Backdoor Treewidth for SAT

Abstract A strong backdoor in a CNF formula is a set of variables such that each possible instantiation of these variables moves the formula into a tractable class. The algorithmic problem of finding a strong backdoor has been the subject of intensive study, mostly within the parameterized complexity framework. Results to date focused primarily on backdoors of small size. In this paper we propose a new approach for algorithmically exploiting strong backdoors for SAT: instead of focusing on small backdoors, we focus on backdoors with certain structural properties. In particular, we consider backdoors that have a certain tree-like structure, formally captured by the notion of backdoor treewidth. First, we provide a fixed-parameter algorithm for SAT parameterized by the backdoor treewidth w. r. t. the fundamental tractable classes Horn, Anti-Horn, and 2CNF. Second, we consider the more general setting where the backdoor decomposes the instance into components belonging to different tractable classes, albeit focusing on backdoors of treewidth 1 (i. e. , acyclic backdoors). We give polynomial-time algorithms for SAT and #SAT for instances that admit such an acyclic backdoor.

Authors

• Robert Ganian
• M. S. Ramanujan 0001
• Stefan Szeider

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