A Note on a Communication Game

We describe a communication game, and a conjecture about this game, whose proof would imply the well-known Sensitivity Conjecture asserting a polynomial relation between sensitivity and block sensitivity for Boolean functions. The author defined this game and observed the connection in Dec. 2013 Jan. 2014. The game and connection were independently discovered by Gilmer, Koucký, and Saks, who also established further results about the game (not proved by us) and published their results in ITCS ‘15 [GKS15]. This note records our independent work, including some observations that did not appear in [GKS15]. Namely, the main conjecture about this communication game would imply not only the Sensitivity Conjecture, but also a stronger hypothesis raised by Chung, Füredi, Graham, and Seymour [CFGS88]; and, another related conjecture we pose about a “query-bounded” variant of our communication game would suffice to answer a question of Aaronson, Ambainis, Balovis, and Bavarian [AABB14] about the query complexity of the “Weak Parity” problem—a question whose resolution was previously shown by [AABB14] to follow from a proof of the Chung et al. hypothesis. 1 The Carmen Sandiego game The work described here1 concerns the following “game” involving three parties: Alice, Bob, and Carmen. Carmen Sandiego is a notorious globe-trotting criminal (created by Brøderbund Software, and now owned by The Learning Company) whose misdeeds inadvertently raise awareness of geography and world cultures. In her latest caper, Carmen makes a single visit to each of n > 1 world cities c1, . . . , cn, in some order of her choosing. Alice and Bob are two sleuths cooperating to catch Carmen; they know the identities of the n cities but not the order in which they will be visited. Bob is hot on the trail and follows Carmen directly from city to city but, due to severe funding restrictions, can’t communicate directly with Alice as he travels. Instead, he is only able to leave a single 0/1 “clue-bit” behind in each city, based on what he has observed of Carmen’s travels so far. (He chooses his clue-bits based upon some deterministic algorithm or “strategy” STRATB agreed upon with Alice in advance.) To make matters worse, when Bob follows Carmen to the final city visited on her tour (let it be cj), she captures him and plants the clue-bit for this city herself, with an intention to confuse. Carmen then “hides out” with her captive somewhere in cj . ∗University of Chicago, Computer Science Department. Email: [email protected]. This research was conducted at the Institute for Advanced Study and was supported by the NSF under agreements Princeton University Prime Award No. CCF-0832797 and Sub-contract No. 00001583. The ideas of the present paper were formulated in Dec. 2013 Jan. 2014, and discussed (in varying detail) in early 2014 with Scott Aaronson, Laci Babai, Sasha Razborov, and Avi Wigderson.