Fair and efficient allocation of indivisible items under category constraints

TL;DR

The paper addresses fair allocation of indivisible items under category constraints with capacities across categories. It extends known two-agent results to arbitrary $n$, proving that a Pareto-optimal allocation exists from which each agent can be made envy-free by reallocating at most $n(n-1)$ items, and provides a polynomial-time algorithm when $n$ is constant. The method combines weighted utilitarian optimization over the simplex, a perturbation to bound the dimension of optimal faces, and a Knaster–Kuratowski–Mazurkiewicz argument to guarantee envy-freeness across all agents. A key consequence is a shared core of at least $m-n(n-1)$ items fixed across the envy-free reallocation plans, preserving Pareto-optimality. The results contribute to fair division under constraints and suggest directions for tightening the envy-bound and improving non-constant-$n$ algorithms.

Abstract

We study the problem of fairly allocating indivisible items 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$[1,1]$. In this paper, we extend the result of Shoshan et al. to $n$ agents, proving the existence of a Pareto-optimal allocation where each agent can be made envy-free by reallocating at most ${n(n-1)}$ items. Furthermore, we present a polynomial-time algorithm to compute such an allocation when the number $n$ of agents is constant.

Theorems & Definitions (25)

• Theorem 1
• Theorem 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Proposition 6
• proof