Abstract

We study the allocation of indivisible goods that form an undirected graph and quantify the loss of fairness when we impose a constraint that each agent must receive a connected subgraph. Our focus is on well-studied fairness notions including envy-freeness and maximin share fairness. We introduce the price of connectivity to capture the largest multiplicative gap between the graph-specific and the unconstrained maximin share and derive bounds on this quantity which are tight for large classes of graphs in the case of two agents and for paths and stars in the general case. For instance, with two agents we show that for biconnected graphs it is possible to obtain at least 3/4 of the maximin share with connected allocations, while for the remaining graphs the guarantee is at most 1/2. In addition, we determine the optimal relaxation of envy-freeness that can be obtained with each graph for two agents and characterize the set of trees and complete bipartite graphs that always admit an allocation satisfying envy-freeness up to one good (EF1) for three agents. Our work demonstrates several applications of graph-theoretic tools and concepts to fair division problems.

Keywords

  1. fair division
  2. envy-freeness
  3. connectivity
  4. graph partition
  5. graph theory

MSC codes

  1. 91B32
  2. 05C40
  3. 68R10

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Information & Authors

Information

Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 1156 - 1186
ISSN (online): 1095-7146

History

Submitted: 27 December 2020
Accepted: 24 December 2021
Published online: 19 May 2022

Keywords

  1. fair division
  2. envy-freeness
  3. connectivity
  4. graph partition
  5. graph theory

MSC codes

  1. 91B32
  2. 05C40
  3. 68R10

Authors

Affiliations

Funding Information

H2020 European Research Council https://doi.org/10.13039/100010663 : 639945
Japan Science and Technology Agency https://doi.org/10.13039/501100002241
Japan Society for the Promotion of Science https://doi.org/10.13039/501100001691 : 18J00997
Ministry of Education - Singapore https://doi.org/10.13039/501100001459 : RG23/20
National University of Singapore https://doi.org/10.13039/501100001352

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