Proof Complexity ( 11 w 5103 ) October 2 - 7 , 2011 MEALS

S (in alphabetic order by speaker surname) 1. Speaker: Albert Atserias (UPC) Title: Sherali-Adams Relaxations and Indistinguishability in Counting Logics Abstract: Two graphs with adjacency matrices A and B are isomorphic if there exists a permutation matrix P for which the identity PTAP = B holds. Multiplying through by P and relaxing the permutation matrix to a doubly stochastic matrix leads to the linear programming relaxation known as fractional isomorphism. We show that the levels of the Sherali-Adams (SA) hierarchy of linear programming relaxations applied to fractional isomorphism interleave in power with the levels of a well-known colorrefinement heuristic for graph isomorphism called the Weisfeiler-Lehman algorithm, or equivalently, with the levels of indistinguishability in a logic with counting quantifiers and a bounded number of variables. This tight connection has quite striking consequences. For example, it follows immediately from a deep result of Grohe in the context of logics with counting quantifiers, that a fixed number of levels of SA suffice to determine isomorphism of planar and minor-free graphs. We also offer applications both in finite model theory and polyhedral combinatorics. First, we show that certain properties of graphs, such as that of having a flow-circulation of a prescribed value, are definable in the infinitary logic with counting with a bounded number of variables. Second, we exploit a lower bound construction due to Cai, Fürer and Immerman in the context of counting logics to give simple explicit instances that show that the SA relaxations of the vertex-cover and cut polytopes do not reach their integer hulls for up to Ω(n) levels, where n is the number of vertices in the graph. Joint work with Elitza Maneva. 2. Speaker: Chris Beck (Princeton) Title: Time-Space Tradeoffs in Resolution: Lower Bounds for Superlinear Space Abstract: We give the first time-space tradeoff lower bounds for Resolution proofs that apply to superlinear space. In particular, we show that there are formulas of size N that have Resolution refutations of space and size each roughly N log2 N (and like all formulas have Resolution refutations of space N) for which any Resolution refutation using space S and length T requires ST ≥ N1.16 log2 N . By downward translation, a similar tradeoff applies to all smaller space bounds. We also show significantly stronger time-space tradeoff lower bounds for Regular Resolution, which are also the first to apply to superlinear space. Namely, for any space bound S at most 2o(N1/4) there are formulas of size N ≤ 2S that have Regular Resolution proofs of space S and slightly larger size T = O(NS), but for which any Regular Resolution proof of space S1−� requires length TΩ(log log N/ log log log N). Joint work with Paul Beame and Russell Impagliazzo. 3. Speaker: Sam Buss (UCSD) Title: An Improved Separation of Regular Resolution from Proof Resolution and Clause Learning. Abstract: