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Node-Max-Cut and the Complexity of Equilibrium in Linear Weighted Congestion Games

Authors Dimitris Fotakis , Vardis Kandiros, Thanasis Lianeas, Nikos Mouzakis, Panagiotis Patsilinakos, Stratis Skoulakis

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ACM Subject Classification
  • Theory of computation → Exact and approximate computation of equilibria
Keywords
  • PLS-completeness
  • Local-Max-Cut
  • Weighted Congestion Games
  • Equilibrium Computation

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Abstract

In this work, we seek a more refined understanding of the complexity of local optimum computation for Max-Cut and pure Nash equilibrium (PNE) computation for congestion games with weighted players and linear latency functions. We show that computing a PNE of linear weighted congestion games is PLS-complete either for very restricted strategy spaces, namely when player strategies are paths on a series-parallel network with a single origin and destination, or for very restricted latency functions, namely when the latency on each resource is equal to the congestion. Our results reveal a remarkable gap regarding the complexity of PNE in congestion games with weighted and unweighted players, since in case of unweighted players, a PNE can be easily computed by either a simple greedy algorithm (for series-parallel networks) or any better response dynamics (when the latency is equal to the congestion). For the latter of the results above, we need to show first that computing a local optimum of a natural restriction of Max-Cut, which we call Node-Max-Cut, is PLS-complete. In Node-Max-Cut, the input graph is vertex-weighted and the weight of each edge is equal to the product of the weights of its endpoints. Due to the very restricted nature of Node-Max-Cut, the reduction requires a careful combination of new gadgets with ideas and techniques from previous work. We also show how to compute efficiently a (1+ε)-approximate equilibrium for Node-Max-Cut, if the number of different vertex weights is constant.

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Dimitris Fotakis, Vardis Kandiros, Thanasis Lianeas, Nikos Mouzakis, Panagiotis Patsilinakos, and Stratis Skoulakis. Node-Max-Cut and the Complexity of Equilibrium in Linear Weighted Congestion Games. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 50:1-50:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.50

Author Details

Dimitris Fotakis
  • National Technical University of Athens, Greece
Vardis Kandiros
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Thanasis Lianeas
  • National Technical University of Athens, Greece
Nikos Mouzakis
  • National Technical University of Athens, Greece
Panagiotis Patsilinakos
  • National Technical University of Athens, Greece
Stratis Skoulakis
  • Singapore University of Technology and Design, Singapore

Funding

This work was supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the "First Call for H.F.R.I. Research Projects to support Faculty members and Researchers and the procurement of high-cost research equipment grant", project: BALSAM (project id: 1424).
  • Skoulakis, Stratis: Supported by NRF 2018, Fellowship NRF-NRFF2018-07. Part of this work was carried out while the author was a PhD student at the National Technical University of Athens.

