Abstract
The Max-Cut problem consists in splitting in two parts the set of vertices of a given graph so as to maximize the sum of weights of the edges crossing the partition. We here address the problem of computing locally maximum cuts in general undirected graphs in a distributed manner. To achieve this, we interpret these cuts as the pure Nash equilibria of a n-player non-zero sum game, each vertex being an agent trying to maximize her selfish interest. A distributed algorithm can then be viewed as the choice of a policy for every agent, describing how to adapt her strategy to other agents’ decisions during a repeated play. In our setting, the only information available to any vertex is the number of its incident edges that cross, or do not cross the cut. In the general, weighted case, computing such an equilibrium can be shown to be PLS-complete, as it is often the case for potential games. We here focus on the (polynomial) unweighted case, but with the additional restriction that algorithms have to be distributed as described above. First, we describe a simple distributed algorithm for general graphs, and prove that it reaches a locally maximum cut in expected time \(4\Delta |E|\), where \(E\) is the set of edges and \(\Delta \) its maximal degree. We then turn to the case of the complete graph, where we prove that a slight variation of this algorithm reaches a locally maximum cut in expected time \(O(\log \log n)\). We conclude by giving experimental results for general graphs.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Adolphs C, Berenbrink P (2012) Distributed selfish load balancing with weights and speeds. In: Proceedings of the 2012 ACM symposium on principles of, distributed computing, pp 135–144
Berenbrink P, Friedetzky T, Hajirasouliha I, Hu Z (2012) Convergence to equilibria in distributed, selfish reallocation processes with weighted tasks. Algorithmica 62:767–786
Chien S, Sinclair A (2007) Convergence to approximate nash equilibria in congestion games. In: Proceedings of SODA, pp 169–178
Christodoulou G, Mirrokni V, Sidiropoulos A (2012) Convergence and approximation in potential games. Theor Comput Sci 438:13–27
Even-Dar E, Kesselman A, Mansour Y (2007) Convergence time to nash equilibrium in load balancing. ACM Trans Algorithms 3(3):32
Fabrikant A, Papadimitriou C, Talwar K (2004) The complexity of pure nash equilibria. In: Proceedings of STOCS, ACM Press, pp 604–612
Goemans M, Williamson D (1995) Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J ACM 42(6):1115–1145
Goldberg P (2004) Bounds for the convergence rate of randomized local search in a multiplayer load-balancing game. In: Proceedings of the twenty-third annual ACM symposium on principles of distributed computing, pp 131–140
Gourvs L, Monnot J (2010) The max k-cut game and its strong equilibria. In: Proceedings of TAMC, Springer, pp 234–246
Khot S, Kindler G, Mossel E (2004) Optimal inapproximability results for max-cut and other 2-variable CSPs. In: Proceedings of FOCS, pp 146–154
Kleinberg J, Tardos E (2005) Algorithm design. Addison-Wesley Longman Publishing Co., Boston
Monderer D, Shapley LS (1996) Potential games. Games Econ Behav 14(1):124–143
Rosenthal RW (1973) A class of games possessing pure-strategy Nash equilibria. Int J Game Theory 2:65–67
Schaffer A, Yannakakis M (1991) Simple local search problems that are hard to solve. SIAM J Comput 20(1):56–87
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this paper
Cite this paper
Auger, D., Cohen, J., Coucheney, P., Rodier, L. (2013). Distributed Selfish Algorithms for the Max-Cut Game. In: Gelenbe, E., Lent, R. (eds) Information Sciences and Systems 2013. Lecture Notes in Electrical Engineering, vol 264. Springer, Cham. https://doi.org/10.1007/978-3-319-01604-7_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-01604-7_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-01603-0
Online ISBN: 978-3-319-01604-7
eBook Packages: Computer ScienceComputer Science (R0)