DNF). His algorithm, which he has called the Harmonic Sieve due to its use of Fourier analysis, builds on work due to Blum, Furst, Jackson, Kearns, Mansour, and Rudich (``Weakly Learning DNF and Characterizing Statistical Query Learning Using Fourier Analysis'') which showed that for any DNF formula with s terms, there exists a parity function which agrees with the DNF formula on roughly a 1/2 +1/s fraction of inputs.

The next step of the algorithm involves a novel application of Boosting algorithms (see this post for more on Boosting) for combining these parity functions to obtain an accurate hypothesis. The output of the Harmonic Sieve is not a DNF formula but a threshold of parity functions.

The Harmonic Sieve is one of the rare examples in computational learning theory of a polynomial-time algorithm for an expressive concept class. It is natural to ask whether the queries are essential for the algorithm. Unfortunately it seems like the answer is yes-- we do not know how to learn decision trees or even juntas (both strictly weaker concept classes than DNF formulas) in polynomial-time with respect to the uniform distribution unless the learner has query access to the unknown function. Removing the dependence on queries would be a real breakthrough.

By the way, the interface on blogger.com is worse than I ever could have imagined. Apologies in advance for formatting errors.

Favorite Theorems: Parallel Repetition

July Edition

Consider a simple Arthur-Merlin game: Arthur probabilistically chooses a string r sends it to Merlin who responds with y and then Arthur runs some algorithm A(r,y) to decide whether to accept. Merlin's goal is to achieve the highest acceptance probability possible p for Arthur. Suppose we run the game twice in parallel, Arthur sends r₁ and r₂ and Merlin sends y₁ and y₂ and Arthur accepts if A(r₁,y₁) AND A(r₂,y₂). The highest possible acceptance probability will be p².

Now consider the MIP model with two Merlins M₁ and M₂ who cannot communicate with each other. Arthur sends u and v to M₁ and M₂ respectively who respond with y and z. Arthur accepts based on some function A(u,v,y,z). Once again M₁ and M₂ try to achieve the highest possible acceptance probability p. Now we run the game twice in parallel, Arthur sending u₁ and u₂ to M₁ and v₁ and v₂ to M₂ receiving y₁ and y₂ from M₁ and z₁ and z₂ from M₂ and accepting if A(u₁,v₁,y₁,z₁) AND A(u₂,v₂,y₂,z₂).

One might assume that the best the provers can achieve is p² (an assumption in fact made in an early paper co-authored by a certain weblog author) but in some circumstances the provers can do better. However Ran Raz shows that if p is less than 1, one can get an exponential decrease in p with a polynomial number of parallel rounds in

A Parallel Repetition Theorem by Ran Raz

This paper settles one of the more perplexing aspects of multiple prover proof systems with a highly complicated proof. The result also plays a critical role in reducing the number of queries in probabilistically checkable proof systems which led to some optimal approximation bounds.

As a side note I am off on vacation tomorrow and Adam Klivans will guest blog in my absence. Enjoy.

Wisdom of Crowds

Keeping with this week's theme of prediction, I just finished reading The Wisdom of Crowds written by New Yorker writer James Surowiecki. The book makes the case that large groups can make great decisions, often better than any individual in a group, if three conditions occur:

1. diversity of the members of the group,
2. independent opinions of the group members, and
3. a method for aggregation of the opinions.

Surowiecki's very readable book gives many examples where group decisions do quite well (sports betting, Google's search techniques based on other's web pages, Linux) and where group decisions fail (stock market bubbles, committee meetings, strong CEOs).

Chapter 8 is devoted to science and how many widely spread scientists developing and criticizing various theories lead to explosive growth in our understanding. He also notes that this ideal world has its flaws as unknown researchers have a harder time selling their work than more established scientists.

I don't agree with all the conclusions drawn by Surowiecki but he does lay out what we need to do and not do to benefit from the pooled knowledge of a group. We can also draw lessons in computer science as computation and information gets more distributed that we need to integrate to find the best solutions we can.

Fun with Information Markets

Just over a year ago the Department of Defense cancelled their program on using markets to predict future world events, an overreaction that stopped funding a potentially powerful prediction tool. Robin Hanson has a comprehensive web page giving a history and plenty of links.

Information markets live in limited academic-based markets like the Iowa Electronic Market and offshore sites like Tradesports. For example the current price on Tradesports for Bush winning the election is 51.5 which translates to a 0.515 probability that Bush will win indicating a very close contest.

For each state, Tradesports has a security on whether Bush will win that state. They also have some bundles of states. The price for Florida is 50.1, Ohio 55.4 and Bush winning both Florida and Ohio is 47.1. This gives a surprising correlation between Florida and Ohio. If you believe the theory there is a very high 0.94 probability that Bush wins Ohio given that he wins Florida and with a 0.89 probability these two very different swing states will go the same way.

Tradesports gives David Vitter a 59 percent chance of becoming a senator from Louisiana. David Vitter is the brother of CS theorist and former SIGACT chair Jeff Vitter.

Micromorts

Currently on airplanes children under two can ride free by sitting on a parent's lap. The FAA is considering whether to require such children to have their own seat in a child seat similar to the ones most states require for cars. Sounds reasonable? One argument against goes as follows: If we require parents to pay for a seat for a children there is a chance they will drive instead greatly increasing their risk.

How can we evaluate risk? Decision scientists have developed a measure called micromorts (μmorts). A μmort is a one-millionth chance of death. Sounds gruesome but by counting micromorts we can analyze the right choices to keep the most people alive.

