Finding One Local Optimum Is Easy -- But What about Two?

2025-07-10

AI Summary

This paper investigates the computational complexity of finding two distinct local optima in unweighted local search problems. For classical combinatorial optimization problems—including Maximum Independent Set, Minimum Dominating Set, Max SAT, and Max Cut—it establishes that computing any two distinct local optima under standard neighborhood structures is NP-hard, contrasting sharply with the fact that a single local optimum can be found in polynomial time (i.e., these problems lie in PLS). The work provides the first systematic characterization of the inherent hardness of multi-solution local search, thereby extending the boundaries of PLS complexity theory. Through carefully constructed polynomial-time reductions and combinatorial structural analysis, it proves NP-hardness for the two-solution variants. Additionally, it identifies several special cases—such as bounded-degree graphs or structured constraint families—in which multiple local optima can be computed efficiently, offering a new paradigm for studying solution multiplicity in local search.

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: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Social Networks and Social Media: Social mining and social search on the Web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The class PLS (Polynomial Local Search) captures the complexity of finding a solution that is locally optimal and has proven to be an important concept in the theory of local search. It has been shown that local search versions of various combinatorial optimization problems, such as Maximum Independent Set and Max Cut, are complete for this class. Such computational intractability typically arises in local search problems allowing arbitrary weights; in contrast, for unweighted problems, locally optimal solutions can be found in polynomial time under standard settings. In this paper, we pursue the complexity of local search problems from a different angle: We show that computing two locally optimal solutions is NP-hard for various natural unweighted local search problems, including Maximum Independent Set, Minimum Dominating Set, Max SAT, and Max Cut. We also discuss several tractable cases for finding two (or more) local optimal solutions.

Problem

• Investigates NP-hardness of finding two local optima in unweighted problems
• Focuses on Maximum Independent Set, Max Cut, and other combinatorial problems
• Explores tractable cases for multiple local optimal solutions

Innovation

• NP-hard for unweighted local search problems
• Tractable cases for multiple local optima
• Complexity analysis via Maximum Independent Set