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. If the graph is a path and the utility functions are monotonic over bundles, we show the existence of EF1 allocations for at most four agents, and the existence of EF2 allocations for any number of agents; our proofs involve discrete analogues of the Stromquist's moving-knife protocol and the Su–Simmons argument based on Sperner's lemma. For identical utilities, we provide a polynomial-time algorithm that computes an EF1 allocation for any number of agents. For the case of two agents, we characterize the class of graphs that guarantee the existence of EF1 allocations as those whose biconnected components are arranged in a path; this property can be checked in linear time.
Highlights
A famous literature considers the problem of cake-cutting [10, 25, 24]
When items are arranged on a path, we prove that connected EF1 allocations exist when there are two, three, or four agents
We have studied the existence of EF1 allocations under connectivity constraints imposed by an undirected graph
Summary
A famous literature considers the problem of cake-cutting [10, 25, 24]. There, a divisible heterogeneous resource (a cake, usually formalized as the interval [0, 1]) needs to be divided among n agents. In simultaneous and independent work, Oh et al [23] designed protocols to find EF1 allocations in the setting without connectivity constraints, aiming for low query complexity They found that adapting cake-cutting protocols to the setting of indivisible items arranged on a path is an especially potent way to achieve low query complexity. This led them to study a discrete version of the cut-and-choose protocol which achieves connected EF1 allocations for two agents, and they found an alternative proof that an EF1 allocation on a path always exists with identical valuations. For the case of three or more agents, it is a challenging open problem to characterize the class of graphs guaranteeing EF1 (or even to find an infinite class of non-traceable graphs that guarantees EF1)
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