Showing papers by "Alexander A. Razborov published in 2016"

A New Kind of Tradeoffs in Propositional Proof Complexity

Abstract: We exhibit an unusually strong tradeoff in propositional proof complexity that significantly deviates from the established pattern of almost all results of this kind. Namely, restrictions on one resource (width, in our case) imply an increase in another resource (tree-like size) that is exponential not only with respect to the complexity of the original problem, but also to the whole class of all problems of the same bit size. More specifically, we show that for any parameter k = k(n), there are unsatisfiable k-CNFs that possess refutations of width O(k), but such that any tree-like refutation of width n1 − e/k must necessarily have doubly exponential size exp (nΩ(k)). This means that there exist contradictions that allow narrow refutations, but in order to keep the size of such a refutation even within a single exponent, it must necessarily use a high degree of parallelism. Our construction and proof methods combine, in a non-trivial way, two previously known techniques: the hardness escalation method based on substitution formulas and expansion. This combination results in a hardness compression approach that strives to preserve hardness of a contradiction while significantly decreasing the number of its variables.

Guest Column: Proof Complexity and Beyond

Abstract: This essay is a highly personal and biased account of some main concepts and several research directions in modern propositional proof complexity. Special attention will be paid to connections with other disciplines.

On the Width of Semi-Algebraic Proofs and Algorithms.

Abstract: In this paper we initiate the study of width in semi-algebraic proof systems and various cut-based procedures in integer programming. We focus on two important systems: Gomory-Chvatal cutting planes and Lovasz-Schrijver lift-and-project procedures. We develop general methods for proving width lower bounds and apply them to random k-CNFs and several popular combinatorial principles like the perfect matching principle and Tseitin tautologies. We also show how to apply our methods to various combinatorial optimization problems. We establish an “ultimate” trade-off between width and rank, that is give an example in which small width proofs are possible but require exponentially many rounds to perform them.