All of three lap children have died in airplane crashes where their parents survived since 1987. The average driver runs the risk of about .02 μmorts/miles. If the average car trip is say 500 miles that translates to about 10 μmorts for each child in the car. Three laptop children have died in airplane crashes where the parent has survived since 1987. This translates to the equivalent of 300,000 car trips or about 15,000/year. About 6 million children ride on laps on airplanes each year, so if more than 0.25% of them were to ride in a car instead because of the higher prices, we would about cost lives by requiring safety seats on planes. My numbers, drawn from various internet sources, don't tell the whole story but nevertheless we can and should do a full analysis before setting policy.

It would be nice to have a list of various activities and how many μmorts they use, say you feel like parachuting, you can get an idea of how dangerous it is compared to say riding a bicycle. But we don't get such lists and people have to use their own judgments and often make the wrong decisions. We can also give a cost amount to a μmort; how much is it worth to save lives?

By finding statistics online you can calculate the risks in your various activities. You need to use about 3 μmort/day on average to keep a 10% chance of accidental death in your life. Spend them wisely.

When to Announce?

Suppose you have some partial solutions of a popular problem. At what point do you announce your results? If you announce your partial results you run the risk of someone else taking your ideas and solving the full problem and you won't get as much credit as you deserve. If you wait and try to extend the work yourself someone else might get the same results you already have and you'll lose or at best have to share the authorship.

If you are completely altruistic you should announce your progress as this will best advance science quickly. But as in the end you need to worry about your own publication record, particularly for a young researcher, the answer isn't so clear. Of course it depends on many factors including your belief that you or others could extend the work as well as when the next conference deadline occurs.

Oddly enough before the internet (in the eighties) such decisions were easier. You could write up a technical report to establish your result and you would have months before your work spread throughout the community. This gives you plenty of time to try and extend the work. The quick spread of information not only improves collaborative work as it does, but forces us to make decisions that we could avoid in the past.

Small Circuits

Fix a constant k. In 1982 Ravi Kannan showed that some Σ₂^p∩Π₂^p language must not have n^k-size (nonuniform) circuits. Here is a proof sketch: A simple counting argument shows there is a function that depends only on the first 5k log n inputs that is not equivalent to a n^k-size circuit. Just by writing out the quantifiers in Σ₄^p you can compute the lexicographically first such function. Now we have two cases:

1. If SAT does not have polynomial-size circuits then SAT then Σ₂^p∩Π₂^p which contains SAT does not have n^k-size circuits.
2. If SAT has polynomial-size circuits then Σ₄^p=Σ₂^p∩Π₂^p (Karp-Lipton) and thus Σ₂^p∩Π₂^p does not have n^k-size circuits.

This is a wonderful example of a non-constructive proof and giving an explicit Σ₂^p∩Π₂^p language without quadratic-size circuits is open. With better known collapses we can improve the result from Σ₂^p∩Π₂^p to S₂^p.

Vinod Variyam recently observed that the class PP which is not known to contain S₂^p also cannot have n^k-size circuits. Here is his proof: If PP has n^k-size circuits then PP is in P/poly which implies the polynomial-time hierarchy and in particular Σ₂^p is in MA which is in PP which has n^k-size circuits contradicting Kannan.

Read Variyam's paper for details and references.

Larry Stockmeyer

We lost a great complexity theorist over the weekend. From Phokion Kolaitis:

It is with great sadness that I write to inform you that Larry Stockmeyer passed away. He died at his home as he had wished when he fell terminally ill a few weeks ago.

Larry was one of the pioneers of computational complexity who made fundamental and lasting contributions to the field. His death creates a void in our community that cannot be filled.

Indeed Stockmeyer developed many of the important early concepts in complexity such as alternation and the polynomial-time hierarchy, concepts that have laid the foundation for many important works in computational complexity. He had a number of great results throughout his career in complexity and nearly all areas of theoretical computer science. Our community has lost one of its giants.

Strangers in the Same Place

Professor X and Professor Y from the same university attend the same conference. At the end of the conference, Professor X says in a surprised tone "That's the most time I have talked with Professor Y all year." He shouldn't be surprised; this is a story I've heard over and over again (and have even told myself).

A professor's life has many responsibilities. Teaching and research of course but also paper writing, grant proposals, meeting with students, and administrative tasks including seemingly endless committee meetings. When I visit another university I leave most of these responsibilities behind so I can focus on research. I also expect the people who invited me to make time in their schedules so we can work together. That way even a short visit can be quite productive.

As the length of the visit increases it becomes harder to avoid these other responsibilities and the amount of research time per day decreases. In the extreme, two people who work at the same university for years end up spending very little time talking research together.

This explains why teleconferencing will never replace traveling no matter how technologically advanced. The social requirements of a short visit require people to spend time together in ways a teleconference cannot. What teleconferencing will do is "allow" me to attend those endless committee meetings wherever I am.

Journal Rankings

An assistant professor writes

In case you need a topic for your weblog: what about journal rankings for theoretical computer science journals? I was looking for something like that for my tenure portfolio. The only web-info I found on the topic was here whose reliability is hard to judge.

Thanks, I am always looking for topics. Journal rankings do not have as strong a perceived ranking in computer science due to the import we give to conferences. Nevertheless, deans like to classify journal articles in computer science like they do for other fields and ask for a ranking.

Here's how I rank theory journals.

1. Journal of the ACM.
2. SIAM Journal on Computing.
3. A large equivalence class of every other major theory journal.
4. Information Processing Letters which publishes short articles that don't merit publication in the above.

Any ordering of journal consistent with this list is okay though even here we have considerable fluctuation. In most theory journals the editor-in-chief rarely overrules the associate editors recommendations and thus have about the same average acceptance criteria. JACM has the tightest quality controls but still occasionally publishes some weaker papers and quite a few mediocre papers appear in SICOMP though I still rate it higher than the rest.

