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Track C: Foundations of Networks and Multi-Agent Systems: Models, Algorithms and Information Management

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We consider the problem of the existence of natural improvement dynamics leading to approximate pure Nash equilibria, with a reasonable small approximation, and the problem of bounding the efficiency of such equilibria in the fundamental framework of weighted congestion game with polynomial latencies of degree at most d >= 1. In this work, by exploiting a simple technique, we firstly show that the game always admits a d-approximate potential function. This implies that every sequence of d-approximate improvement moves by the players always leads the game to a d-approximate pure Nash equilibrium. As a corollary, we also obtain that, under mild assumptions on the structure of the players' strategies, the game always admits a constant approximate potential function. Secondly, by using a simple potential function argument, we are able to show that in the game there always exists a (d+delta)-approximate pure Nash equilibrium, with delta in [0,1], whose cost is 2/(1+delta) times the cost of an optimal state.

Ioannis Caragiannis and Angelo Fanelli. On Approximate Pure Nash Equilibria in Weighted Congestion Games with Polynomial Latencies. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 133:1-133:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{caragiannis_et_al:LIPIcs.ICALP.2019.133, author = {Caragiannis, Ioannis and Fanelli, Angelo}, title = {{On Approximate Pure Nash Equilibria in Weighted Congestion Games with Polynomial Latencies}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {133:1--133:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.133}, URN = {urn:nbn:de:0030-drops-107095}, doi = {10.4230/LIPIcs.ICALP.2019.133}, annote = {Keywords: Congestion games, approximate pure Nash equilibrium, potential functions, approximate price of stability} }

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**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

We study the existence of allocations of indivisible goods that are envy-free up to one good (EF1), under the additional constraint that each bundle needs to be connected in an underlying item graph G. When the items are arranged in a path, we show that EF1 allocations are guaranteed to exist for arbitrary monotonic utility functions over bundles, provided that either there are at most four agents, or there are any number of agents but they all have identical utility functions. Our existence proofs are based on classical arguments from the divisible cake-cutting setting, and involve discrete analogues of cut-and-choose, of Stromquist's moving-knife protocol, and of the Su-Simmons argument based on Sperner's lemma. Sperner's lemma can also be used to show that on a path, an EF2 allocation exists for any number of agents. Except for the results using Sperner's lemma, all of our procedures can be implemented by efficient algorithms. Our positive results for paths imply the existence of connected EF1 or EF2 allocations whenever G is traceable, i.e., contains a Hamiltonian path. For the case of two agents, we completely characterize the class of graphs G that guarantee the existence of EF1 allocations as the class of graphs whose biconnected components are arranged in a path. This class is strictly larger than the class of traceable graphs; one can check in linear time whether a graph belongs to this class, and if so return an EF1 allocation.

Vittorio Bilò, Ioannis Caragiannis, Michele Flammini, Ayumi Igarashi, Gianpiero Monaco, Dominik Peters, Cosimo Vinci, and William S. Zwicker. Almost Envy-Free Allocations with Connected Bundles. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 14:1-14:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bilo_et_al:LIPIcs.ITCS.2019.14, author = {Bil\`{o}, Vittorio and Caragiannis, Ioannis and Flammini, Michele and Igarashi, Ayumi and Monaco, Gianpiero and Peters, Dominik and Vinci, Cosimo and Zwicker, William S.}, title = {{Almost Envy-Free Allocations with Connected Bundles}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {14:1--14:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.14}, URN = {urn:nbn:de:0030-drops-101078}, doi = {10.4230/LIPIcs.ITCS.2019.14}, annote = {Keywords: Envy-free Division, Cake-cutting, Resource Allocation, Algorithmic Game Theory} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We revisit fundamental problems in undirected and directed graphs, such as the problems of computing spanning trees, shortest paths, steiner trees, and spanning arborescences of minimum cost. We assume that there are d different cost functions associated with the edges of the input graph and seek for solutions to the resulting multidimensional graph problems so that the p-norm of the different costs of the solution is minimized. We present combinatorial algorithms that achieve very good approximations for this objective. The main advantage of our algorithms is their simplicity: they are as simple as classical combinatorial graph algorithms of Dijkstra and Kruskal, or the greedy algorithm for matroids.

Vittorio Bilò, Ioannis Caragiannis, Angelo Fanelli, Michele Flammini, and Gianpiero Monaco. Simple Greedy Algorithms for Fundamental Multidimensional Graph Problems. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 125:1-125:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bilo_et_al:LIPIcs.ICALP.2017.125, author = {Bil\`{o}, Vittorio and Caragiannis, Ioannis and Fanelli, Angelo and Flammini, Michele and Monaco, Gianpiero}, title = {{Simple Greedy Algorithms for Fundamental Multidimensional Graph Problems}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {125:1--125:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.125}, URN = {urn:nbn:de:0030-drops-74669}, doi = {10.4230/LIPIcs.ICALP.2017.125}, annote = {Keywords: multidimensional graph problems, matroids, shortest paths, Steiner trees, arborescences} }

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