Abstract
We study the complexity of local search in the max-cut problem with FLIP neighborhood, in which exactly one node changes the partition. We introduce a technique of constructing instances which enforce certain sequences of improving steps. Using our technique we can show that already graphs with maximum degree four satify the following two properties.
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1
There are instances with initial solutions for which every local search takes exponential time to converge to a local optimum.
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1
The problem of computing a local optimum reachable from a given solution by a sequence of improving steps is PSPACE-complete.
Schäffer and Yannakakis (JOC ’91) showed via a so called “tight” PLS-reduction that the properties (1) and (2) hold for graphs with unbounded degree. Our improvement to the degree four is the best possible improvement since Poljak (JOC ’95) showed for cubic graphs that every sequence of improving steps has polynomial length, whereby his result is easily generalizable to arbitrary graphs with maximum degree three. In his paper Poljak also asked whether (1) holds for graphs with maximum degree four, which is settled by our result. Many tight PLS-reductions in the literature are based on the max-cut problem. Via some of them our constructions carry over to other problems and show that the properties (1) and (2) already hold for very restricted sets of feasible inputs of these problems.
Since our paper provides the two results that typically come along with tight PLS-reductions it does naturally put the focus on the question whether it is even PLS-complete to compute a local optimum on graphs with maximum degree four – a question that was recently asked by Ackermann et al. We think that our insights might be helpful for tackling this question.
This work was partially supported by the DFG Schwerpunktprogramm 1307 “Algorithm Engineering.”
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References
Anshelevich, E., Dasgupta, A., Kleinberg, J., Tardos, E., Wexler, T., Roughgarden, T.: The price of stability for network design with fair cost allocation. In: FOCS, pp. 295–304 (2004)
Ackermann, H., Röglin, H., Vöcking, B.: On the Impact of Combinatorial Structure on Congestion Games. JACM Art. 25, 55(6) (2008)
Awerbuch, B., Azar, Y., Epstein, A., Mirokkni, V.S., Skopalik, A.: Fast Convergence to Nearly Optimal Solutions in Potential Games. In: Conference on Electronic Commerce (EC), pp. 264–273 (2008)
Dunkel, J., Schulz, A.S.: On the Complexity of Pure-Strategy Nash Equilibria in Congestion and Local-Effect Games. In: Spirakis, P.G., Mavronicolas, M., Kontogiannis, S.C. (eds.) WINE 2006. LNCS, vol. 4286, pp. 62–73. Springer, Heidelberg (2006)
Christodoulou, G., Mirrokni, V.S., Sidiropoulos, A.: Convergence and approximation in potential games. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 349–360. Springer, Heidelberg (2006)
Englert, M., Röglin, H., Vöcking, B.: Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP. In: SODA, pp. 1295–1304 (2006)
Fabrikant, A., Papadimitriou, C., Talwar, K.: The complexity of pure Nash Equilibria. In: STOC, pp. 604–612 (2004)
Floreen, P., Orponen, P.: Complexity Issues in Discrete Hopfield Networks. NeuroCOLT Technical Report Series, NC-TR-94-009 (October 1994)
Garey, M., Johnson, D.: Computers and Intractability, A Guide to the Theory of NP-Completeness. Freeman, New York (1979)
Goemans, M., Mirokkni, V., Vetta, A.: Sink Equilibria and Convergence. In: FOCS, pp. 142–154 (2005)
Haken, A.: Connectionist Networks that need exponential time to stabilize. Unpublished manuscript, Dept. of Computer Science, University of Toronto (1989)
Haken, A., Luby, M.: Steepest Descent Can Take Exponential Time for Symmetric Connection Networks. Complex Systems 2, 191–196 (1988)
Johnson, D., Papadimitriou, C., Yannakakis, M.: How easy is local search? JCSS 37(1), 79–100 (1988)
Krentel, M.: Structure in locally optimal solutions. In: FOCS, pp. 216–221 (1989)
Ladner, R.: The circuit value problem is log space complete for P. SIGACT News 7(1), 18–20 (1975)
Loebl, M.: Efficient maximal cubic graph cuts. In: Leach Albert, J., Monien, B., Rodríguez-Artalejo, M. (eds.) ICALP 1991. LNCS, vol. 510, pp. 351–362. Springer, Heidelberg (1991)
Michiels, W., Aarts, E., Korst, J.: Theoretical aspects of local search. Springer, New York (2007)
Poljak, S.: Integer linear programs and local search for max-cut. SIAM Journal on Computing 21(3), 450–465 (1995)
Poljak, S., Tuza, Z.: Maximum cuts and largest bipartite subgraphs. In: Combinatorial Optimization, pp. 181–244. American Mathematical Society, Providence (1995)
Rosenthal, R.: A Class of Games Possessing Pure-Strategy Nash Equilibria. International Journal of Game Theory 2, 65–67 (1973)
Savani, R., von Stengel, B.: Hard-To-Solve Bimatrix Games. Econometrica 74, 397–429 (2006)
Schäffer, A., Yannakakis, M.: Simple local search problems that are hard to solve. SIAM Journal on Computing 20(1), 56–87 (1991)
Vattani, A.: k-means Requires Exponentially Many Iterations Even in the Plane. In: Proceedings of the 25th annual symposium on Computational geometry, pp. 324–332 (2009)
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Monien, B., Tscheuschner, T. (2010). On the Power of Nodes of Degree Four in the Local Max-Cut Problem. In: Calamoneri, T., Diaz, J. (eds) Algorithms and Complexity. CIAC 2010. Lecture Notes in Computer Science, vol 6078. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13073-1_24
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