Computational Complexity

Tuesday, July 27, 2010

 
This Post is Quite Different then Any You've Every Read!

Posted by GASARCH

I recently a letter from WETA (public TV) which I quote from:

This letter is quite different from any we've ever sent to you. For years we wrote to you about WETA's great programs and the need they filled in your life. Today, I must write to you about WETA's needs. And if friends like you don't respond to them, there will be far less programs to enjoy.

There is something wrong with this letter: I have gotten the exact same letter from them for about 4 years now. Hence the statement This letter is quite different from any we've send to you is not just false but verifiably false.

While I expect letters to exaggerate I do not expect to have easily verifiable lies that do not even help their cause. So why did they do this? I do not know. But whatever the reason, it is sheer incompetency. Hence I will not give to them. This raises the following question:

If a charity (or whatever Public TV is) asks for money they can exaggerate how much they need it. Should they?

1. Some readers will say GOSH, they really need the money! I better give!
2. Some readers will say They always need money. I am not going to bother.

We also have here a societal problem. Since many (legitimate) charities exaggerate about how dire their situation is or how serious their problem is, after a while it all gets tuned out. We have here a a Prisoners Dilemma problem- each one thinks (correctly?) that if they exaggerate their problems they will get more money. But if they all do it the public gets cynical and gives less money overall. (NOTE- I do not know this to be true, but I am curious. If anyone does know then let me know.)

How can they get out of this trap? I do not know. However, the least they can do is to not say things that are obviously false. There may be a well defined Game Theory or EC problem here. Or it may be a public policy problem. We won't know until its solved.

12:04 PM # 19 comments

Monday, July 26, 2010

 
A Seventh Mil. Problem

Posted by GASARCH

Richard Lipton had a wonderful post asking for a seventh Millennium Prize now that Poincare's conjecture has been solved. I posted a suggestion on his blog but got no comments. I'll expand on it and see if I get any comments here.

HISTORY: The original proof of VDW's theorem , in 1927, yields INSANE (not primitive recursive ) bounds on the VDW numbers. (Shelah (1988) later got primitive recursive bounds and Gowers (2001) got bounds you can actually write down!) Inspired by VDW's proof Erdos and Turan (1936), made two conjectures:

1. If A is a subset of N of positive upper density then A has arbitrarily long arithmetic sequences. Proven by Szemeredi in 1975 (see here for more.)
2. If &Sigma_{x ∈ A} 1/x diverges then A has arbitrarily long arithmetic sequences. (This conjecture implies the first one.)

A proof of either of these yields a proof of VDW theorem. The hope was that it would lead to a proof with better bounds. Szemeredi's proof of the first conjecture did not yield better bounds; however, Gowers proved the first conjecture a different way that did yield better bounds on the VDW numbers.

The second conjecture is still worthwhile since it may yield even better bounds and because it is interesting in its own right. So, I propose the second conjecture of Erdos-Turan as the 7th Millennium problem. (It might need a snazzier name. The Divergence Conjecture? The k-AP Conjecture? Suggestions are welcome!)

1. Greene and Tao have already shown that the primes have arbitrarily large arithmetic progressions.
2. The work that has gone into Szemeredi's theorem and the Greene-Tao theorem spanned many areas of mathematics. Hence this is not just an isolated problem.
3. The problem has been open since 1936. Hence it is a hard problem.
4. Will more connections to other parts of math be made? Is the problem too hard? A NO answer to both of these would make it not that good a problem.
5. The converse to the conjecture is not true. Note the following set:
   
   A = &cup_{k&isin N} {2^k + i : 0 &le i < k }
   
   The set A has arbitrarily long arithmetic sequences but If &Sigma_{x ∈ A} 1/x converges.
6. Is there a plausible condition that characterizes the sets that have arbitrarily long arithmetic sequences?
7. There is already (I think) a 3000 dollar bounty on the second conjecture. So the Clay Math Institute will have to just give 997,000 dollars.

11:35 AM # 2 comments

Friday, July 23, 2010

 
CRA Snowbird Part II

Posted by Lance

Considerable discussion about funding at CRA Snowbird. Ken Gabriel, Deputy Director of DARPA, talked about how DARPA is restructuring its programs to become more university-friendly. They've made great progress though there are still some sticky issues of project-oriented proposals and security clearances. On a related note DARPA recently announced a new Crypto Program that may be of interest to the theory community.

Peter Harsha, the CRA director of public affairs, talked about NSF funding and how the renewal of the COMPETES act almost got derailed over pornography. NSF and CISE in particular did well in the administration's budget request but there is some uncertainty as we head into the fall elections.

The best part of the snowbird meeting is networking, talking to a number of CS leaders especially at the meals and breaks. The last session was small group meetings with current deans on how to deal with our own deans. Even though I'm not a chair I do find myself dealing with my dean and his staff quite often and we were able to get some good advice on quite a range of specific issues. Our group got lucky in matching up with  Dan Huttenlocher, Dean of Computing and Information Science at Cornell, and Martha Pollack, former Dean of the School of Information at Michigan. The best general advice: have a good working relationship with your dean and don't just ask or complain but really make the case on how the particular resource you need will benefit your school.

