On Almost Fair and Equitable Allocations of Indivisible Items for Non-monotone Valuations

In this work, we revisit well-studied problems of fair allocation of indivisible items among agents with general, non-monotone valuations. We explore the existence and efficient computation of allocations that satisfy either fairness or equity constraints. The fairness notions we consider ensure that each agent values her bundle at least as much as others', allowing for (any or some) item removal, while the equity guarantees roughly equal valuations among agents, with similar adjustments. For objective valuations where items are classified as either goods or chores, we present a pseudo-polynomial local-search algorithm computing an ``equitable-up-to-any-good-or-any-chore'' (EQX*) allocation, a weaker version of an ``equitable-up-to-any-item" (EQX) allocation. Additionally, we provide a polynomial-time greedy algorithm that computes an ``equitable-up-to-one-item" (EQ1) allocation, and a similar algorithm returning an EQX* allocation when the valuations are also additive. As a key technical contribution of this work, by leveraging fixed-point theorems (such as Sperner's Lemma and its variants), we establish the existence of ``equitable-up-to-one-good-and-one-chore'' (EQ1*) and ``envy-free-up-to-one-good-and-one-chore'' (EF1*) allocations for non-negative (and possibly non-objective and non-monotone) valuations. This holds even when items are arranged in a path and bundles must form connected sub-paths. Additionally, we present a polynomial-time dynamic-programming algorithm that computes an EQ1* allocation. Finally, we extend the EF1* and EQ1* results to non-positive valuations using a novel multi-coloring variant of Sperner's lemma, a combinatorial result of independent interest. For monotone non-increasing valuations and path-connected bundles, this implies the existence of EF1 and EQ1 allocations, with EQ1 allocations being efficiently computable.