Complexity of Local Search for CSPs Parameterized by Constraint Difference

TL;DR

This work introduces ImproveMaxCSP, a parameterized local-search framework for CSPs where the parameter k measures the symmetric-difference distance between a given candidate set of constraints and an underlying near-feasible solution. It provides a complete complexity dichotomy for symmetric boolean CSPs: the problems ImproveMaxCSP({SymLang-N}(r,S)) are either fixed-parameter tractable or W[1]-hard; the only nontrivial tractable cases are r-AND and 2AE, with explicit FPT algorithms, while rAE for r \ge 3 and ≤1-out-of-r-SAT remain W[1]-hard. The tractable results hinge on color-coding and densest-subhypergraph techniques, including a novel MaximumInducedSubhypergraphwithVertexWeights approach, while the hardness results derive from reductions from MinCSPs and paired-cut problems. This characterization clarifies how near-perfect instances influence parameterized tractability in CSP local search and connects CSP structural properties to precise complexity outcomes. The work also draws connections to ImproveMaxCut-Uncut and provides a coherent framework for future extensions to broader CSP languages and structural graph problems.

Abstract

In this paper, we study the parameterized complexity of local search, whose goal is to find a good nearby solution from the given current solution. Formally, given an optimization problem where the goal is to find the largest feasible subset $S$ of a universe $U$, the new input consists of a current solution $P$ (not necessarily feasible) as well as an ordinary input for the problem. Given the existence of a feasible solution $S^*$, the goal is to find a feasible solution as good as $S^*$ in parameterized time $f(k) \cdot n^{O(1)}$, where $k$ denotes the distance $|PΔS^*|$. This model generalizes numerous classical parameterized optimization problems whose parameter $k$ is the minimum number of elements removed from $U$ to make it feasible, which corresponds to the case $P = U$. We apply this model to widely studied Constraint Satisfaction Problems (CSPs), where $U$ is the set of constraints, and a subset $U'$ of constraints is feasible if there is an assignment to the variables satisfying all constraints in $U'$. We give a complete characterization of the parameterized complexity of all boolean-alphabet symmetric CSPs, where the predicate's acceptance depends on the number of true literals.

Theorems & Definitions (63)

• Theorem 1.1: Main Theorem (Informal)
• Claim 2.1
• proof
• Theorem 2.2: Main Theorem (Formal)
• Theorem 2.3
• Remark 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8