On the Smoothed Complexity of Combinatorial Local Search

Authors Yiannis Giannakopoulos , Alexander Grosz , Themistoklis Melissourgos

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Yiannis Giannakopoulos
  • School of Computing Science, University of Glasgow, UK
Alexander Grosz
  • School of Computation, Information and Technology, Technical University of Munich, Germany
Themistoklis Melissourgos
  • School of Computer Science and Electronic Engineering, University of Essex, UK


Y. Giannakopoulos is grateful to Diogo Poças for many useful discussions and inspiration during the early stages of this project.

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Yiannis Giannakopoulos, Alexander Grosz, and Themistoklis Melissourgos. On the Smoothed Complexity of Combinatorial Local Search. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 72:1-72:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We propose a unifying framework for smoothed analysis of combinatorial local optimization problems, and show how a diverse selection of problems within the complexity class PLS can be cast within this model. This abstraction allows us to identify key structural properties, and corresponding parameters, that determine the smoothed running time of local search dynamics. We formalize this via a black-box tool that provides concrete bounds on the expected maximum number of steps needed until local search reaches an exact local optimum. This bound is particularly strong, in the sense that it holds for any starting feasible solution, any choice of pivoting rule, and does not rely on the choice of specific noise distributions that are applied on the input, but it is parameterized by just a global upper bound ϕ on the probability density. The power of this tool can be demonstrated by instantiating it for various PLS-hard problems of interest to derive efficient smoothed running times (as a function of ϕ and the input size). Most notably, we focus on the important local optimization problem of finding pure Nash equilibria in Congestion Games, that has not been studied before from a smoothed analysis perspective. Specifically, we propose novel smoothed analysis models for general and Network Congestion Games, under various representations, including explicit, step-function, and polynomial resource latencies. We study PLS-hard instances of these problems and show that their standard local search algorithms run in polynomial smoothed time. Further applications of our framework to a wide range of additional combinatorial problems can be found in the full version of our paper.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithmic game theory
  • Mathematics of computing → Combinatorial optimization
  • Theory of computation → Computational complexity and cryptography
  • Smoothed Analysis
  • local search
  • better-response dynamics
  • PLS-hardness
  • combinatorial local optimization
  • congestion games
  • pure Nash equilibria


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