Width-Parameterized SAT: Time-Space Tradeoffs

Abstract

Alekhnovich and Razborov (2002) presented an algorithm that solves SAT on instances φ of size n and tree-width TW(φ), using time and space bounded by 2O(TW(φ))nO(1). Although several follow-up works appeared over the last decade, the first open question of Alekhnovich and Razborov remained essentially unresolved: Can one check satisfiability of formulas with small tree-width in polynomial space and time as above? We essentially resolve this question, by (1) giving a polynomial space algorithm with a slightly worse run-time, (2) providing a complexity-theoretic characterization of bounded tree-width SAT, which strongly suggests that no polynomial-space algorithm can run significantly faster, and (3) presenting a spectrum of algorithms trading off time for space, between our PSPACE algorithm and the fastest known algorithm. First, we give a simple algorithm that runs in polynomial space and achieves run-time 3TW(φ) lognnO(1), which approaches the run-time of Alekhnovich and Razborov (2002), but ∗Supported by National Science Foundation grants CCF-0832787 and CCF-1064785. †Supported by National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301, and the National Natural Science Foundation of China Grant 61033001, 61061130540, 61073174, 61250110577. ACM Classification: F.1.3, F.2.2 AMS Classification: 68Q15, 68Q25