I'll be mostly on vacation and off the net for the next couple of weeks. Be nice to Bill while I'm gone.

7:05 AM # 0 comments

Thursday, July 22, 2010

 
CRA Snowbird Part I

Posted by Lance

I just returned home from my first trip to the CRA Snowbird Conference, the biennial meeting of CS chairs and other leaders in the CS community. I really enjoyed the short meeting and saw many old friends who are now chairs, some people I've only known by email and many I've talked to for the first time. There are a few theorists who have become chairs and deans, and, in the case of Jeff Vitter, provost at Kansas. Unlike theory conferences where I am usually one of the old people, most CS chairs are just about my age.

I, as I had to remind most people I met, am not a chair. I attended as a speaker on the Peer Review in Computing Research panel giving my usual spiel on how the current publication culture hurts the community-building aspects of conferences. Jeannette Wing made a great argument of how our deadline-driven research and conservative program committees may lead our field to lose its "vibrancy, excitement and relevance". 

A number of people talked about the big projects they work on which make me almost rethink having gone into theory. Almost. The need for better algorithms shows up in many of these talks. Yoky Matsuoka from U. Washington talked about the artificial hand her group developed that has the full range of motions of a human hand but they lack the algorithms to have the hand do even some simple natural tasks. Illah Nourbakhsh from CMU talked about building electric cars and his ideas of using a supercapacitor as an energy cache for batteries so the batteries become smaller and cheaper but hits challenging cache optimizing issues. The group is running a contest, best algorithm wins an electric car. 

Sally Fincher from the University of Kent gave a surprisingly strong talk Why Can't Teaching Be More Like Research? We get judged by our research based on how we compare to the global community but teaching is much more local and Sally talked about the implications of this distinction.

Most disappointing was the discussion on the NRC Rankings. Charlotte Kuh, who served on the NRC committee putting together the "soon" to be released report, said it will not give a specific ranking of each department but rather a range, like University of Southern North Dakota is ranked between 8th and 36th. And not just one range but five ranges based on different weights of the various criteria. And you can create other rankings based on your own choice of weights. All based on 2005-6 data. And they used citation data from the ISI which doesn't include most CS conferences. The CRA board talked them out of that but now the CS data and rankings will use no citation information at all. But even outside of CS, with multiple ranking ranges and old data, the NRC report will be of little value.

7:44 AM # 7 comments

Wednesday, July 21, 2010

 
Do you want to review a book

Posted by GASARCH

I will be sending my next book review column for SIGACT NEWS off on July 28, 2010. It has LOTS of books on its BOOKS I WANT REVIEWED list. YOU get a chance to look at the book list and request one to review before the list goes out (if you do this then I will modify the list).

Here is a link to the list of books I want reviewed HERE. If you see a book you want then email me at gasarch@cs.umd.edu the name of the book and your postal address. We will then work out details over email- when the review is due, and whether me or the publisher sends it to you. I would like you to email me before July 28 so I can take those books off the list, but if you email after that date and the book you want is still available, that will be fine.

Before volunteering you should read my advice for reviewers.

Here is a LaTeX Template.

10:15 AM # 11 comments

Monday, July 19, 2010

 
Factors for getting a job- Arbitrary, random, and complex

Posted by GASARCH

The Job Market in Theory (likely in all of academia) has more randomness and arbitrariness (are those the same?) then people may realize. Especially young PhD's who have never been to a faculty meeting where these things are discussed. The process is quite complex (is that the same as random and arbitrary?). I am NOT saying that merit plays no role. I am saying that its very hard to make a clean statement like Person X got a job an school Y because of Z.

With that in mind, I list out some criteria I have heard played a role in a hiring decision. My point is NOT to help you game the system (note that some of the factors I've heard cut both ways) nor to argue that the system is good, bad, or ugly. My point is only that the process is more arbitrary-random-complex then you might think. While its not all merit, its also not all who-you-know or politics.

I invite you to give leave comments on factors you have heard of, but to keep it civil please do not mention the people or schools involved.