Special issues rank higher, especially those devoted to the best papers of a strong conference. On the other hand, I put no faith on the quality of theory papers that appear in non-theory and especially non-CS journals no matter how they are ranked in their respective field. More than a few rather weak CS papers have appeared in Science, the gold standard for many other scientific disciplines.

NSF Budget

The US House Appropriations committee has passed the NSF budget at a 2% ($111 Million) cut. There are still many more phases in the budget process to go but this cannot be viewed as good news for science. For various reasons, the NSF is lumped in the same budgetary group as Veteran's Affairs and the veterans lobby better than scientists.

The American Institute of Physics has a detailed report and perspective. Here also is a statement from the Coalition for National Science Funding and some comments from the Computing Research Policy Blog.

Favorite Theorems: Superlinear Bounds on Branching Programs

June Edition

Branching programs give us a nice way to model time and space bounds for Boolean functions in a simple non-uniform model. A branching program is a directed acyclic graph where every non-leaf node is labeled by a variable and has two edges labeled One and Zero. All of the leaves are labeled Accept or Reject. Given an input, one follows a path taking the One edge on a node labeled i if the ith input bit is one and the Zero edge otherwise.

The depth (length of the longest path) of the branching program represents time and log of the size represents space. Lower bounds on branching programs give us lower bounds on unrestricted computation.

In 1999, Miklós Ajtai gave the first polynomial-time computable Boolean function for which any subexponential-size deterministic branching program requires superlinear length.

A Non-Linear Time Lower Bound for Boolean Branching Programs by Miklós Ajtai In other words there exists a specified easily computable function that cannot be solved in linear-time unless one uses nearly linear space.

Ajtai creates a function based on quadratic forms and builds on techniques used in his slightly earlier paper.

For more details I recommend the paper Time-space tradeoff lower bounds for randomized computation of decision problems by Beame, Saks, Sun and Vee which gives a nice history of the problem and the techniques to solve it and generalizes Ajtai's work to the probabilistic setting.

Carl Smith 1950-2004

Maryland Professor Carl Smith passed away last night losing his year and a half battle with brain cancer. He was an expert in inductive inference and an active member of the computational learning community. He traveled extensively making many connections in Holland, Germany, Japan and especially Latvia where he is a foreign member of the Latvian Academy of Science.

Carl Smith also played an important role in the computational complexity community. He organized conferences in the early 80's at Perdue and Maryland on Recursion Theoretic Aspects of Computer Science, precursors to the current IEEE Conference on Computational Complexity. He also co-organized the third Complexity (then called Structures) conference in Georgetown in 1988.

Carl was a colleague and a good friend. We both had sabbaticals in Amsterdam in 1996-7, wrote some papers together and often visited each other afterwards. We shared a love of beer and baseball; I would plan my trips to Maryland around the Orioles home schedule.

I always enjoyed the time I spent with Carl and the many interesting discussions we've had. I, my family, and the entire theory community will miss him greatly.

Extracting Randomness

In this post I will describe some recent results in extracting randomness in terms of Kolmogorov complexity since I (and some others) find Kolmogorov complexity more intuitive than entropy. If you would like a background in Kolmogorov complexity, here are some notes from a short course I taught a few years ago. I have not verified that the Kolmogorov results listed below actually follow from the extractor results but it should be straightforward.

Let K(x) be the smallest program generating x. We say a string x is random if K(x)≥|x|. For this post we ignore O(log n) additive factors to avoid various coding issues.

The optimal extractor paper of Lu, Reingold, Vadhan and Wigderson gives us the following. Let n=|x| and K(x)≥k. For all α>0, there is a polynomial-time computable f such that f(x) outputs a polynomial list of strings of length (1-α)k such that most of these strings are random. Using probabilistic constructions of extractors, if one only requires f to be computable, we can set α=0 for k≤n/2.

Barak, Impagliazzo and Wigderson have a new result (mentioned here) on extracting randomness from independent sources. For any constant δ>0, there exists a k polynomial in 1/δ and a polynomial-time computable f such that if we have x₁,…,x_k with

1. |x_i|=n for all i,
2. K(x_i)≥δn for all i, and
3. K(x₁x₂…x_k)=K(x₁)+K(x₂)+…+K(x_k) (the x_i's are independent)

then f(x₁…x_k) is a random string of length n.

Even more recently Barak, Kindler, Shaltiel, Sudakov and Wigderson have even a stronger result in this direction (mentioned here). For any constant δ>0, there exists a ε>0 and a polynomial-time computable f such that if we have x₁,…,x₇ with

1. |x_i|=n for all i,
2. K(x_i)≥δn for all i, and
3. K(x₁x₂…x₇)=K(x₁)+K(x₂)+…+K(x₇) (the x_i's are independent)

then f(x₁…x₇) is a random string of length εn.

Some Links and Random Thoughts

Michael Nielsen is in the midst of a long series of posts on Principles of Effective Research. Much of what he says seems obvious but the obvious often needs to be pointed out. Update 7/27: Complete Principles now available.

Nielsen mentions a new Erdös number eBay auction. We shouldn't use eBay to get people to pay us to do our research; that's what we have graduate students for.

A couple of computational geometers Suresh Venkatasubramanian and Jeff Erickson have been quite active on their weblogs. Check them out.

Finally for some music to prove theorems by, the BBC has put the entire Beethoven sonata cycle with Portuguese pianist Artur Pizarro online.

Why are CS Conferences so Important?

In nearly every scientific discipline conferences play a minor role. Most conferences have a few plenary speakers mixed with massive parallel sessions where nearly everyone who wants to present can present. The vetting of papers occurs in journals and the quality of one's research is measured much by which journal the work appears.

