From High Girth Graphs to Hard Instances

  • Authors:

  • Carlos Ansótegui;Ramón Béjar;César Fernàndez;Carles Mateu

  • Affiliations:

  • Dept. of Computer Science, Universitat de Lleida, Spain;Dept. of Computer Science, Universitat de Lleida, Spain;Dept. of Computer Science, Universitat de Lleida, Spain;Dept. of Computer Science, Universitat de Lleida, Spain

  • Venue:
  • CP '08 Proceedings of the 14th international conference on Principles and Practice of Constraint Programming
  • Year:
  • 2008

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Abstract

In this paper we provide a new method to generate hard k-SAT instances. Basically, we construct the bipartite incidence graph of a k-SAT instance where the left side represents the clauses and the right side represents the literals of our Boolean formula. Then, the clauses are filled by incrementally connecting both sides while keeping the girth of the graph as high as possible. That assures that the expansion of the graph is also high. It has been shown that high expansion implies high resolution width w. The resolution width characterizes the hardness of an instance Fof nvariables since if every resolution refutation of Fhas width wthen every resolution refutation requires size $2^{\Omega(w^2/n)}$. We have extended this approach to generate hard n-ary CSP instances. Finally, we have also adapted this idea to increase the expansion of the system of linear equations used to generate XOR-SAT instances, being able to produce harder satisfiable instances than former generators.