research-article

Smoothed Analysis of Local Search for the Maximum-Cut Problem

Authors:

Michael Etscheid,

Heiko RöglinAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 13, Issue 2

Article No.: 25, Pages 1 - 12

https://doi.org/10.1145/3011870

Published: 21 March 2017 Publication History

Abstract

Even though local search heuristics are the method of choice in practice for many well-studied optimization problems, most of them behave poorly in the worst case. This is, in particular, the case for the Maximum-Cut Problem, for which local search can take an exponential number of steps to terminate and the problem of computing a local optimum is PLS-complete. To narrow the gap between theory and practice, we study local search for the Maximum-Cut Problem in the framework of smoothed analysis in which inputs are subject to a small amount of random noise. We show that the smoothed number of iterations is quasi-polynomial, that is, it is bounded from above by a polynomial in n^{log n} and ϕ, where n denotes the number of nodes and ϕ denotes the perturbation parameter. This shows that worst-case instances are fragile, and it is a first step in explaining why they are rarely observed in practice.

References

[1]

Heiner Ackermann, Heiko Röglin, and Berthold Vöcking. 2008. On the impact of combinatorial structure on congestion games. J. ACM 55, 6 (2008).

Digital Library

[2]

David Arthur, Bodo Manthey, and Heiko Röglin. 2011. Smoothed analysis of the k-means method. J. ACM 58, 5 (2011), 19.

Digital Library

[3]

David Arthur and Sergei Vassilvitskii. 2009. Worst-case and smoothed analysis of the ICP algorithm, with an application to the k-means method. SIAM J. Comput. 39, 2 (2009), 766--782.

Digital Library

[4]

Baruch Awerbuch, Yossi Azar, Amir Epstein, Vahab S. Mirrokni, and Alexander Skopalik. 2008. Fast convergence to nearly optimal solutions in potential games. In Proceedings of the ACM Conference on Electronic Commerce (EC). 264--273.

Digital Library

[5]

Pavel Berkhin. 2002. Survey of Clustering Data Mining Techniques. Technical report. Accrue Software, San Jose, CA.

[6]

Anand Bhalgat, Tanmoy Chakraborty, and Sanjeev Khanna. 2010. Approximating pure nash equilibrium in cut, party affiliation, and satisfiability games. In Proceedings of the 11th ACM Conference on Electronic Commerce (EC). 73--82.

Digital Library

[7]

George Christodoulou, Vahab S. Mirrokni, and Anastasios Sidiropoulos. 2012. Convergence and approximation in potential games. Theoretical Computer Science 438 (2012), 13--27.

Digital Library

[8]

Robert Elsässer and Tobias Tscheuschner. 2011. Settling the complexity of local max-cut (almost) completely. In Proceedings of the 38th International Colloquium on Automata, Languages and Programming (ICALP). 171--182.

Digital Library

[9]

Matthias Englert, Heiko Röglin, and Berthold Vöcking. 2014. Worst case and probabilistic analysis of the 2-opt algorithm for the TSP. Algorithmica 68, 1 (2014), 190--264.

[10]

Michael Etscheid and Heiko Röglin. 2015. Smoothed analysis of the squared Euclidean maximum-cut problem. In Algorithms-ESA 2015. Springer, 509--520.

[11]

Alex Fabrikant, Christos H. Papadimitriou, and Kunal Talwar. 2004. The complexity of pure Nash equilibria. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC). 604--612.

Digital Library

[12]

David S. Johnson and Lyle A. McGeoch. 1997. The traveling salesman problem: A case study in local optimization. In Local Search in Combinatorial Optimization, E. H. L. Aarts and J. K. Lenstra (Eds.). John Wiley and Sons.

[13]

Jon M. Kleinberg and Éva Tardos. 2006. Algorithm Design. Addison-Wesley. I--XXIII, 1--838 pages.

[14]

Bodo Manthey and Heiko Röglin. 2011. Smoothed analysis: Analysis of algorithms beyond worst case. Inf. Technol. 53, 6 (2011), 280--286.

[15]

Bodo Manthey and Heiko Röglin. 2013. Worst-case and smoothed analysis of k-means clustering with bregman divergences. J. Comput. Geom. 4, 1 (2013), 94--132.

[16]

Christos H. Papadimitriou. 1994. On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. Syst. Sci. 48, 3 (1994), 498--532.

Digital Library

[17]

Heiko Röglin. 2008. The Complexity of Nash Equilibria, Local Optima, and Pareto-Optimal Solutions. Ph.D. Dissertation. RWTH Aachen University.

[18]

Alejandro A. Schäffer and Mihalis Yannakakis. 1991. Simple local search problems that are hard to solve. SIAM J. Comput. 20, 1 (1991), 56--87.