Computer science conferences are much more selective and the quality of one's work is measured by which conference the work appears. Journals play a far lesser role and many important papers never appear in a journal at all. Why is computer science different?

The answer is technological, namely airplanes. Before air travel conferences were much more difficult to attend and drew from a much more regional audience. Those who made the great effort and time to attend a conference were allowed to present. But presenting your paper at such a conference would not reach the majority of your colleagues. Journals were the most efficient way to broadly publicize your research and took on the more important role and have kept that role for historical reasons.

Computer science started as a field during the jet age. Many more people from a wider geographical base could attend a conference. One could now widely disseminate their research through conferences well before a paper appeared in a journal. Journals still played an important role for refereeing, editing and archiving but never held the importance in computer science as conferences do.

Since then we've seen another technological revolution and the internet easily trumps conferences for quickly distributing your results. Perhaps some new scientific field starting today would have a different internet-based system for judging research. But conferences will remain the primary focus for computer science as journals do for the older scientific disciplines.

Time and Space Hierarchies

What the world needs are the time and space hierarchies clearly spelled out in one place.

A function t is time-constructible if there is a Turing machine M such that on input 1ⁿ outputs 1^t(n) in time O(t(n)). Space constructible functions are defined similarly. All the natural functions are time and space constructible.

DTIME(t(n)) are the set of problems computable by a multi-tape Turing machine in deterministic time O(t(n)) on inputs of length n. NTIME (nondeterministic time), DSPACE and NSPACE are defined similarly.

Let t₁ and t₂ be time-constructible functions and s₁ and s₂ space-constructible function. We let "⊂" denote strict subset. A function f(n)=o(g(n)) if lim_n→∞f(n)/g(n)=0.

1. If t₁(n)log t₁(n)=o(t₂(n)) then DTIME(t₁(n))⊂DTIME(t₂(n)).
2. If t₁(n+1)=o(t₂(n)) then NTIME(t₁(n))⊂NTIME(t₂(n)).
3. If s₁(n)=o(s₂(n)) then DSPACE(s₁(n))⊂DSPACE(s₂(n)).
4. If s₁(n)=o(s₂(n)) then NSPACE(s₁(n))⊂NSPACE(s₂(n)).

The DSPACE hierarchy is straightforward diagonalization. For DTIME the proof is similar but we lose log t₁(n) in the simulation of a k-tape machine by a 2-tape machine.

Straightforward diagonalization does not work directly for nondeterministic computation because one need to negate the answer. For NSPACE we easily get around this problem by using Immerman-Szelepcsényi.

The NTIME hierarchy has the most interesting proof that leads to requiring the "+1" in t₁(n+1). This can make a big difference for t₁(n) larger than 2ⁿ².

An NTIME hierarchy was first proved by Cook and in the strongest form by Seiferas, Fischer and Meyer. We sketch a simple proof due to Zàk.

Let M₁,… be an enumeration of nondeterministic Turing machines. We define a nondeterministic machine M that acts as follows on input w=1ⁱ01^m01^k:

• If k<m^t₁(m) then simulate M_i on input 1ⁱ01^m01^k+1 for t₂(|w|) steps.
• If k=m^t₁(m) then accept if 1ⁱ01^m0 rejects which we can do quickly as a function of the current input size.

This machine uses time O(t₂(n)). If NTIME(t₁(n))=NTIME(t₂(n)) then there is an equivalent machine M_i using time O(t₁(n)).

Since t₁(n+1)=o(t₂(n)) we have for sufficiently large m,

1ⁱ01^m0 in L(M) ⇔ 1ⁱ01^m01 in L(M) ⇔ … ⇔ 1ⁱ01^m01^mt₁(m) in L(M) ⇔ 1ⁱ01^m0 not in L(M) a contradiction.

Bringing Families to Conferences

When we have a conference or a workshop in a tourist location, like Banff, many of the participants bring their non-computer scientist spouses and sometimes their whole families. I rarely do so. The main purpose in attending conferences and workshops is not the talks but to meet with your fellow researchers. The main purpose of a family vacation is to spend time with the family. These conflicting goals would make me feel guilty during the whole conference no matter how I split my time. Sometimes I will bring the wife or the family a week before or after or between conferences but conference time is science time.

Still I cannot fault my fellow scientists who bring their families to conferences. I would much rather they attend the conference with their families than not come at all. Every professional has a major challenge in balancing family and work life and they need to find the right mix that works for them.

Final Notes from Banff

Some final notes from the Banff workshop. First a few lemmas used in Wigderson's talk.

Lemma 1: Let G=(V,E) with n vertices and m edges and m≥4n. Let cr(G) be the number of edge crossings in any planer layout of G. Then cr(G)≥m³/64n².

Lemma 2 (Trotter-Szemérdi): Suppose we have a set of points P and lines L in the plane. Let n=|P| and m=|L|. Let I be the number of indices, i.e. the number of pairs (p,l) with p in P, l in L and line l contains the point p. Then |I| ≤ 4((mn)^2/3+m+n).

Guy Kindler talked about a brand new set of results with Barak, Shaltiel, Sudakov and Wigderson. Among other things they improve on the Barak-Impagliazzo-Wigderson result I mentioned earlier by showing that for any constant δ>0, one can take seven independent sources of n bits each with δn min-entropy and combine them to get O(δn) bits of randomness.

Mario Szegedy talked about his recent work showing that the quantum hitting time of a symmetric ergodic Markov chains is the square root of the classical hitting time, a result that becomes a powerful tool in developing quantum algorithms.

Update 7/12: Group Photo now online.

RESULTAPHOBIA!

