Resolution and the binary encoding of combinatorial principles.

Res(s) is an extension of Resolution working on s-DNFs. We prove tight n (k) lower bounds for the size of refutations of the binary version of the k-Clique Principle in Res(o(log log n)). Our result improves that of Lauria, Pudlák et al. [27] who proved the lower bound for Res(1), i.e. Resolution. The exact complexity of the (unary) k-Clique Principle in Resolution is unknown. To prove the lower bound we do not use any form of the Switching Lemma [35], instead we apply a recursive argument specific for binary encodings. Since for the k-Clique and other principles lower bounds in Resolution for the unary version follow from lower bounds in Res(log n) for their binary version we start a systematic study of the complexity of proofs in Resolution-based systems for families of contradictions given in the binary encoding. We go on to consider the binary version of the weak Pigeonhole Principle Bin-PHPmn for m > n. Using the the same recursive approach we prove the new result that for any > 0, Bin-PHPmn requires proofs of size 2n1− in Res(s) for s = o(log1/2 n). Our lower bound is almost optimal since for m 2 p n log n there are quasipolynomial size proofs of Bin-PHPmn in Res(log n). Finally we propose a general theory in which to compare the complexity of refuting the binary and unary versions of large classes of combinatorial principles, namely those expressible as first order formulae in 2-form and with no finite model

Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs

We show exponential lower bounds on resolution proof length for pigeonhole principle (PHP) formulas and perfect matching formulas over highly unbalanced, sparse expander graphs, thus answering the challenge to establish strong lower bounds in the regime between balanced constant-degree expanders as in [Ben-Sasson and Wigderson '01] and highly unbalanced, dense graphs as in [Raz '04] and [Razborov '03, '04]. We obtain our results by revisiting Razborov's pseudo-width method for PHP formulas over dense graphs and extending it to sparse graphs. This further demonstrates the power of the pseudo-width method, and we believe it could potentially be useful for attacking also other longstanding open problems for resolution and other proof systems

Certifying Solvers for Clique and Maximum Common (Connected) Subgraph Problems

An algorithm is said to be certifying if it outputs, together with a solution to the problem it solves, a proof that this solution is correct. We explain how state of the art maximum clique, maximum weighted clique, maximal clique enumeration and maximum common (connected) induced subgraph algorithms can be turned into certifying solvers by using pseudo-Boolean models and cutting planes proofs, and demonstrate that this approach can also handle reductions between problems. The generality of our results suggests that this method is ready for widespread adoption in solvers for combinatorial graph problems

Proof Complexity and Beyond

Proof complexity is a multi-disciplinary intellectual endeavor that addresses questions of the general form “how difficult is it to prove certain mathematical facts?” The current workshop focused on recent advances in our understanding of logic-based proof systems and on connections to algorithms, geometry and combinatorics research, such as the analysis of approximation algorithms, or the size of linear or semidefinite programming formulations of combinatorial optimization problems, to name just two important examples