Constant-factor approximation of MinCostCSP with a conservative majority polymorphism

2026-07-12

AI Summary

This study investigates whether the Minimum-Cost Constraint Satisfaction Problem (MinCostCSP) admits a constant-factor approximation algorithm over relational structures endowed with a conservative majority polymorphism—specifically, a 3-near-unanimity operation. By integrating algebraic structure analysis with approximation algorithm theory, the work establishes the first dichotomy theorem for constant-factor approximability of MinCostCSP on such structures: either a constant-factor approximation algorithm exists, or none exists under standard complexity-theoretic assumptions. Beyond providing a complete classification, the paper demonstrates that this approximability cannot be fully characterized by purely algebraic conditions alone, thereby revealing inherent limitations of algebraic approaches in the context of approximation algorithms for constraint satisfaction problems.

Abstract

For a relational structure A, the Minimum Cost Constraint Satisfaction Problem is the following problem denoted by MinCostCSP(A): Given an instance of CSP(A) with rational costs on variable-value pairs, find a solution to the instance minimizing the sum of the chosen costs. For the exact minimization, a classification of MinCostCSP(A) in terms of A was established by Takhanov [STACS'10]. We focus on constant-factor approximations of MinCostCSP(A). DeHaan, Huang, and Lee recently showed that if A fails to admit a conservative near-unanimity polymorphism then MinCostCSP(A) is not constant-factor approximable [APPROX'25]. We provide a first step towards a classification, by proving a dichotomy for structures A admitting a conservative majority (also known as 3-near-unanimity) polymorphism. Our dichotomy criterion is not in terms of an algebraic condition on A but we show that this is unavoidable. We include a simple argument proving that no such condition exists.

Problem

• MinCostCSP
• constant-factor approximation
• conservative majority polymorphism
• constraint satisfaction problem
• dichotomy

Innovation

• MinCostCSP
• constant-factor approximation
• conservative majority polymorphism
• dichotomy
• algebraic obstructions