A Guest Post by Bill Gasarch and Brian Postow

In the 1970's there was some hope that deep techniques from Computability theory might crack P vs NP. Some nice results came out of this (e.g., Ladner's theorem that if P ≠ NP then there is a set inbetween). Then the oracle results seemed to say these techniques (whatever that means) would not work.

In the 1980's there was some hope that deep techniques from Combinatorics might crack P vs NP. Some nice results came out of this (e.g., PARITY not in AC₀, and the monotone circuits lower bounds). Then the Natural Proofs framework seemed to say these techniques (whatever that means) would not work.

So where are we now? Fortnow and Homer's paper on the History of Complexity Theory seems to say that we have no ideas at this time. A recent talk at Complexity seemed to say "we didn't work on this aspect of the problem since, if we solved it, we would have P ≠ NP."

We as a community seemed to be afraid of big separation results. We are almost scared of working on hard problems since they might not pan out. Is this wise? There are stories (some apocryphal some not) about people solving problems because they didn't know they were hard. (Examples below)

I recognize that working on problems with little hope of success is dangerous. But to shy away from a line of research BECAUSE it may lead to a big result seems... odd.

EXAMPLE ONE: Neil Immerman tells a story about Robert Szelepcsényi. Szelepcsényi's result that Context Sensitive Languages are closed under complement was announced in an issue of EATCS (in the same issue, two other articles mentioned Immerman's own proof that NSPACE is closed under complement, an effectively equivalent result). Szelepcsényi was an undergrad at the time, and his adviser gave him the famous problem as a challenge, probably not really expecting him to actually solve it. He did solve it, perhaps because he was never told that it was an old open problem that others had failed to solve.

EXAMPLE TWO: A prominent researcher (who told me about this, so its verified) was working on Σ₂-SPACE(n) = Π₂SPACE(n) but stopped since it might lead to the absurd result that Σ₁-SPACE(n)=Π₁=SPACE(n).

DEBUNKING: There is a RUMOR that Umesh Vazarani would have had Quantum factoring in P but didn't get it since it was obviously false. He has denied this. (I put this in so that someone doesn't post a comment about it.)

Are there more cases of either people solving a problem because they didn't know it was open OR of people NOT working on a problem because they thought it was hard (and it wasn't that hard)? I'm sure there there are. If you know of any that have been verified please post to comments or email to gasarch@cs.umd.edu and postow@acm.org.

Gems of Additive Number Theory

Yesterday Avi Wigderson gave a talk entitled Gems of Additive/Combinatorial Number Theory where he presented three interesting results about the sizes of sets when you add all the possible numbers from one set with another.

Let A and B be subsets of an Abelian group G and define A+B = {a+b | a in A and B in B}. We define AxB as the same with multiplication when we work over a field. Let |A|=|B|=m.

1. Erdös-Szemerédi: Let A be a subset of the reals. Either |A+A|≥m^5/4 or |AxA|≥m^5/4.
2. Ruzsa: For all k, if |A+B|≤km then |A+A|≤k²m.
3. Gowers: Let E be a set of pairs (a,b) with a in A and b in B. Let A+_EB be the set of values a+b with (a,b) in E. For any δ and k, if |E|≥δm² and |A+_EB|≤km then there is an A'⊆A and B'⊆B with |A'|,|B'|≥δ²m and |A'+B'|≤mk³/δ⁵.

Later Russell Impagliazzo showed how to use finite field versions of these results in his upcoming FOCS paper with Barak and Wigderson. They show how to convert poly(1/δ) independent sources of distributions with δn min entropy to n nearly uniform random bits.

Howdy from Banff

Another fourth of July out of the states, this time at the Banff International Research Station (BIRS), a Canadian mathematical conference center similar in spirit to Oberwolfach and Dagstuhl. BIRS is hosting a workshop on Advances in Complexity Theory with a pretty impressive collection of researchers.

Today's talks focused on PCPs and their applications. Guy Kindler gave an interesting presentation on his work with Khot, Mossel and O'Donnell showing that under a few believable assumptions, the Goemans-Williamson Max-Cut approximation is optimal.

Took some time off to see Greece win Euro2004. Sorry Luis.

JCSS To Pay Editors, Possibly Referees

The Elsevier owned Journal of Computer and System Sciences will pay an honorarium of $100 to a cognizant editor for each paper handled. According to Editor-in-Chief Ed Blum the intent "is to establish the practice of compensating editors for their scholarly contribution to scientific publication and is a partial response to the complaint that we scholars do all the work and the publishers reap all the rewards. This practice will apply to Guest Editors of Special Issues. I am aware that it does not address the problem of journal pricing. I am still working on that. I am pretty sure that we can also give a $50, honorarium to referees, but am awaiting a final OK [from Elsevier] on that." (Thanks to Bill Gasarch for this information.)

Update 8/7/04: JCSS is rescinding this new policy of awarding honoraria for papers handled.

Lessons from Economics

Rakesh Vohra pointed me to some interesting takes on journals in economics. Economics runs on a different model than computer science; conferences are less selective and economists are judged more on the quality of the journals where their papers appear.

The Berkeley Electronic Press offers an electronic subscription-based system for their journals. Look at the B.E. Journals in Theoretical Economics. Here you submit to all four journals at once and your paper gets accepted to one with the highest quality rating that the editors decide is appropriate for your paper.

NAJ Economics is Not A Journal but offers reviews of economics papers. One cannot submit papers but a strong rotating editorial board just finds papers freely available on the internet and post reviews of those they feel are worthy. From the FAQ:

The purpose of NAJ Economics is to work towards replacing the existing commercial system of scientific publication. Because papers published in printed journals are less available than working papers, which are freely available on the Internet, publication in the traditional sense inhibits scientific communication. It also generates additional costs as most printed journals charge high subscription fees, in particular to libraries. However, it does serve the useful purpose of certifying the scientific quality of published work. It also assures that articles remain available regardless of the idiosyncrasies of individual websites and links. Our immediate goal is to provide some of the useful certification functions of current journals at a negligible cost by reviewing papers that we think have substantial merit.

