Michael Etscheid
Heiko Röglin
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 nlog 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.
- Heiner Ackermann, Heiko Röglin, and Berthold Vöcking. 2008. On the impact of combinatorial structure on congestion games. J. ACM 55, 6 (2008). Google Scholar
Digital Library
- David Arthur, Bodo Manthey, and Heiko Röglin. 2011. Smoothed analysis of the k-means method. J. ACM 58, 5 (2011), 19. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- Pavel Berkhin. 2002. Survey of Clustering Data Mining Techniques. Technical report. Accrue Software, San Jose, CA.Google Scholar
- 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. Google ScholarDigital Library
- George Christodoulou, Vahab S. Mirrokni, and Anastasios Sidiropoulos. 2012. Convergence and approximation in potential games. Theoretical Computer Science 438 (2012), 13--27. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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.Google ScholarCross Ref
- Michael Etscheid and Heiko Röglin. 2015. Smoothed analysis of the squared Euclidean maximum-cut problem. In Algorithms-ESA 2015. Springer, 509--520.Google Scholar
- 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. Google ScholarDigital Library
- 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.Google Scholar
- Jon M. Kleinberg and Éva Tardos. 2006. Algorithm Design. Addison-Wesley. I--XXIII, 1--838 pages.Google Scholar
- Bodo Manthey and Heiko Röglin. 2011. Smoothed analysis: Analysis of algorithms beyond worst case. Inf. Technol. 53, 6 (2011), 280--286.Google ScholarCross Ref
- 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.Google Scholar
- 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. Google ScholarDigital Library
- Heiko Röglin. 2008. The Complexity of Nash Equilibria, Local Optima, and Pareto-Optimal Solutions. Ph.D. Dissertation. RWTH Aachen University.Google Scholar
- Alejandro A. Schäffer and Mihalis Yannakakis. 1991. Simple local search problems that are hard to solve. SIAM J. Comput. 20, 1 (1991), 56--87. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- Andrea Vattani. 2011. k-Means requires exponentially many iterations even in the plane. Discrete Comput. Geom. 45, 4 (2011), 596--616. Google ScholarDigital Library
Index Terms
- Smoothed Analysis of Local Search for the Maximum-Cut Problem
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