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DOI: 10.4230/LIPIcs.ITCS.2019.14
URN: urn:nbn:de:0030-drops-101078
URL: http://drops.dagstuhl.de/opus/volltexte/2018/10107/
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BilÚ, Vittorio ; Caragiannis, Ioannis ; Flammini, Michele ; Igarashi, Ayumi ; Monaco, Gianpiero ; Peters, Dominik ; Vinci, Cosimo ; Zwicker, William S.

Almost Envy-Free Allocations with Connected Bundles

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LIPIcs-ITCS-2019-14.pdf (0.5 MB)


Abstract

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.

BibTeX - Entry

@InProceedings{bil_et_al:LIPIcs:2018:10107,
  author =	{Vittorio Bil{\`o} and Ioannis Caragiannis and Michele Flammini and Ayumi Igarashi and Gianpiero Monaco and Dominik Peters and Cosimo Vinci and William S. Zwicker},
  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 =	{2018},
  volume =	{124},
  editor =	{Avrim Blum},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/10107},
  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}
}

Keywords: Envy-free Division, Cake-cutting, Resource Allocation, Algorithmic Game Theory
Seminar: 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)
Issue Date: 2018
Date of publication: 21.12.2018


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