Improved Fixed-Parameter Algorithm for the Minimum Weight 3-SAT Problem

Abstract

The problem of finding a satisfying assignment for a CNF formula that minimizes the weight (the number of variables that are set to 1) is NP-complete even if the formula is a 2-CNF formula. It generalizes the well-studied problem of finding the smallest hitting set for a family of sets, which can be modeled using a CNF formula with no negative literals. The natural parameterized version of the problem asks for a satisfying assignment of weight at most k.

It is known that when the input instance is a 2-CNF formula, the problem actually is equivalent (in terms of parameterized and exact complexity) to the vertex cover (or 2-hitting set) problem. In this paper, we present the first non-trivial fixed-parameter algorithm for the problem when the given input is a 3-CNF formula.

We give an 2.85^k n ^O(1) algorithm for determining whether a given 3-CNF formula on n variables has a satisfying assignment with weight at most k. This improves the trivial 3^k n ^O(1) time algorithm for the problem and answers a question asked in an earlier paper. This implies that within the same time, we can test whether a given 3-CNF formula has a weak backdoor on k variables, to a 0-valid formula – i.e. whether there are k variables such that there exists an assignment to these variables that results in a 0-valid formula (formulas that are satisfiable by an all 0 assignment). This improves the naive 6^k n ^O(1) algorithm for the problem.