On the Hardness of Fair Allocation under Ternary Valuations

TL;DR

This paper proves that finding fair allocations based on Nash and egalitarian welfare with agents having ternary valuations is computationally hard, except in specific cases, advancing understanding of the problem's complexity.

Contribution

It establishes APX-hardness results for maximizing Nash and egalitarian welfare under ternary valuations, clarifying the computational difficulty of these fair allocation problems.

Findings

Maximizing Nash welfare is APX-hard for distinct non-negative values.

Maximizing egalitarian welfare is APX-hard except when one value is zero.

Resolves open questions on complexity of fair allocations under bivalued and ternary valuations.

Abstract

We study the problem of fair allocation of indivisible items when agents have ternary additive valuations -- each agent values each item at some fixed integer values $a$, $b$, or $c$ that are common to all agents. The notions of fairness we consider are max Nash welfare (MNW), when $a$, $b$, and $c$ are non-negative, and max egalitarian welfare (MEW). We show that for any distinct non-negative $a$, $b$, and $c$, maximizing Nash welfare is APX-hard -- i.e., the problem does not admit a PTAS unless P = NP. We also show that for any distinct $a$, $b$, and $c$, maximizing egalitarian welfare is APX-hard except for a few cases when $b = 0$ that admit efficient algorithms. These results make significant progress towards completely characterizing the complexity of computing exact MNW allocations and MEW allocations. En route, we resolve open questions left by prior work regarding the complexity of computing MNW allocations under bivalued valuations, and MEW allocations under ternary mixed manna.