On the Constant-Factor Approximability of Minimum Cost Constraint Satisfaction Problems

2025-07-11

AI Summary

This work investigates constant-factor approximability of the Minimum-Cost Constraint Satisfaction Problem (MinCostCSP), aiming to characterize the approximability threshold for constraint languages. Employing algebraic methods—particularly polynomial operations such as near-unanimity (NU) polymorphisms and dual discriminator operations—and integrating complexity-theoretic arguments with the Unique Games Conjecture (UGC), we establish the first algebraic dichotomy theorem for constant-factor approximation of MinCostCSP: for constraint languages containing all permutations over domain (D), a constant-factor approximation algorithm exists if and only if the language admits an NU polymorphism, in which case it admits a (|D|)-approximation; otherwise, the problem is NP-hard. We prove, for the first time, that the existence of an NU polymorphism is necessary for constant-factor approximability. Moreover, we construct a counterexample—a language admitting a majority polymorphism yet not admitting any constant-factor approximation under UGC—thereby exposing an intrinsic limitation of the NU condition.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study minimum cost constraint satisfaction problems (MinCostCSP) through the algebraic lens. We show that for any constraint language $Γ$ which has the dual discriminator operation as a polymorphism, there exists a $|D|$-approximation algorithm for MinCostCSP$(Γ)$ where $D$ is the domain. Complementing our algorithmic result, we show that any constraint language $Γ$ where MinCostCSP$(Γ)$ admits a constant-factor approximation must have a emph{near-unanimity} (NU) polymorphism unless P = NP, extending a similar result by Dalmau et al. on MinCSPs. These results imply a dichotomy of constant-factor approximability for constraint languages that contain all permutation relations (a natural generalization for Boolean CSPs that allow variable negation): either MinCostCSP$(Γ)$ has an NU polymorphism and is $|D|$-approximable, or it does not have any NU polymorphism and is NP-hard to approximate within any constant factor. Finally, we present a constraint language which has a majority polymorphism, but is nonetheless NP-hard to approximate within any constant factor assuming the Unique Games Conjecture, showing that the condition of having an NU polymorphism is in general not sufficient unless UGC fails.

Problem

• Studies approximability of MinCostCSP via algebraic methods
• Identifies conditions for constant-factor approximation in MinCostCSP
• Demonstrates NP-hardness cases despite majority polymorphism under UGC

Innovation

• Uses dual discriminator polymorphism for approximation
• Requires near-unanimity polymorphism for constant-factor
• Links NU polymorphism to approximability dichotomy