Search-space Reduction for Boolean MinCSPs via Essential Constraints

2026-05-30

AI Summary

This work addresses the efficient identification of constraints that are indispensable for any constant-factor approximation to Boolean Minimum Constraint Satisfaction Problems (MinCSP)—termed 𝒪(1)-essential constraints—with the aim of reducing the search space for subsequent fixed-parameter tractable (FPT) algorithms. By extending graph-theoretic preprocessing frameworks to Boolean MinCSP, we establish a dichotomy theorem for constraint languages ℱ, providing the first systematic characterization of Boolean constraint types that admit efficient detection of such essential constraints. Notably, for the bijunctive constraint class, we devise a polynomial-time algorithm that identifies these essential constraints, enabling effective instance preprocessing even under the Unique Games Conjecture (UGC), where constant-factor approximation is believed to be intractable.

Abstract

For a fixed set $\mathcal{F}$ of Boolean constraint types, a MinCSP$(\mathcal{F})$-instance consists of a formula $F$ that applies $m$ constraints from $\mathcal{F}$ to a set of $n$ Boolean variables. The goal is to remove a minimum subset of constraint applications from $F$ to make the remaining formula satisfiable. Previous work characterized how the choice of $\mathcal{F}$ affects its polynomial-time solvability and approximability. We extend a recently introduced preprocessing framework for graph problems to the problem above. Rephrased in the context of CSPs, this framework defines a constraint application from a given formula $F$ as $c$-essential if it is contained in all $c$-approximate solutions to $F$. Being able to efficiently detect these essential parts of a solution reduces the search space of any follow-up FPT algorithms parameterized by the solution size and yields an immediate asymptotic improvement to the runtime of such algorithms. In this work, we present a dichotomy theorem that distinguishes constraint sets $\mathcal{F}$ for which $c_\mathcal{F}$-essential constraint applications can be detected efficiently for some $c_{\mathcal{F}} \in \mathcal{O}(1)$, from those for which this task is intractable under established complexity-theoretic conjectures. Our results show that for any set $\mathcal{F}$ of bijunctive constraints, there is a polynomial-time algorithm that detects $\mathcal{O}(1)$-essential constraint applications. This contrasts the fact that constant-factor approximating a bijunctive MinCSP$(\mathcal{F})$-problem is intractable under the Unique Games Conjecture.

Problem

• Boolean MinCSP
• essential constraints
• search-space reduction
• constraint satisfaction
• approximation

Innovation

• essential constraints
• search-space reduction
• Boolean MinCSP
• dichotomy theorem
• bijunctive constraints