Fair allocation of conflicting items

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 $$1/\Delta $$ 1 / Δ -approximate MMS allocation for n agents may be found in polynomial time when $$n>\Delta >2$$ n > Δ > 2 , for any conflict graph with maximum degree $$\Delta$$ Δ , and that, if $$n > \Delta $$ n > Δ , a 1/3-approximate MMS allocation always exists.

Weighted Envy-freeness in Indivisible Item Allocation

We introduce and analyze new envy-based fairness concepts for agents with weights that quantify their entitlements in the allocation of indivisible items. We propose two variants of weighted envy-freeness up to one item (WEF1): strong , where envy can be eliminated by removing an item from the envied agent’s bundle, and weak , where envy can be eliminated either by removing an item (as in the strong version) or by replicating an item from the envied agent’s bundle in the envying agent’s bundle. We show that for additive valuations, an allocation that is both Pareto optimal and strongly WEF1 always exists and can be computed in pseudo-polynomial time; moreover, an allocation that maximizes the weighted Nash social welfare may not be strongly WEF1, but it always satisfies the weak version of the property. Moreover, we establish that a generalization of the round-robin picking sequence algorithm produces in polynomial time a strongly WEF1 allocation for an arbitrary number of agents; for two agents, we can efficiently achieve both strong WEF1 and Pareto optimality by adapting the adjusted winner procedure. Our work highlights several aspects in which weighted fair division is richer and more challenging than its unweighted counterpart.

Approximating nash social welfare under rado valuations

The Nash social welfare problem asks for an allocation of indivisible items to agents in order to maximize the geometric mean of agents' valuations. We give an overview of the constant-factor approximation algorithm for the problem when agents have Rado valuations [Garg et al. 2021]. Rado valuations are a common generalization of the assignment (OXS) valuations and weighted matroid rank functions. Our approach also gives the first constant-factor approximation algorithm for the asymmetric Nash social welfare problem under the same valuations, provided that the maximum ratio between the weights is bounded by a constant.

Almost Group Envy-free Allocation of Indivisible Goods and Chores

We consider a multi-agent resource allocation setting in which an agent's utility may decrease or increase when an item is allocated. We take the group envy-freeness concept that is well-established in the literature and present stronger and relaxed versions that are especially suitable for the allocation of indivisible items. Of particular interest is a concept called group envy-freeness up to one item (GEF1). We then present a clear taxonomy of the fairness concepts. We study which fairness concepts guarantee the existence of a fair allocation under which preference domain. For two natural classes of additive utilities, we design polynomial-time algorithms to compute a GEF1 allocation. We also prove that checking whether a given allocation satisfies GEF1 is coNP-complete when there are either only goods, only chores or both.

Fairness Towards Groups of Agents in the Allocation of Indivisible Items

In this paper, we study the problem of matching a set of items to a set of agents partitioned into types so as to balance fairness towards the types against overall utility/efficiency. We extend multiple desirable properties of indivisible goods allocation to our model and investigate the possibility and hardness of achieving combinations of these properties, e.g. we prove that maximizing utilitarian social welfare under constraints of typewise envy-freeness up to one item (TEF1) is computationally intractable. We also define a new concept of waste for this setting, show experimentally that augmenting an existing algorithm with a marginal utility maximization heuristic can produce a TEF1 solution with reduced waste, and also provide a polynomial-time algorithm for computing a non-wasteful TEF1 allocation for binary agent-item utilities.