Almost Envy-Free Allocations with Connected Bundles
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by
Vittorio Bilò, Ioannis Caragiannis, Michele Flammini, Ayumi
Igarashi, Gianpiero Monaco, Dominik Peters, Cosimo Vinci, William S. Zwicker
2018
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 be check in
linear time whether a graph belongs to this class, and if so return an EF1
allocation.
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