Generating Hard SAT/CSP Instances Using Expander Graphs

In this paper we provide a new method to generate hard k-SAT instances. We incrementally construct a high girth bipartite incidence graph of the k-SAT instance. Having high girth assures high expansion for the graph, and high expansion implies high resolution width. We have extended this approach to generate hard n-ary CSP instances and we have also adapted this idea to increase the expansion of the system of linear equations used to generate XORSAT instances, being able to produce harder satisfiable instances than former generators. Introduction Providing challenging benchmarks for the SAT and the CSP problems is of a great significance for both the experimental evaluation of SAT and CSP solvers and for the theoretical computer science community. Every year new benchmarks are submitted to the SAT and CSP competitions. Our aim is to provide a method for generating hard k-SAT and n-ary CSP instances. In order to do that we look at the field of propositional proof complexity, where it turns out that graph expansion has been established as a key to hard formulas for resolution (e.g. (Atserias 2004)), but also for other proof systems like the polynomial calculus. Roughly speaking, an expander graph is a graph G=(V ,E) that, for any, not too big, subset of vertexes S, its set of neighbors in V \ S is big, compared with |S|. We have compared our approach against other methods in the SAT Community (Bayardo & Schrag 1996; Boufkhad et al. 2005) which try to get hard SAT instances by balancing the occurrences of literals, and thus the degrees of the vertexes at the literal incidence graph become also balanced. Our empirical results confirm that our method generates harder instances. We have also modified the underlying system of linear equations used in regular kXORSAT (Järvisalo 2006) by using our High girth bipartite graphs, instead of the original random regular bipartite ∗Research partially supported by projects TIN2006-15662C02-02, TIN2007-68005-C04-02 and José Castillejo 2007 program funded by the Ministerio de Educación y Ciencia Copyright c © 2008, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. graphs, and show that the hardness of the instances increases by orders of magnitude. Preliminaries Definition 1 The expansion of a subset X from the vertexes of G = (L ∪ R,E) is defined to be the ratio |N(X)|/|X|, where N(X) = {w ∈ (L ∪R) \X | ∃v ∈ X, {v, w} ∈ E} is the set of outside neighbors of X . When all the neighbors of X are inside X , we have expansion 0. We consider a set high expanding when its expansion is greater than 1, that means that the set of different outside neighbors of X is larger than X , so it is well connected with the rest of the graph. Definition 2 A left (α, c)-expander is a bipartite graph (L∪ R,E) such that every subset of L of size at most α|L| has expansion at least c. For this work, the following three concepts are the main tools used to link complexity with structural properties of k-SAT and n-ary CSP instances. Definition 3 Given a k-SAT instance F with set of clauses C, set of variables V and set of literals L, G(F ) = (C ∪ V,E) is its bipartite variable incidence graph such that (c, v) ∈ E if and only if variable v appears in clause c. LG(F ) = (C ∪L,E) is its bipartite literal incidence graph where (c, l) ∈ E if and only if literal l appears in clause c. Observe that if LG(F ) = (C ∪ L,E) is a left (α, c)expander, then G(F ) = (C ∪ V,E), will be, at least, a left (α, c/2)-expander. High girth bipartite graphs Probabilistic methods have been used to show that regular graphs are almost surely very good expanders.The particular case of k-regular or (k1, k2)-regular bipartite graphs have received special attention in the communications community and such bipartite graphs are good expanders almost always. However, the balance of the degrees does not provide a complete characterization of good expander graphs. Consider the graphs (a) and (b) of Figure 1, that are both equally balanced. Graph (a) has several cycles of length 4, and thus girth 4, and its expansion for some left subsets of size 4 is 5/4. By contrast, in graph (b) the minimum expansion for Proceedings of the Twenty-Third AAAI Conference on Artificial Intelligence (2008)