Complexity of Local Search for CSPs Parameterized by Constraint Difference

TL;DR

This paper investigates the parameterized complexity of local search algorithms for CSPs, providing a complete classification for boolean symmetric cases based on the constraint predicate's properties.

Contribution

It offers a comprehensive complexity characterization of local search for boolean symmetric CSPs, extending the understanding of parameterized algorithms in this domain.

Findings

Complete complexity classification for boolean symmetric CSPs

Identification of tractable and intractable cases based on predicate properties

Extension of local search analysis to a broad class of CSPs

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 $|P\Delta S^*|$. 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.