Tree-width is a well-studied parameter of structures that measures their similarity to a tree. Many important NP-complete problems, such as Boolean satisfiability (SAT), are tractable on bounded tree-width instances. In this paper we focus on the canonical PSPACE-complete problem QBF, the fully-quantified version of SAT. It was shown by Pan and Vardi [LICS 2006] that this problem is PSPACE-complete even for formulas whose tree-width grows extremely slowly. Vardi also posed the question of whether the problem is tractable when restricted to instances of bounded treewidth. We answer this question by showing that QBF on instances with constant tree-width is PSPACE-complete.

Abstract-The relativized weak pigeonhole principle states that if at least 2n out of n 2 pigeons fly into n holes, then some hole must be doubly occupied. We prove that every DNF-refutation of the CNF encoding of this principle requires size 2 (log n) 3/2− for every > 0 and every sufficiently large n. For its proof we need to discuss the existence of unbalanced low-degree bipartite expanders satisfying a certain robustness condition.

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Abstract-The relativized weak pigeonhole principle states that if at least 2n out of n 2 pigeons fly into n holes, then some hole must be doubly occupied. We prove that every DNF-refutation of the CNF encoding of this principle requires size 2 (log n) 3/2− for every > 0 and every sufficiently large n. For its proof we need to discuss the existence of unbalanced low-degree bipartite expanders satisfying a certain robustness condition.