(Almost) Envy-Free, Proportional and Efficient Allocations of an Indivisible Mixed Manna

TL;DR

This paper develops polynomial-time algorithms for fair and efficient allocations of indivisible mixed items among agents, addressing complex fairness notions like EFX and PropMX in various instance types.

Contribution

It introduces new algorithms for achieving strong fairness and efficiency guarantees in mixed manna allocation problems, including separable and restricted mixed goods cases.

Findings

Polynomial-time PropMX$_0$ allocation for separable instances.

PropMX and EFX+PropMX allocations for restricted mixed goods.

Strengthened guarantees for binary mixed goods with EFX$_0$ and PropMX$_0$.

Abstract

We study the problem of finding fair and efficient allocations of a set of indivisible items to a set of agents, where each item may be a good (positively valued) for some agents and a bad (negatively valued) for others, i.e., a mixed manna. As fairness notions, we consider arguably the strongest possible relaxations of envy-freeness and proportionality, namely envy-free up to any item (EFX and EFX$_0$), and proportional up to the maximin good or any bad (PropMX and PropMX$_0$). Our efficiency notion is Pareto-optimality (PO). We study two types of instances: (i) Separable, where the item set can be partitioned into goods and bads, and (ii) Restricted mixed goods (RMG), where for each item $j$, every agent has either a non-positive value for $j$, or values $j$ at the same $v_j>0$. We obtain polynomial-time algorithms for the following: (i) Separable instances: PropMX$_0$ allocation. (ii) RMG instances: Let pure bads be the set of items that everyone values negatively. - PropMX allocation for general pure bads. - EFX+PropMX allocation for identically-ordered pure bads. - EFX+PropMX+PO allocation for identical pure bads. Finally, if the RMG instances are further restricted to binary mixed goods where all the $v_j$'s are the same, we strengthen the results to guarantee EFX$_0$ and PropMX$_0$ respectively.