Complexity of Local Search for CSPs Parameterized by Constraint Difference

2025-12-02

AI Summary

This paper investigates the parameterized complexity of local search for Boolean symmetric constraint satisfaction problems (CSPs) under the “constraint difference” parameter $k = |P riangle S^*|$, where $P$ is an initial assignment and $S^*$ an optimal feasible solution. Methodologically, it integrates structural analysis of symmetric Boolean predicates with tailored parameterized algorithm design, yielding distance-aware optimization strategies. The main contribution is a complete complexity dichotomy for all Boolean symmetric CSPs under this parameter: it systematically classifies their local search tractability, unifies criteria for fixed-parameter tractability (FPT) and W[1]-hardness, and thereby establishes the first comprehensive local search complexity landscape for symmetric Boolean CSPs with respect to constraint-difference distance. This resolves a longstanding gap in the parameterized complexity theory of local search.

Technology Category

• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Applications of semantic technologies for improving search, browsing, recommendation, personalization

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 $|PDelta S^*|$. 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.

Problem

• Parameterized complexity of local search for CSPs
• Finding feasible solutions near given current solution
• Characterizing boolean symmetric CSPs by constraint difference

Innovation

• Parameterized local search for CSPs
• Constraint difference as distance parameter
• Complete complexity classification for symmetric CSPs