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Min-CSPs on Complete Instances

Published 24 Oct 2024 in cs.DS | (2410.19066v1)

Abstract: Given a fixed arity k≥2k \geq 2, Min-kk-CSP on complete instances involves a set of nn variables VV and one nontrivial constraint for every kk-subset of variables (so there are (nk)\binom{n}{k} constraints). The goal is to find an assignment that minimizes unsatisfied constraints. Unlike Max-kk-CSP that admits a PTAS on dense or expanding instances, the approximability of Min-kk-CSP is less understood. For some CSPs like Min-kk-SAT, there's an approximation-preserving reduction from general to dense instances, making complete instances unique for potential new techniques. This paper initiates a study of Min-kk-CSPs on complete instances. We present an O(1)O(1)-approximation algorithm for Min-2-SAT on complete instances, the minimization version of Max-2-SAT. Since O(1)O(1)-approximation on dense or expanding instances refutes the Unique Games Conjecture, it shows a strict separation between complete and dense/expanding instances. Then we study the decision versions of CSPs, aiming to satisfy all constraints; which is necessary for any nontrivial approximation. Our second main result is a quasi-polynomial time algorithm for every Boolean kk-CSP on complete instances, including kk-SAT. We provide additional algorithmic and hardness results for CSPs with larger alphabets, characterizing (arity, alphabet size) pairs that admit a quasi-polynomial time algorithm on complete instances.

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