Abstract
We study fair allocation of indivisible items, where the items are furnished with a set of conflicts, and agents are not permitted to receive conflicting items. This kind of constraint captures, for example, participating in events that overlap in time, or taking on roles in the presence of conflicting interests. We demonstrate, both theoretically and experimentally, that fairness characterizations such as EF1, MMS and MNW still are applicable and useful under item conflicts. Among other existence, non-existence and computability results, we show that a 1/Delta -approximate MMS allocation for n agents may be found in polynomial time when n>Delta >2, for any conflict graph with maximum degree Delta, and that, if n > Delta , a 1/3-approximate MMS allocation always exists.
Highlights
We are interested in the problem of allocating a set of indivisible items among a set of agents with additive valuations, and beyond finding an efficient solution, where the total utility is high, we want the allocation to be fair, in some sense—a problem that has been studied extensively in the last couple of decades [8]
We present a series of results on maximin share (MMS) allocation with item conflicts (Sect. 3.3), with the main results being that, if the maximum degree Δ of the conflict graph is lower than the number of agents,1 (i) there exists an -approximate MMS allocation, with α > 1∕3 (Theorem 1); and (ii) an -approximate MMS allocation may be found in polynomial time, with α > 1∕Δ when Δ > 2 (Theorem 2)
We examine the behavior of various fairness measures in practice, that is, empirically studying how random allocation, EF1, MMS, proportionality and maximum Nash welfare (MNW) are affected by the introduction of item conflicts, on 18,629 randomly generated graphs (Sect. 4), with the main conclusion being that fairness is largely unharmed, with random allocation improving as an MMS approximation, all instances exhibiting EF1 and MMS, and MNW producing EF1 in almost all cases, with a tight approximation of MMS
Summary
We are interested in the problem of allocating a set of indivisible items among a set of agents with additive valuations, and beyond finding an efficient solution, where the total utility is high, we want the allocation to be fair, in some sense—a problem that has been studied extensively in the last couple of decades [8]. If the items are structured as a matroid, one may require that the set of all allocated items form a basis [18], or that each bundle be an independent set of the matroid [4]. We look at yet another form of constraint, where items may be in conflict with each other, and an agent may receive at most one of any two conflicting items. This is a situation that may occur in many realistic allocation scenarios. Such conflicts arise naturally in scheduling problems where the items represent participation in
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