Journal of Computer and System Sciences - 26th IEEE Conference on Foundations of Computer Science, October 21-23, 1985
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SIAM Journal on Computing
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ICALP '91 Proceedings of the 18th International Colloquium on Automata, Languages and Programming
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STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
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STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Beyond Hirsch Conjecture: Walks on Random Polytopes and Smoothed Complexity of the Simplex Method
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Smoothed analysis of integer programming
Mathematical Programming: Series A and B
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SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
On the Hardness and Smoothed Complexity of Quasi-Concave Minimization
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Structure in locally optimal solutions
SFCS '89 Proceedings of the 30th Annual Symposium on Foundations of Computer Science
On the impact of combinatorial structure on congestion games
Journal of the ACM (JACM)
Smoothed analysis: an attempt to explain the behavior of algorithms in practice
Communications of the ACM - A View of Parallel Computing
k-Means Has Polynomial Smoothed Complexity
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Local search: simple, successful, but sometimes sluggish
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Computing stable outcomes in hedonic games
SAGT'10 Proceedings of the Third international conference on Algorithmic game theory
On the power of nodes of degree four in the local max-cut problem
CIAC'10 Proceedings of the 7th international conference on Algorithms and Complexity
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We consider the problem of finding a local optimum for the Max-Cut problem with FLIP-neighborhood, in which exactly one node changes the partition. Schäffer and Yannakakis (SICOMP, 1991) showed PLS-completeness of this problem on graphs with unbounded degree. On the other side, Poljak (SICOMP, 1995) showed that in cubic graphs every FLIP local search takes O(n2) steps, where n is the number of nodes. Due to the huge gap between degree three and unbounded degree, Ackermann, Röglin, and Vöcking (JACM, 2008) asked for the smallest d such that on graphs with maximum degree d the local MAX-CUT problem with FLIP-neighborhood is PLS-complete. In this paper, we prove that the computation of a local optimum on graphs with maximum degree five is PLS-complete. Thus, we solve the problem posed by Ackermann et al. almost completely by showing that d is either four or five (unless PLS ⊆ P). On the other side, we also prove that on graphs with degree O(log n) every FLIP local search has probably polynomial smoothed complexity. Roughly speaking, for any instance, in which the edge weights are perturbated by a (Gaussian) random noise with variance &sigma2, every FLIP local search terminates in time polynomial in n and σ-1, with probability 1-n-Ω(1). Putting both results together, we may conclude that although local MAX-CUT is likely to be hard on graphs with bounded degree, it can be solved in polynomial time for slightly perturbated instances with high probability.