Min-CSPs on Complete Instances

TL;DR

This paper explores the complexity and approximation algorithms for Min-$k$-CSPs on complete instances, revealing new algorithms and hardness results that distinguish these from dense or expanding instances.

Contribution

It introduces an $O(1)$-approximation for Min-2-SAT on complete instances and a quasi-polynomial time algorithm for Boolean $k$-CSPs, establishing new complexity boundaries.

Findings

An $O(1)$-approximation algorithm for Min-2-SAT on complete instances.

A quasi-polynomial time algorithm for Boolean $k$-CSPs on complete instances.

Characterization of (arity, alphabet size) pairs with quasi-polynomial algorithms.

Abstract

Given a fixed arity $k \geq 2$, Min-$k$-CSP on complete instances involves a set of $n$ variables $V$ and one nontrivial constraint for every $k$-subset of variables (so there are $\binom{n}{k}$ constraints). The goal is to find an assignment that minimizes unsatisfied constraints. Unlike Max-$k$-CSP that admits a PTAS on dense or expanding instances, the approximability of Min-$k$-CSP is less understood. For some CSPs like Min-$k$-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-$k$-CSPs on complete instances. We present an $O(1)$-approximation algorithm for Min-2-SAT on complete instances, the minimization version of Max-2-SAT. Since $O(1)$-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 $k$-CSP on complete instances, including $k$-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.