Thursday, January 18, 2018

A Sensitive Game

Last month I posted about the sensitivity conjecture and today I would like to talk about an interesting game developed by Gilmer, Koucký and Saks and independently by Drucker that could yield a proof.

Consider the following game on n bits among three players, Alice, Bob and Carol. The game works as follows: Carol picks a bit position and Alice sets that bit to 0 or 1. Carol then picks another unused bit position and Alice sets that bit as well. This repeats until the last bit which Carol gets to set herself. The bits are then revealed to Bob who has to give a set of bit positions that includes the bit Carol set at the end. Alice and Bob can work out a strategy ahead of time with the goal of minimizing the size of the set.

If Carol can force Bob to give a set of nε bits for some ε>0, this would prove the sensitivity conjecture! Basically a family of functions that give a counterexample to the sensitivity conjecture would give a strategy for Alice and Bob that yields a set of size no(1). You can find the full proof in the papers above.

How well can Alice and Bob do? They can use the following strategy to achieve n1/2: Break up the input positions into n1/2 groups of n1/2 bits each. Whenever Carol picks a bit position, Alice sets that bit to zero unless it is the last bit in that group in which case she sets it to one. Carol can set the last bit in some group anyway she wants. Bob will either see every group having one 1 and will give the set of the positions of all the 1's, or Bob will see a group of all zeros and give the positions in that
group.

Mario Szegedy uses error-correcting codes to improve the upper bound to O(n0.4732). A Georgia Tech undergraduate DeVon Ingram improves the upper bound to O(n0.4696) by generalizing Szegedy's techniques. Ingram's proof comes down to finding a one-to-one function mapping subsets of {1,...,15} of size 4 to perfect codes of length 15 with the property that the bits of the code are 1 for the indices of the subset. Perhaps one could find a clever mapping that has this property but Ingram wrote code using a matching algorithm. A true computer assisted proof. Longer code lengths alas won't yield direct improvements.

The connection to sensitivity doesn't go both ways. An upper bound of no(1) wouldn't necessarily yield a counterexample to the sensitivity conjecture though would tell us the limitation of this game approach. Nevertheless I find it cool that a game that doesn't talk about functions at all could help settle an important question about them.