Settling the Computational Complexity of Max-Min Allocation with Ternary Valuations

2026-10-04

AI Summary

This study investigates the computational complexity of computing maximin share allocations under ternary valuations. For additive and submodular valuation functions, we systematically characterize the computational boundaries across various settings by leveraging maximum-weight perfect matching reductions, combinatorial optimization analysis, and query complexity theory. Our primary contributions are threefold. First, we propose efficient polynomial-time algorithms for specific cases and establish corresponding approximation guarantees. Second, we resolve a related open conjecture by proving exact inapproximability results with a unit gap. Finally, we precisely determine the computational complexity and approximation ratio limits for each problem variant. Collectively, this work provides a comprehensive theoretical framework for fair division problems under restricted valuation classes.

Abstract

We study the problem of computing an allocation of indivisible items that maximizes egalitarian welfare, i.e., the utility of the worst-off agent, when agents'item values or marginal values belong to a small set. For additive valuations with values in $\{p,q\}$, where $q>p>0$ and $\gcd(p,q)=1$, we give a polynomial-time algorithm when $p=2$ and prove constant-gap hardness when $p\geq3$, already with exactly three high-valued goods per agent. We also give an $\sqrt{3/2}$-approximation for common positive bi-valued additive valuations. For mixed additive valuations in $\{-p,0,c\}$, where $p\in\{1,2\}$ and $c$ is a positive integer, a reduction to maximum-weight perfect matching resolves the conjectured tractability of $\{-2,0,c\}$-valuations. For submodular valuations with marginals in $\{-2,0,c\}$, where $c$ is odd, we establish an exact unit-gap hardness result and exponential value-query lower bounds, even when all but one agent are additive. Finally, for $\{-1,0,1\}$-submodular valuations, we prove that no finite multiplicative approximation exists unless $\p=\np$. Together, our results resolve open questions and provide a complete picture of the computational complexity of max-min allocation with ternary valuations.

Problem

• Max-Min Allocation
• Computational Complexity
• Egalitarian Welfare
• Ternary Valuations
• Indivisible Items

Innovation

• Max-Min Allocation
• Computational Complexity
• Ternary Valuations
• Submodular Valuations
• Approximation Algorithms