I have some quibbles about the service. Without submissions a lesser known author might have trouble getting his paper reviewed. The editors will have a nightmare keeping links up to date, especially since they seem to link to papers on people's homepages. They also don't have the ability to force improvements in the papers they review the way a journal can.

But perhaps in this age of the internet one needs to separate the refereeing and distribution aspects of a journal. NAJEcon is an interesting step in that direction.

FOCS Accepted Papers

The list of accepted papers for the upcoming FOCS conference in Rome is out. [Thanks Suresh]

A few complexity papers to note: Ran Raz finds easy languages with no log-depth multilinear circuits. Andris Ambainis and Mario Szegedy have separate papers showing nice applications of quantum "random" walks. Barak, Impagliazzo and Wigderson show how to do extract nearly uniform distributions from multiple independent random sources as opposed to one random source and a few truly random bits. And lots more.

Don't Make it Too Easy or Too Much

Two easy ways to improve your paper but lessen your chances of acceptance at a conference: Add more results and simplify your proofs. Adding a result could only increase the usefulness of a paper but program committees see many results in a paper and conclude that none of them could be very strong. One of our students a few years ago had a paper rejected at STOC, he removed one of his two main theorems and won the best student paper award at FOCS.

Given the same theorem, the community benefits from a simple proof over a complicated proof. Program committees look for hard results so if they see a very simple proof, it can count against you.

You need to play the game. If you have many results depending on the situation, you can either split the paper or highlight one result and bury the others. It's a bit unethical to use a hard proof where you know an easy one but many people make an easy proof look harder by adding an unnecessary level of detail or proving a more general but less interesting theorem.

You do what you need to do, within ethical standards, to get your paper accepted. After you get it accepted, remember you have a rewrite for a proceedings version to get the paper written the way it should.

Note from Vereshchagin

I received the following from Nikolay Vereshchagin.

The combinatorial question I have discussed last summer at the rump session at Computational complexity (about partitioning a planar set into a small number of uniform parts) has been answered almost immediately by Ilan Newman and Gabor Tárdos. They have found a pure combinatorial proof. Recently I have written a note on the subject.

Complexity Conference Recap

The Complexity Conference ended yesterday. You can already find the papers on the IEEE site and if you don't have access you can often find versions of the papers on author's homepages.

We had a strong turnout and a nice variety of papers on many different areas of complexity with particularly strong showings in quantum complexity and structural complexity making a comeback.

Amit Chakrabarti asked about group isomorphism. Arvind and Torán showed that solvable group isomorphism is "almost" in NP∩co-NP.

Although I did not have my own talk in the conference, I presented a paper by Buhrman and Torenvliet since they unfortunately could not be in Amherst. I like giving talks on other people's work since you can be honest about the strengths of a paper without having to brag. My favorite result in their paper showed that if you take a many-one complete set for EXP, remove any easily computable set of subexponential size, what remains is Turing-complete for EXP. The proof is a clever recursive algorithm using the set itself to find safe places to map the reduction.

Next year we have our 20th conference in San Jose followed by Prague in 2006.

Rump Session Redux

This week I am in Amherst at the University of Massachusetts for the 19th IEEE Conference on Computational Complexity. Lots of fun papers and complexity theorists. This is complexity heaven.

Like last year, we had a number of interesting new results described at the rump session. Let me describe a couple of them to you.

Scott Aaronson follows up on his guest post about the complexity of agreement. Aumann has a famous theorem that two players who communicate cannot agree to disagree on the probability of some state of the world; after some discussion they will converge to a common probability. Aaronson looked at the complexity of this process and found that convergence comes relatively fast. He defined a notion of (ε,δ)-agreement where the probabilities are within ε of correct with a confidence of 1-δ and shows that such an agreement happens after polynomial in 1/ε and 1/δ rounds.

Neeraj Kayal looked at the complexity of the problem #RA, the number of automorphisms of a ring given by generators. He showed that factoring and graph isomorphism reduce to #RA and #RA sits in AM∩co-AM. As an open question he wondered about the complexity of determining whether a ring has nontrivial automorphisms where one is given tables for addition and multiplication. It remains open even for commutative rings.

Update 6/23: Kayal tells me I didn't accurately capture his rump session talk and sent me the following summary.

We have an algorithm that determines whether a ring has a nontrivial isomorphism even when the ring is given in the form of generators for its additive group and pairwise product of the generators expressed as a linear combination of the generators. (We get this by getting a characterization of all finite rigid rings and it turns out that we can test whether a ring follows this characterization or not without solving integer factoring.) Unfortunately however we do not know of a reduction from Graph automorphism to ring automorphism although we have found a cute reduction from Graph Isomorphism to Ring Isomorphism!

The open problem that I would love to solve is to decide whether two rings are isomorphic or not when they are given in the form of tables (one table each for addition and multiplication.) I do not know how to do this even for commutative rings.

Shimon Even (1935-2004)

Shimon Even was born in Israel on June 15th, 1935. He died on May 1st, 2004. In addition to his pioneering research contributions (most notably to Graph Algorithms and Cryptography), Shimon is known for having been a highly influential educator. He played a major role in establishing computer science education in Israel (e.g., at the Weizmann Institute and the Technion). He served as a source of professional inspiration and as a role model for generations of young students and researchers. Two notable avenues of influence were his PhD students and his books Algorithmic Combinatorics (Macmillan, 1973) and Graph Algorithms (Computer Science Press, 1979).

