The Complexity of Making Unique Choices: Approximating 1-in-k SAT

Abstract

Abstract. We study the approximability of 1-in-kSAT, the variant of Max kSAT where a clause is deemed satisfied when precisely one of its literals is satisfied. We also investigate different special cases of the problem, including those obtained by restricting the literals to be unnegated and/or all clauses to have size exactly k. Our results show that the 1-in-kSAT problem exhibits some rather peculiar phenomena in the realm of constraint satisfaction problems. Specifically, the problem becomes substantially easier to approximate with perfect completeness as well as when negations of literals are not allowed.

Tight Bounds on the Approximability of Almost-satisfiable Horn SAT and Exact Hitting Set

“…An instance of Max 1-in-k-HS consists of a universe U and a collection C of subsets of U of size at most k, and the goal is to find a subset of U that intersects the maximum number of sets in C at a unique element. We prove that Max 1-in-k-HS is hard to approximate within a factor of O(1/ log k) for every fixed integer k. This matches (up to constant factors) an easy factor Ω(1/ log k) approximation algorithm for the problem, and resolves a question posed in [GT05]. It is crucial for the above hardness that sets of size up to k are allowed; indeed, when all sets have size k, there is a simple factor 1/e-approximation algorithm.…”

“…The complexity of this problem was investigated in [GT05] and [DFHS08], where applications of the problem to pricing, computing ad-hoc selective families for radio broadcasting, etc. are also discussed.…”

“…Our interest in this problem stems in part from the following peculiar approximability behavior of Max 1-in-k-HS, as pointed out in [GT05]. The Max 1-in-k-HS problem appears to be much easier to approximate on "satisfiable instances" (where a hitting set intersecting all subsets exactly once exists) or when all sets have size exactly equal to k (instead of at most k).…”

“…The Max 1-in-k-HS problem appears to be much easier to approximate on "satisfiable instances" (where a hitting set intersecting all subsets exactly once exists) or when all sets have size exactly equal to k (instead of at most k). In both these cases, there is a factor 1/eapproximation algorithm, and obtaining a (1/e + ε)-approximation is NP-hard even when both restrictions hold simultaneously [GT05].…”

“…We also investigate another problem, the "exact hitting set" problem for set sizes bounded by k, which has a very peculiar approximation behavior [GT05]. It admits a much better approximation algorithm on satisfiable instances, as well as when sets all have size exactly (or close to) k. We prove that these restrictions are inherent, and relaxing these rules out a constant factor approximation algorithm (again, under the UGC).…”