Min-CSPs on Complete Instances
Aditya Anand, Euiwoong Lee, Amatya Sharma

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 , Min--CSP on complete instances involves a set of variables and one nontrivial constraint for every -subset of variables (so there are constraints). The goal is to find an assignment that minimizes unsatisfied constraints. Unlike Max--CSP that admits a PTAS on dense or expanding instances, the approximability of Min--CSP is less understood. For some CSPs like Min--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--CSPs on complete instances. We present an -approximation algorithm for Min-2-SAT on complete instances, the minimization version of Max-2-SAT. Since -approximation on dense or expanding instances refutes the Unique Games Conjecture, it shows a strict separation…
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Taxonomy
TopicsLogic, programming, and type systems