From a memorial page by Oded Goldreich.

Visa Problems Continue

Wisconsin Professor Dieter van Melkebeek has a paper at the ICALP conference but cannot go to Finland to present it. Why not? Delayed processing of his green card application has led to problems with his current visa putting him in some temporary state of visa hell. Dieter would actually have no trouble attending ICALP; he would just have problems coming back.

Dieter is one of many stories of people changing travel plans and missing conferences because of America's tougher requirements and slower processing of foreign immigration applications. An Indian graduate student with a paper at next week's Complexity conference could not get a visa in time. I would not be surprised if many graduate students will not start the fall semester on time awaiting my government's blessing to come to study here.

This is a story I have told before and will likely tell again. I understand the need for security but most scientific progress happens through collaboration and preventing or delaying this collaboration holds back the advancement of knowledge. Not since the 80's have we seen such a limitation on traveling though this time in reverse. During the cold war several countries would not let many of their best scientists out; these days we don't allow many of the world's best scientists in.

Riemann Hypothesis and Computational Complexity

A commenter asks a good question for a bad reason: Would a proof of the Riemann Hypothesis have any impact on complexity theory?

Rather surprisingly the answer is yes, particularly in the area of computational number theory. In the most famous example, Gary Miller in 1975 gave a polynomial-time algorithm for primality whose correctness could be proven by assuming the Extended Riemann Hypothesis (ERH). Of course in 2002 we had a polynomial-time primality algorithm with no assumption. However the original analysis of the algorithm gave a constant which depends on how ERH is resolved.

There are still many other problems in computational number theory that require ERH. For example, according to Eric Bach, the only polynomial-time algorithm computing square roots modulo p, when p is large relies on ERH. "The idea is to combine an algorithm that uses a quadratic nonresidue, such as Shanks's algorithm (this in Knuth v. 2 I am pretty sure) with a bound on the least quadratic nonresidue mod p (e.g. in my thesis it is proved to be <= 2 (ln p)^2 if ERH is true)."

Special Issues

Journals dominate the non-research talk at STOC. We had a long discussion at the business meeting about the special issue of STOC. A little background: For the past 24 years the STOC program committee selects 6-10 papers from the conference and one of the PC members serves as editor of a special issue of a journal where all these papers are invited to appear. The Journal of Computer and System Sciences (JCSS) has always hosted the special issue for STOC as well as a few other conferences including FOCS and Complexity.

JCSS became an Elsevier journal a few years ago when Elsevier bought Academic Press. Elsevier has come under attack over the past few years in our field for their pricing policies, an issue discussed in this weblog before. Some editorial boards have resigned and many others are considering it. The current PC chair (and fellow U. Chicago Professor) Laszlo Babai has strong negative feelings towards Elsevier and spearheaded the issue at the conference.

The STOC Executive Board has final say on the future of the special issue but based on the business meeting discussion, the special issue for STOC will likely move to SIAM Journal on Computing (SICOMP) perhaps as early as this year.

My concern, which I expressed at the meeting, is that we already have a culture where too many papers never appear in a journal, i.e., never get written with full proofs and go through a rigorous refereeing process. The more negative press we give towards journals the more likely authors will take the easy solution of no journal. When was the last time you downloaded the journal paper never written?

Update 6/18: Hal Gabow, chair of SIGACT, has set up a website containing additional information on the meeting and subsequent procedures.

STOC Business Meeting

STOC got underway Sunday with a full slate of talks and a lengthy business meeting last night. I do not have time for a long post now so I will just bring you up to date on some facts from the business meeting.

The attendance was 261 (242 paid + 19 local helpers). Later today I will update the contest post with the results.

STOC 2005 will be in Baltimore May 22-24 and STOC 2006 will be in Seattle. There were announcements of three new journals, the previously mentioned ACM Transactions on Algorithms and two on-line open-access journals Logical Methods in Computer Science and Theory of Computing.

Most of the business meeting was devoted to the future of the special issue and I left around 11 PM last night before this discussion had ended. This discussion will require a post of its own in the near future.

STOC runs through Tuesday. Much more as the week goes on.

Favorite Theorems: Connections

May Edition

I have always loved results that find connections between previously-thought different areas of complexity. This month we highlight one of the best.

Extractors and Pseudorandom Generators by Luca Trevisan

Informally a pseudorandom generator takes a small random seed and generates strings that can fool every probabilistic algorithm. To describe an extractor we start with some distribution D over strings of length n. Let p be the maximum probability of any string in D and let k = log(1/p). An extractor uses D and a small number of truly random bits to create a new uniform distribution of strings of length close to k.

Both pseudorandom generators and extractors have many uses in complexity and many papers in the field show various constructions to improve the parameters of both. Trevisan showed that one can view any pseudorandom generator as an extractor and then derives better extractors from known pseudorandom generator constructions.

Pseudorandom generators fool resource-bounded algorithms while extractors nearly uniform distributions in an information-theoretic sense. That makes this connection all the more amazing. Trevisan's paper has affected the how researchers think about and prove results in both areas.

Win a Gmail Account

My first weblog contest. Guess the paid attendance (including students and postdocs) at next week's STOC conference. Closest to the correct answer receives an invitation for a Beta Gmail account (donated by weblog friend Meridel).

Rules: Send your guess in the subject of an email to stocguess@fortnow.com. Include your name and email in the body of the message. One guess per person. All guesses must be sent by Saturday noon CDT. Closest guess to the attendance announced at the business meeting Sunday night will receive an invitation to open a Gmail account (still in Beta testing). In case of tie, first closest guess received will win. Anyone involved in STOC organization is ineligible. Not responsible for delayed or undelivered email. My decision of the winner is final. Contest not sponsored or affiliated with Google or ACM SIGACT.

