Ayumi Igarashi

Ayumi Igarashi, Frédéric Meunier

We study the problem of fairly allocating indivisible goods and chores under category constraints. Specifically, there are $n$ agents and $m$ indivisible items which are partitioned into categories with associated capacities. An allocation is considered feasible if each bundle satisfies the capacity constraints of its respective categories. For the case of two agents, Shoshan et al. (2023) recently developed a polynomial-time algorithm to find a Pareto-optimal allocation satisfying a relaxed version of envy-freeness, called EF$[1,1]$. Extending such guarantees beyond two agents has remained open. We make progress toward this question by proving that for any number $n$ of agents, there always exists a Pareto-optimal allocation in which each agent can be made envy-free by reallocating at most $\min \{k+1,n\}(n-1)$ items. Moreover, when the number of agents is constant, we give a polynomial-time algorithm to compute such an allocation. Our results rely on a new application of the Knaster-Kuratowski-Mazurkiewicz (KKM) lemma to a simplex of agent weights, which may be of independent interest for fair division problems involving indivisible items.

Nawal Benabbou, Mithun Chakraborty, Ayumi Igarashi et al.

In this paper, we present new results on the fair and efficient allocation of indivisible goods to agents whose preferences correspond to {\em matroid rank functions}. This is a versatile valuation class with several desirable properties (such as monotonicity and submodularity), which naturally lends itself to a number of real-world domains. We use these properties to our advantage; first, we show that when agent valuations are matroid rank functions, a socially optimal (i.e. utilitarian social welfare-maximizing) allocation that achieves envy-freeness up to one item (EF1) exists and is computationally tractable. We also prove that the Nash welfare-maximizing and the leximin allocations both exhibit this fairness/efficiency combination, by showing that they can be achieved by minimizing any symmetric strictly convex function over utilitarian optimal outcomes. To the best of our knowledge, this is the first valuation function class not subsumed by additive valuations for which it has been established that an allocation maximizing Nash welfare is EF1. Moreover, for a subclass of these valuation functions based on maximum (unweighted) bipartite matching, we show that a leximin allocation can be computed in polynomial time. Additionally, we explore possible extensions of our results to fairness criteria other than EF1 as well as to generalizations of the above valuation classes.

Mithun Chakraborty, Ayumi Igarashi, Warut Suksompong et al.

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 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.