A Parameterized-Complexity Framework for Finding Local Optima

2026-01-02

Information Technology Convergence and Services

AI Summary

This work addresses the limitations of traditional complexity frameworks—such as PLS—in capturing the core computational challenges inherent in designing efficient pivoting rules for local search. The authors propose a novel framework that requires algorithms to output not only a locally optimal solution but also the complete improvement path leading to it. By integrating parameterized complexity theory with the entire trajectory of local search and focusing on improvement chains rather than individual steps, this approach more accurately models the computational hardness of pivoting rules. Using a new form of reduction, the study analyzes the fixed-parameter tractability of canonical problems—including Subset Weight Optimization and Weighted Circuit—under c-swap and flip neighborhoods. It shows that these problems are efficiently solvable when parameterized by the number of weights, yet become intractable when parameterized by the distance to an optimal solution.

Technology Category

• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Search
• 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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• local search
• computational complexity
• parameterized complexity
• local optima
• pivoting rule

Innovation

• parameterized complexity
• local search
• pivoting rule
• fixed-parameter tractability
• local improvement chain