Good luck.

Results Update 6/14: Total paid attendance was 242. The closest at 254 was Nanda Raghunathan, second place at 223 was Kamalika Chaudhuri and third at 265 was Chandra Chekuri. We have some extra invites so we've decided to give gmail accounts to all three. Congratulations and thanks to everyone who participated.

Professional Societies

Professional Societies perform valuable roles in academics. They give awards, sponsor conference and publish reasonably-priced journals as well as bulletins, newsletters and reviews. Societies disseminate information among researchers about future activities and the state of the field. They form an advocacy group representing the scientists in government and universities. Most importantly they give a focal point that lets us identify as a community.

Unfortunately in theoretical computer science no single group plays all these roles and thus one interacts with a large number of professional societies during an academic career. Let's look at some of them.

First most comes the Association for Computing Machinery (ACM) as the largest society devoted to computer issues. ACM tries to cover the entire computing profession so computer science research issues do not get center stage. They do publish several journals and give many of the important awards such as the Turing award.

ACM has a number of special interest groups (SIGs). SIGACT, the Special Interest Group on Algorithms and Computation Theory, is the main organization devoted to theoretical computer science in the US. They sponsor STOC and other conferences and publish SIGACT News. Many theorists join SIGACT without joining ACM.

The IEEE Computer Society also deals with computer issues and has a Technical Committee on Mathematical Foundations of Computer Science that sponsors conferences including FOCS and Computational Complexity. Why do we need both a Computer Society and ACM and a SIGACT and a TC-MFCS? Perhaps for the competition?

None of these societies serve as a strong advocate for computer science research and so we have the Computing Research Association. The CRA has as its members not individuals but academic departments and research labs. They have a newsletter, advocate and keep us informed on government policy on computer science, and collect information such as the Taulbee Surveys giving salary and job information in CS research. The CRA also has a strong focus on women's issues in CS research.

Let's not forget the Society for Industrial and Applied Mathematics (SIAM) that helps sponsor some conferences (SODA) and publishes the well-respected Journal on Computing.

The European Association for Theoretical Computer Science (EATCS) covers not just Europe but captures theory from an international perspective. They sponsor conferences like ICALP and publish a hefty bulletin three times a year. Also many countries have their own computer science and/or theoretical computer science societies.

Then based on my research interests I have now or at some time been a member of AMS, MAA, ASL, SIGecom and the Game Theory Society. Where does it all end?

BEATCS Complexity Column

With the June issue, Jacobo Torán takes over the editorial duties of the BEATCS Complexity Column. Following with tradition, he wrote his first column, Space and Width in Propositional Resolution. A strong start to what should be a great run of columns.

Survey Papers

Let's end this week how we started it, with a survey paper. Luca Trevisan has recently posted on ECCC a new survey Some Applications of Coding Theory in Computational Complexity. The survey gives a rather in-depth look at several different types of codes with some connections to private information retrieval, average-case complexity and probabilistically checkable proofs. Trevisan gives a broader and more in-depth look at coding theory than an earlier yet also excellent survey by Madhu Sudan focusing on list decoding.

Survey papers play a valuable role in our field. As computational complexity has broadened over the years, one cannot hope to keep on top of all of the many areas. A survey paper written by an expert in the field can perform many valuable tasks including

• Putting the main results of an area in a common framework. Early work often uses different notation and definitions making it hard to compare one paper to another. Fixing the notation and definitions allow us to easily compare different results. A well-liked survey can also influence future notation.
• Proofs get easier over time and a survey can give easier-to-follow proofs of old results. A survey can also develop a common proof technique useful for many result in the area.
• Giving the author's informed opinion to the importance of different results in an area.
• Stating open problems and directing future research in that area.

In case I've managed to put the survey bug in you, here are two topics where we've seen several recent research papers but lack good surveys that I know of.

1. The complexity of Nash Equilibrium
2. ε-biased Sets

Complexity Registration Deadline

Tomorrow is the last day for early registration for this year's Complexity Conference in Amherst. I promise a good time will be had by all.

IEEE Fellowship at the State Department

Are you an American IEEE member? Now you can help guide American foreign policy. IEEE-USA has announced an Engineering and Diplomacy Fellowship where IEEE members can serve as a Fellow in the U.S. State Department and continue to advise them afterwards. These fellowships are being offered for a few professional societies; perhaps the ACM should try to get in on this.

Some more background from FYI.

Impagliazzo's Five Worlds

Boaz Barak in a comment last week mentioned one of my favorite survey papers, Russell Impagliazzo's A Personal View of Average-Case Complexity presented at the 1995 Complexity Conference. In that paper he describes five possible worlds and their implications to computer science.

• Algorithmica: P = NP or something "morally equivalent" like fast probabilistic algorithms for NP. This was the world I described last week but looking back at Impagliazzo's paper, he does a nicer job.
• Heuristica: NP problems are hard in the worst case but easy on average.
• Pessiland: NP problems hard on average but no one-way functions exist. We can easily create hard NP problems, but not hard NP problems where we know the solution. This is the worst of all possible worlds, since not only can we not solve hard problems on average but we apparantly do not get any cryptographic advantage from the hardness of these problems.
• Minicrypt: One-way functions exist but we do not have public-key cryptography.
• Cryptomania: Public-key cryptography is possible, i.e. two parties can exchange secret messages over open channels.

Impagliazzo does not guess which world we live in. Most computer scientists would say Cryptomania or Minicrypt.
The paper goes on to give one of the better justifications for Levin's definition of average-case complexity.