Digital Library

[19]

Daniel A. Spielman and Shang-Hua Teng. 2004. Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time. J. ACM 51, 3 (2004), 385--463.

Digital Library

[20]

Daniel A. Spielman and Shang-Hua Teng. 2009. Smoothed analysis: An attempt to explain the behavior of algorithms in practice. Commun. ACM 52, 10 (2009), 76--84.

Digital Library

[21]

Andrea Vattani. 2011. k-Means requires exponentially many iterations even in the plane. Discrete Comput. Geom. 45, 4 (2011), 596--616.

Digital Library

Cited By

Gao L(2024)Finding local Max-Cut in graphs in randomized polynomial timeSoft Computing - A Fusion of Foundations, Methodologies and Applications10.1007/s00500-023-09230-528:4(3029-3048)Online publication date: 1-Feb-2024
https://dl.acm.org/doi/10.1007/s00500-023-09230-5
Christ MYannakakis MSaha BServedio R(2023)The Smoothed Complexity of Policy Iteration for Markov Decision ProcessesProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585220(1890-1903)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585220
Liu CWei F(2023)Phase transition of degeneracy in minor-closed familiesAdvances in Applied Mathematics10.1016/j.aam.2023.102489146(102489)Online publication date: May-2023
https://doi.org/10.1016/j.aam.2023.102489
Show More Cited By

Index Terms

Smoothed Analysis of Local Search for the Maximum-Cut Problem
1. Mathematics of computing
  1. Probability and statistics
2. Theory of computation
  1. Design and analysis of algorithms
    1. Graph algorithms analysis
  2. Randomness, geometry and discrete structures

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Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 13, Issue 2

Special Issue on SODA'15 and Regular Papers

April 2017

316 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/3040971

Editor:
Aravind Srinivasan
University of Maryland, USA

Issue’s Table of Contents

Copyright © 2017 ACM.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 21 March 2017

Accepted: 01 October 2016

Received: 01 March 2016

Published in TALG Volume 13, Issue 2

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Cited By

Gao L(2024)Finding local Max-Cut in graphs in randomized polynomial timeSoft Computing - A Fusion of Foundations, Methodologies and Applications10.1007/s00500-023-09230-528:4(3029-3048)Online publication date: 1-Feb-2024
https://dl.acm.org/doi/10.1007/s00500-023-09230-5
Christ MYannakakis MSaha BServedio R(2023)The Smoothed Complexity of Policy Iteration for Markov Decision ProcessesProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585220(1890-1903)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585220
Liu CWei F(2023)Phase transition of degeneracy in minor-closed familiesAdvances in Applied Mathematics10.1016/j.aam.2023.102489146(102489)Online publication date: May-2023
https://doi.org/10.1016/j.aam.2023.102489
Erickson Jvan der Hoog IMiltzow T(2022)Smoothing the Gap Between NP and ERSIAM Journal on Computing10.1137/20M1385287(FOCS20-102-FOCS20-138)Online publication date: 7-Apr-2022
https://doi.org/10.1137/20M1385287
Bhaskara AChen APerreault AVijayaraghavan A(2022)Smoothed analysis for tensor methods in unsupervised learningMathematical Programming: Series A and B10.1007/s10107-020-01577-z193:2(549-599)Online publication date: 1-Jun-2022
https://dl.acm.org/doi/10.1007/s10107-020-01577-z
Babichenko YRubinstein AKhuller SVassilevska Williams V(2021)Settling the complexity of Nash equilibrium in congestion gamesProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3406325.3451039(1426-1437)Online publication date: 15-Jun-2021
https://dl.acm.org/doi/10.1145/3406325.3451039
Chen XGuo CVlatakis-Gkaragkounis EYannakakis MZhang XMakarychev KMakarychev YTulsiani MKamath GChuzhoy J(2020)Smoothed complexity of local max-cut and binary max-CSPProceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3357713.3384325(1052-1065)Online publication date: 22-Jun-2020
https://dl.acm.org/doi/10.1145/3357713.3384325
Erickson Jvan der Hoog IMiltzow T(2020)Smoothing the gap between NP and ER2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS46700.2020.00099(1022-1033)Online publication date: Nov-2020
https://doi.org/10.1109/FOCS46700.2020.00099
Zhang XZhan ZZhang J(2020)A Fast Efficient Local Search-Based Algorithm for Multi-Objective Supply Chain Configuration ProblemIEEE Access10.1109/ACCESS.2020.29834738(62924-62931)Online publication date: 2020
https://doi.org/10.1109/ACCESS.2020.2983473
Molla AShur D(2020)Smoothed Analysis of Leader Election in Distributed NetworksStabilization, Safety, and Security of Distributed Systems10.1007/978-3-030-64348-5_14(183-198)Online publication date: 18-Nov-2020
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