A Parameterized-Complexity Framework for Finding Local Optima

TL;DR

This paper introduces a new parameterized-complexity framework for local search that captures the entire process of finding local optima and chains of improvements, providing both lower bounds and tractability results.

Contribution

It proposes a novel local search framework that models the full optimization process and applies parameterized complexity to analyze its computational difficulty.

Findings

Established fixed-parameter tractability results for specific problems.

Ruling out tractability when parameterized by distance to the optimum.

Provided new lower bounds for local search complexity.

Abstract

Local search is a fundamental optimization technique that is both widely used in practice and deeply studied in theory, yet its computational complexity remains poorly understood. The traditional frameworks, PLS and the standard algorithm problem, introduced by Johnson, Papadimitriou, and Yannakakis (1988) fail to capture the methodology of local search algorithms: PLS is concerned with finding a local optimum and not with using local search, while the standard algorithm problem restricts each improvement step to follow a fixed pivoting rule. In this work, we introduce a novel formulation of local search which provides a middle ground between these models. In particular, the task is to output not only a local optimum but also a chain of local improvements leading to it. With this framework, we aim to capture the challenge in designing a good pivoting rule. Especially, when combined with the parameterized complexity paradigm, it enables both strong lower bounds and meaningful tractability results. Unlike previous works that combined parameterized complexity with local search, our framework targets the whole task of finding a local optimum and not only a single improvement step. Focusing on two representative meta-problems -- Subset Weight Optimization Problem with the $c$-swap neighborhood and Weighted Circuit with the flip neighborhood -- we establish fixed-parameter tractability results related to the number of distinct weights, while ruling out an analogous result when parameterized by the distance to the nearest optimum via a new type of reduction.