On the Smoothed Complexity of Combinatorial Local Search

Authors Yiannis Giannakopoulos , Alexander Grosz , Themistoklis Melissourgos



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.72.pdf
  • Filesize: 0.8 MB
  • 19 pages

Document Identifiers

Author Details

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

Acknowledgements

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

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.ICALP.2024.72

Abstract

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
Keywords
  • Smoothed Analysis
  • local search
  • better-response dynamics
  • PLS-hardness
  • combinatorial local optimization
  • congestion games
  • pure Nash equilibria

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Emile Aarts and Jan Karel Lenstra, editors. Local Search in Combinatorial Optimization. John Wiley, 1997. Google Scholar
  2. Heiner Ackermann, Heiko Röglin, and Berthold Vöcking. On the impact of combinatorial structure on congestion games. Journal of the ACM, 55(6):1-22, 2008. URL: https://doi.org/10.1145/1455248.1455249.
  3. Rene Beier and Berthold Vöcking. Typical properties of winners and losers in discrete optimization. SIAM Journal on Computing, 35(4):855-881, 2006. URL: https://doi.org/10.1137/s0097539705447268.
  4. Shant Boodaghians, Rucha Kulkarni, and Ruta Mehta. Smoothed efficient algorithms and reductions for network coordination games. In 11th Innovations in Theoretical Computer Science Conference (ITCS), volume 151 of LIPIcs, pages 73:1-73:15, 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.73.
  5. Barun Chandra, Howard Karloff, and Craig Tovey. New results on the old k-opt algorithm for the traveling salesman problem. SIAM Journal on Computing, 28(6):1998-2029, 1999. URL: https://doi.org/10.1137/s0097539793251244.
  6. Constantinos Daskalakis and Christos Papadimitriou. Continuous local search. In Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2011. URL: https://doi.org/10.1137/1.9781611973082.62.
  7. Robert Elsässer and Tobias Tscheuschner. Settling the complexity of local max-cut (almost) completely. In Luca Aceto, Monika Henzinger, and Jiří Sgall, editors, International Colloquium on Automata, Languages, and Programming (ICALP), pages 171-182, 2011. URL: https://doi.org/10.1007/978-3-642-22006-7_15.
  8. Matthias Englert, Heiko Röglin, and Berthold Vöcking. Smoothed analysis of the 2-opt algorithm for the general TSP. ACM Transactions on Algorithms, 13(1):1-15, 2016. URL: https://doi.org/10.1145/2972953.
  9. Michael Etscheid and Heiko Röglin. Smoothed analysis of local search for the maximum-cut problem. ACM Transactions on Algorithms, 13(2):25:1-25:12, 2017. URL: https://doi.org/10.1145/3011870.
  10. Alex Fabrikant, Christos Papadimitriou, and Kunal Talwar. The complexity of pure nash equilibria. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC), pages 604-612, 2004. URL: https://doi.org/10.1145/1007352.1007445.
  11. John Fearnley, Spencer Gordon, Ruta Mehta, and Rahul Savani. Unique end of potential line. Journal of Computer and System Sciences, 114:1-35, 2020. URL: https://doi.org/10.1016/j.jcss.2020.05.007.
  12. Yiannis Giannakopoulos, Alexander Grosz, and Themistoklis Melissourgos. On the smoothed complexity of combinatorial local search, 2022. URL: https://arxiv.org/abs/2211.07547v3.
  13. David S. Johnson, Christos H. Papadimitriou, and Mihalis Yannakakis. How easy is local search? Journal of Computer and System Sciences, 37(1):79-100, 1988. URL: https://doi.org/10.1016/0022-0000(88)90046-3.
  14. Mark W. Krentel. Structure in locally optimal solutions. In Proceedings of the 30th Annual Symposium on Foundations of Computer Science (FOCS), pages 216-221, 1989. URL: https://doi.org/10.1109/SFCS.1989.63481.
  15. Wil Michiels, Jan Korst, and Emile Aarts. Theoretical Aspects of Local Search. Springer, 2007. URL: https://doi.org/10.1007/978-3-540-35854-1.
  16. Dov Monderer and Lloyd S. Shapley. Potential games. Games and Economic Behavior, 14(1):124-143, 1996. URL: https://doi.org/10.1006/game.1996.0044.
  17. James B. Orlin, Abraham P. Punnen, and Andreas S. Schulz. Approximate local search in combinatorial optimization. SIAM Journal on Computing, 33(5):1201-1214, 2004. URL: https://doi.org/10.1137/s0097539703431007.
  18. Robert W. Rosenthal. A class of games possessing pure-strategy Nash equilibria. International Journal of Game Theory, 2(1):65-67, 1973. URL: https://doi.org/10.1007/BF01737559.
  19. Tim Roughgarden. Pure Nash Equilibria and PLS-Completeness, pages 261-278. Cambridge University Press, 2016. URL: https://doi.org/10.1017/CBO9781316779309.020.
  20. Tim Roughgarden, editor. Beyond the Worst-Case Analysis of Algorithms. Cambridge University Press, 2021. URL: https://doi.org/10.1017/9781108637435.
  21. Heiko Röglin. The Complexity of Nash Equilibria, Local Optima, and Pareto-Optimal Solutions. PhD thesis, RWTH Aachen University, 2008. URL: https://publications.rwth-aachen.de/record/50046.
  22. Heiko Röglin and Berthold Vöcking. Smoothed analysis of integer programming. Mathematical Programming, 110(1):21-56, 2007. URL: https://doi.org/10.1007/s10107-006-0055-7.
  23. Alejandro A. Schäffer and Mihalis Yannakakis. Simple local search problems that are hard to solve. SIAM J. Comput., 20(1):56-87, 1991. URL: https://doi.org/10.1137/0220004.
  24. Daniel A. Spielman and Shang-Hua Teng. Smoothed analysis of algorithms. Journal of the ACM, 51(3):385-463, 2004. URL: https://doi.org/10.1145/990308.990310.
  25. Daniel A. Spielman and Shang-Hua Teng. Smoothed analysis: An attempt to explain the behavior of algorithms in practice. Commun. ACM, 52(10):76-84, 2009. URL: https://doi.org/10.1145/1562764.1562785.
  26. Mihalis Yannakakis. Computational complexity. In Emile Aarts and Jan Karel Lenstra, editors, Local Search in Combinatorial Optimization, chapter 2. John Wiley, 1997. Google Scholar