1. Merit: This is itself ambiguous. More papers? Longer papers? Co-authors? How about take a sum where each summand is (number of citations)*(importance of paper)*(number of pages)/(number of coauthors). And then there is grant potential.
2. Does a postdoc have a better chance then a fresh PhD? A postdoc has had more time to increase his weighted sum mentioned under Merit, but the school KNOWS he has had more time.
3. Two-body-problems can be GOOD or BAD. Often a school does not have two positions.
4. Being a women can be GOOD or BAD (for getting a job).
5. I would like to say Being black can be GOOD or BAD (for getting a job) however since there are so few black PhD's in computer scientists applying for academic jobs, I have not heard any stories about this. (NOTE- I use the term black instead of African American since they need not be American.)
6. Being socially inept can be GOOD or BAD. How could it be good? It plays to the stereotype. He's so socially inept, he must be a genius. I DO NOT recommend pretending you are socially inept.
7. Speaking your mind can be GOOD or BAD. He'll be a leader in the community or He'll be a pain in the ass
8. Having a Blog- the jury is still out on this one. I have heard of a case where having a blog helped someone get an interview. (It was a serious blog about the field he was in. I can't imagine that complexity blog would help me get a job if I was looking.)
9. Spending too much time deciding what to have for lunch can be a negative. If he can't figure out what to have for lunch then how can he formulate a coherent research plan.
10. Area can be a factor- does a school need or want someone doing area X? This is sometimes formalized, for example the school has money targeted to hire someone who does, a particular area. (Side topic- Targeted positions- Good or Bad? Good in that you are not going to have to compare people in diff areas. Bad in that there may be someone really good in another area who you want to hire but can't.)
11. Subarea is a factor- for example, we want the kind of theorist who can talk to our people in systems.
12. Having a champion in the dept who is helping push your case HELPS unless the person doing the pushing is a jerk or not good at pushing a case.
13. Being an ex-convict probably hurts. If there was a theory genius who served 10 years in jail, that might be a negative. Might be mitigated if he could pull in some serious grant money. However, if the jail term was for embezzling grant money, then maybe not. Actually, the whole question might depend on what he was in jail for and is he now reformed.
14. A very big factor is how good is the job market when you get out. This may be a much bigger factor than anything else on this list.

11:09 AM # 65 comments

Friday, July 16, 2010

 
How I Find Homework Problems

Posted by Lance

What do you get out of this paragraph (from Charlie Stross' The Atrocity Archives via Daniel Lemire)

The [Turing] theorem is a hack on discrete number theory that simultaneously disproves the Church-Turing hypothesis (wave if you understood that) and worse, permits NP-complete problems to be converted into P-complete ones. This has several consequences, starting with screwing over most cryptography algorithms—translation: all your bank account are belong to us—and ending with the ability to computationally generate a Dho-Nha geometry curve in real time.

I get a new homework question.

Show that NP-complete = P-complete if and only if NP = L.

Wave if you solve it.

8:48 AM # 14 comments

Thursday, July 15, 2010

 
Sparse problems in NP thought to not be in P

Posted by GASARCH

(This post is similar to this old post. I am posting this anyway since when I first posted I made fundamental mistake. I fixed it the point I was trying to make get lost.)

About once a semester I get asked

What are the natural problems that are in NP, not known to be in P, and not known to be NPC? What is known about them?

And I give the standard answers:

1. Graph Isomorphism. If its NPC then PH collapses so it is though to NOT be NPC. There is no real consensus about its status with respect to P.
2. Factoring. If its NPC then NP=coNP so it is thought to NOT be NPC. Most people think that it is NOT in P since people really want to solve this one and have not been able to. This is not a rigorous argument. Factoring is in QP. If quantum computers are ever practical then we may need to rethink how we think about these things.
3. Discrete Log. Actually, are there reasons to think this is NOT NPC?

The answer above are standard and I suspect most of my readers know them. However, there is a large source of problems that are in NP, likely not NPC, likely not in P, that people don't seem to talk about much: SPARSE PROBLEMS that are thought to be hard. The ones I know about are from Ramsey Theory. I give one example but there are many like it:

{(1ⁿ,1^m,c) : the n×m grid can be c-colored and not have any mono squares}

1. This is clearly in NP since the coloring itself is the witness.
2. Since this is a sparse set it is not NPC unless P=NP. (Actually it can be coded as a tally set.)
3. I want to say People think this is hard. You may ask Name Three of Them! Indeed, the number of people who work on Computational Ramsey Theory is fairly small. However, Ramsey Theory itself has many people working on it and nobody has anything close to a result indicating that this problem is in P. I personally think that it is hard.
4. For a fixed c there is a finite number of grids G1,...,Gp such that the n×m grid is c-colorable iff none of G1,...,Gp can fit inside the n×m grid. Hence, for fixed c, the problem is in P. (So the problem is Fixed Parameter Tractable.)
5. Everything said above holds if you replace squares with rectangles or other configurations in the above.

Questions:

1. Are there other sparse problems in NP that we think are NOT in P? (that is, ones that DO NOT come from Ramsey Theory).
2. Should we be teaching these in our complexity classes as examples of possible intermediary problems? The proof that they are likely not NPC is easy. However, to argue that these problems are likely not in P is hard. Also, these problems may appear unnatural to some students. (They may appear unnatural to some of my readers.)
3. It is easy to argue that Factoring is probably not in P since you can point to the fact that people REALLY want to solve it quickly and so far have not. This is not a rigorous argument but I do count it as evidence. Is there a similar argument for GI? Is there a similar argument for my Ramsey Problems?
4. Is there a mathematical reason to think that Factoring, GI, or my Ramsey Problems are not in P?

10:20 AM # 10 comments

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