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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

We study truthful mechanisms for allocation problems in graphs, both for the minimization (i.e., scheduling) and maximization (i.e., auctions) setting. The minimization problem is a special case of the well-studied unrelated machines scheduling problem, in which every given task can be executed only by two pre-specified machines in the case of graphs or a given subset of machines in the case of hypergraphs. This corresponds to a multigraph whose nodes are the machines and its hyperedges are the tasks. This class of problems belongs to multidimensional mechanism design, for which there are no known general mechanisms other than the VCG and its generalization to affine minimizers. We propose a new class of mechanisms that are truthful and have significantly better performance than affine minimizers in many settings. Specifically, we provide upper and lower bounds for truthful mechanisms for general multigraphs, as well as special classes of graphs such as stars, trees, planar graphs, k-degenerate graphs, and graphs of a given treewidth. We also consider the objective of minimizing or maximizing the L^p-norm of the values of the players, a generalization of the makespan minimization that corresponds to p = ∞, and extend the results to any p > 0.

George Christodoulou, Elias Koutsoupias, and Annamária Kovács. Truthful Allocation in Graphs and Hypergraphs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 56:1-56:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{christodoulou_et_al:LIPIcs.ICALP.2021.56, author = {Christodoulou, George and Koutsoupias, Elias and Kov\'{a}cs, Annam\'{a}ria}, title = {{Truthful Allocation in Graphs and Hypergraphs}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {56:1--56:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.56}, URN = {urn:nbn:de:0030-drops-141256}, doi = {10.4230/LIPIcs.ICALP.2021.56}, annote = {Keywords: Algorithmic Game Theory, Scheduling Unrelated Machines, Mechanism Design} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

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.

George Christodoulou, Martin Gairing, Yiannis Giannakopoulos, Diogo Poças, and Clara Waldmann. Existence and Complexity of Approximate Equilibria in Weighted Congestion Games. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{christodoulou_et_al:LIPIcs.ICALP.2020.32, author = {Christodoulou, George and Gairing, Martin and Giannakopoulos, Yiannis and Po\c{c}as, Diogo and Waldmann, Clara}, title = {{Existence and Complexity of Approximate Equilibria in Weighted Congestion Games}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {32:1--32:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.32}, URN = {urn:nbn:de:0030-drops-124392}, doi = {10.4230/LIPIcs.ICALP.2020.32}, annote = {Keywords: Atomic congestion games, existence of equilibria, pure Nash equilibria, approximate equilibria, hardness of equilibria} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We give exponential lower bounds on the Price of Stability (PoS) of weighted congestion games with polynomial cost functions. In particular, for any positive integer d we construct rather simple games with cost functions of degree at most d which have a PoS of at least Omega(Phi_d)^{d+1}, where Phi_d ~ d/ln d is the unique positive root of equation x^{d+1}=(x+1)^d. This essentially closes the huge gap between Theta(d) and Phi_d^{d+1} and asymptotically matches the Price of Anarchy upper bound. We further show that the PoS remains exponential even for singleton games. More generally, we also provide a lower bound of Omega((1+1/alpha)^d/d) on the PoS of alpha-approximate Nash equilibria, even for singleton games. All our lower bounds extend to network congestion games, and hold for mixed and correlated equilibria as well.
On the positive side, we give a general upper bound on the PoS of alpha-approximate Nash equilibria, which is sensitive to the range W of the player weights and the approximation parameter alpha. We do this by explicitly constructing a novel approximate potential function, based on Faulhaber's formula, that generalizes Rosenthal's potential in a continuous, analytic way. From the general theorem, we deduce two interesting corollaries. First, we derive the existence of an approximate pure Nash equilibrium with PoS at most (d+3)/2; the equilibrium's approximation parameter ranges from Theta(1) to d+1 in a smooth way with respect to W. Secondly, we show that for unweighted congestion games, the PoS of alpha-approximate Nash equilibria is at most (d+1)/alpha.

George Christodoulou, Martin Gairing, Yiannis Giannakopoulos, and Paul G. Spirakis. The Price of Stability of Weighted Congestion Games. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 150:1-150:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{christodoulou_et_al:LIPIcs.ICALP.2018.150, author = {Christodoulou, George and Gairing, Martin and Giannakopoulos, Yiannis and Spirakis, Paul G.}, title = {{The Price of Stability of Weighted Congestion Games}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {150:1--150:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.150}, URN = {urn:nbn:de:0030-drops-91541}, doi = {10.4230/LIPIcs.ICALP.2018.150}, annote = {Keywords: Congestion games, price of stability, Nash equilibrium, approximate equilibrium, potential games} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

In this paper, we study contention resolution protocols from a game-theoretic perspective. We focus on acknowledgment-based protocols, where a user gets feedback from the channel only when she attempts transmission. In this case she will learn whether her transmission was successful or not. Users that do not transmit will not receive any feedback. We are interested in equilibrium protocols, where no player has an incentive to deviate.
The limited feedback makes the design of equilibrium protocols a hard task as best response policies usually have to be modeled as Partially Observable Markov Decision Processes, which are hard to analyze. Nevertheless, we show how to circumvent this for the case of two players and present an equilibrium protocol. For many players, we give impossibility results for a large class of acknowledgment-based protocols, namely age-based and backoff protocols with finite expected finishing time. Finally, we provide an age-based equilibrium protocol, which has infinite expected finishing time, but every player finishes in linear time with high probability.

George Christodoulou, Martin Gairing, Sotiris Nikoletseas, Christoforos Raptopoulos, and Paul Spirakis. Strategic Contention Resolution with Limited Feedback. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 30:1-30:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{christodoulou_et_al:LIPIcs.ESA.2016.30, author = {Christodoulou, George and Gairing, Martin and Nikoletseas, Sotiris and Raptopoulos, Christoforos and Spirakis, Paul}, title = {{Strategic Contention Resolution with Limited Feedback}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {30:1--30:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.30}, URN = {urn:nbn:de:0030-drops-63813}, doi = {10.4230/LIPIcs.ESA.2016.30}, annote = {Keywords: contention resolution, acknowledgment-based protocols, game theory} }

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