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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.

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