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Settling the Complexity of Local Max-Cut (Almost) Completely

  • Robert Elsässer
  • Tobias Tscheuschner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)

Abstract

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 \(\mathcal{PLS}\)-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(n 2) 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 \(\mathcal{PLS}\)-complete. In this paper, we prove that the computation of a local optimum on graphs with maximum degree five is \(\mathcal{PLS}\)-complete. Thus, we solve the problem posed by Ackermann et al. almost completely by showing that d is either four or five (unless \(\mathcal{PLS}\)\(\mathcal{P}\)).

On the other side, we also prove that on graphs with degree O(logn) 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 σ 2, 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.

Keywords

Max-Cut PLS graphs local search smoothed complexity 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Robert Elsässer
    • 1
  • Tobias Tscheuschner
    • 1
  1. 1.Faculty of Computer Science, Electrical Engineering and MathematicsUniversity of PaderbornGermany

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