References

  1. Heiner Ackermann, Heiko Röglin, and Berthold Vöcking. On the impact of combinatorial structure on congestion games. J. ACM, 55(6):25:1-25:22, 2008. Google Scholar
  2. Omer Angel, Sébastien Bubeck, Yuval Peres, and Fan Wei. Local Max-Cut in Smoothed Polynomial Time. In Proc. of the 49th ACM SIGACT Symposium on Theory of Computing (STOC 2017), pages 429-437, 2017. Google Scholar
  3. Anand Bhalgat, Tanmoy Chakraborty, and Sanjeev Khanna. Approximating pure Nash equilibrium in cut, party affiliation, and satisfiability games. In Proc. of the 11th ACM Conference on Electronic Commerce (EC 2010), pages 73-82, 2010. Google Scholar
  4. Ioannis Caragiannis and Angelo Fanelli. On Approximate Pure Nash Equilibria in Weighted Congestion Games with Polynomial Latencies. In Proc. of the 46th International Colloquium on Automata, Languages, and Programming, (ICALP 2019), volume 132 of LIPIcs, pages 133:1-133:12, 2019. Google Scholar
  5. Ioannis Caragiannis, Angelo Fanelli, Nick Gravin, and Alexander Skopalik. Efficient Computation of Approximate Pure Nash Equilibria in Congestion Games. In Proc. of the IEEE 52nd Symposium on Foundations of Computer Science, (FOCS 2011), pages 532-541, 2011. Google Scholar
  6. Ioannis Caragiannis, Angelo Fanelli, Nick Gravin, and Alexander Skopalik. Approximate Pure Nash Equilibria in Weighted Congestion Games: Existence, Efficient Computation, and Structure. ACM Transactions on Economics and Computation, 3(1):2:1-2:32, 2015. Google Scholar
  7. Xi Chen, Chenghao Guo, Emmanouil-Vasileios Vlatakis-Gkaragkounis, Mihalis Yannakakis, and Xinzhi Zhang. Smoothed complexity of local Max-Cut and binary Max-CSP. In Proc. of the 52nd Annual ACM Symposium on Theory of Computing (STOC 2020), 2020. Google Scholar
  8. Steve Chien and Alistair Sinclair. Convergence to approximate Nash equilibria in congestion games. Games and Economic Behavior, 71(2):315-327, 2011. Google Scholar
  9. Robert Elsässer and Tobias Tscheuschner. Settling the Complexity of Local Max-Cut (Almost) Completely. In Proc. of the 38th International Colloquium on Automata, Languages and Programming (ICALP 2011), pages 171-182, 2011. Google Scholar
  10. Michael Etscheid and Heiko Röglin. Smoothed analysis of local search for the maximum-cut problem. ACM Transactions on Algorithms, 13(2):25:1-25:12, 2017. Google Scholar
  11. Eyal Even-Dar, Alexander Kesselman, and Yishay Mansour. Convergence time to nash equilibria. In Proc. of the 30th International Colloquium on Automata, Languages and Programming (ICALP 2003), pages 502-513, 2003. Google Scholar
  12. Alex Fabrikant, Christos H. Papadimitriou, and Kunal Talwar. The Complexity of Pure Nash Equilibria. In Proc. of the 36th Annual ACM Symposium on Theory of Computing (STOC 2004), pages 604-612, 2004. Google Scholar
  13. Angelo Fanelli and Luca Moscardelli. On best response dynamics in weighted congestion games with polynomial delays. In Proc. of the 5th Workshop on Internet and Network Economics (WINE 2009), pages 55-66, 2009. Google Scholar
  14. Matthias Feldotto, Martin Gairing, Grammateia Kotsialou, and Alexander Skopalik. Computing approximate pure nash equilibria in shapley value weighted congestion games. In Proc. of the 13th International Conference on Web and Internet Economics (WINE 2017), pages 191-204, 2017. Google Scholar
  15. Dimitris Fotakis. A selective tour through congestion games. In Algorithms, Probability, Networks, and Games: Scientific Papers and Essays Dedicated to Paul G. Spirakis on the Occasion of His 60th Birthday, pages 223-241, 2015. Google Scholar
  16. Dimitris Fotakis, Spyros C. Kontogiannis, Elias Koutsoupias, Marios Mavronicolas, and Paul G. Spirakis. The structure and complexity of Nash equilibria for a selfish routing game. Theoretical Computer Science, 410(36):3305-3326, 2009. Google Scholar
  17. Dimitris Fotakis, Spyros C. Kontogiannis, and Paul G. Spirakis. Selfish unsplittable flows. Theoretical Computer Science, 348(2-3):226-239, 2005. Google Scholar
  18. Dimitris Fotakis, Spyros C. Kontogiannis, and Paul G. Spirakis. Symmetry in Network Congestion Games: Pure Equilibria and Anarchy Cost. In Proc. of the 3rd Workshop on Approximation and Online Algorithms, Revised Papers (WAOA 2005), pages 161-175, 2005. Google Scholar
  19. Martin Gairing, Thomas Lücking, Marios Mavronicolas, and Burkhard Monien. Computing Nash equilibria for scheduling on restricted parallel links. In Proc. of the 36th Annual ACM Symposium on Theory of Computing (STOC 2004), pages 613-622, 2004. Google Scholar
  20. Martin Gairing and Rahul Savani. Computing Stable Outcomes in Hedonic Games. In Proc. of the 3rd Symposium on Algorithmic Game Theory (SAGT 2010), pages 174-185, 2010. Google Scholar
  21. Yiannis Giannakopoulos, Georgy Noarov, and Andreas S. Schulz. An improved algorithm for computing approximate equilibria in weighted congestion games. CoRR, abs/1810.12806, 2018. URL: http://arxiv.org/abs/1810.12806.
  22. Paul W. Goldberg. Bounds for the convergence rate of randomized local search in a multiplayer load-balancing game. In Proc. of the 23rd Annual ACM Symposium on Principles of Distributed Computing (PODC 2004), pages 131-140, 2004. Google Scholar
  23. Tobias Harks and Max Klimm. On the Existence of Pure Nash Equilibria in Weighted Congestion Games. In Proc. of the 37th International Colloquium on Automata, Languages and Programming (ICALP 2010), pages 79-89, 2010. Google Scholar
  24. Tobias Harks, Max Klimm, and Rolf H. Möhring. Characterizing the existence of potential functions in weighted congestion games. Theory Computing Systems, 49(1):46-70, 2011. Google Scholar
  25. David S. Johnson, Christos H. Papadimitriou, and Mihalis Yannakakis. How easy is local search? Journal of Computer and System Sciences, 37(1):79-100, 1988. Google Scholar
  26. Richard M. Karp. Reducibility among combinatorial problems. In Proc. of Symposium on the Complexity of Computer Computations, The IBM Research Symposia Series, pages 85-103, 1972. Google Scholar
  27. Pieter Kleer and Guido Schäfer. Potential Function Minimizers of Combinatorial Congestion Games: Efficiency and Computation. In Proc. of the 2017 ACM Conference on Economics and Computation (EC 2017), pages 223-240, 2017. Google Scholar
  28. Wil Michiels, Emile Aarts, and Jan Korst. Theoretical Aspects of Local Search. EATCS Monographs in Theoretical Computer Science. Springer, 2007. Google Scholar
  29. Dov Monderer and Lloyd S. Shapley. Potential Games. Games and economic behavior, 14(1):124-143, 1996. Google Scholar
  30. Panagiota N. Panagopoulou and Paul G. Spirakis. Algorithms for pure Nash equilibria in weighted congestion games. ACM Journal of Experimental Algorithmics, 11, 2006. Google Scholar
  31. Svatopluk Poljak. Integer Linear Programs and Local Search for Max-Cut. SIAM Journal on Computing, 21(3):450-465, 1995. Google Scholar
  32. R.W. Rosenthal. A Class of Games Possessing Pure-Strategy Nash Equilibria. International Journal of Game Theory, 2:65-67, 1973. Google Scholar
  33. Alejandro A. Schäffer and Mihalis Yannakakis. Simple Local Search Problems That are Hard to Solve. SIAM Journal on Computing, 20(1):56-87, 1991. Google Scholar
  34. Alexander Skopalik and Berthold Vöcking. Inapproximability of Pure Nash Equilibria. In Proc. of the 40th Annual ACM Symposium on Theory of Computing (STOC 2008), pages 355-364, 2008. Google Scholar
  35. J. Valdez, R.E. Tarjan, and E.L. Lawler. The Recognition of Series-Parallel Digraphs. SIAM J. Comput., 11(2):298-313, 1982. Google Scholar
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