2026-08-11T08:34:48Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:12439 2024-03-06T10:50:08Z ddc:004 open_access

Existence and Complexity of Approximate Equilibria in Weighted Congestion Games Christodoulou, George Gairing, Martin Giannakopoulos, Yiannis Poças, Diogo Waldmann, Clara Atomic congestion games existence of equilibria pure Nash equilibria approximate equilibria hardness of equilibria We study the existence of approximate pure Nash equilibria (α-PNE) in weighted atomic congestion games with polynomial cost functions of maximum degree d. Previously it was known that d-approximate equilibria always exist, while nonexistence was established only for small constants, namely for 1.153-PNE. We improve significantly upon this gap, proving that such games in general do not have Θ̃(√d)-approximate PNE, which provides the first super-constant lower bound. Furthermore, we provide a black-box gap-introducing method of combining such nonexistence results with a specific circuit gadget, in order to derive NP-completeness of the decision version of the problem. In particular, deploying this technique we are able to show that deciding whether a weighted congestion game has an Õ(√d)-PNE is NP-complete. Previous hardness results were known only for the special case of exact equilibria and arbitrary cost functions. The circuit gadget is of independent interest and it allows us to also prove hardness for a variety of problems related to the complexity of PNE in congestion games. For example, we demonstrate that the question of existence of α-PNE in which a certain set of players plays a specific strategy profile is NP-hard for any α < 3^(d/2), even for unweighted congestion games. Finally, we study the existence of approximate equilibria in weighted congestion games with general (nondecreasing) costs, as a function of the number of players n. We show that n-PNE always exist, matched by an almost tight nonexistence bound of Θ̃(n) which we can again transform into an NP-completeness proof for the decision problem. Schloss Dagstuhl – Leibniz-Zentrum für Informatik George Christodoulou and Martin Gairing and Yiannis Giannakopoulos and Diogo Poças and Clara Waldmann 2020 Is Part Of LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2020.32 urn:nbn:de:0030-drops-124392 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.32 eng https://creativecommons.org/licenses/by/3.0/legalcode