Almost Envy-Free Allocations with Connected Bundles

Authors Vittorio Bilò, Ioannis Caragiannis, Michele Flammini, Ayumi Igarashi, Gianpiero Monaco, Dominik Peters, Cosimo Vinci, William S. Zwicker

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Author Details

Vittorio Bilò
  • University of Salento, Lecce, Italy
Ioannis Caragiannis
  • University of Patras, Rion-Patras, Greece
Michele Flammini
  • Gran Sasso Science Institute and University of L'Aquila, L'Aquila, Italy
Ayumi Igarashi
  • Kyushu University, Fukuoka, Japan
Gianpiero Monaco
  • University of L'Aquila, L'Aquila, Italy
Dominik Peters
  • University of Oxford, Oxford, U.K.
Cosimo Vinci
  • University of L'Aquila, L'Aquila, Italy
William S. Zwicker
  • Union College, Schenectady, USA

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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)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithmic game theory
  • Mathematics of computing → Combinatoric problems
  • Mathematics of computing → Graph theory
  • Envy-free Division
  • Cake-cutting
  • Resource Allocation
  • Algorithmic Game Theory


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