Maximin Shares in Hereditary Set Systems

TL;DR

The paper advances fair division under maximin share (MMS) by studying hereditary set system valuations, introducing a lone-divider framework that yields a $1/2$-approximate MMS allocation (indeed $n/(2n-1)$) and establishing a nonexistence result for $(2/3+\epsilon)$-approximate MMS allocations when $n\ge 2$. It provides a constructive, though MMS-partition-dependent, existence proof and a polynomial-time $2/5$-approximation algorithm using valuation oracles, with extensions to entitled valuations and various constrained allocation settings. The work also demonstrates improvements for constrained problems such as budget constraints, conflicting items, and interval scheduling, linking MMS guarantees to broader fair-division contexts. Open questions remain regarding the gap between existence and approximation ratios and the extension to more general valuation models, including asymmetries and parametric improvements to $\alpha$-approximation schemes.

Abstract

We consider the problem of fairly allocating a set of indivisible items under the criteria of the maximin share guarantee. Specifically, we study approximation of maximin share allocations under hereditary set system valuations, in which each valuation function is based on the independent sets of an underlying hereditary set systems. Using a lone divider approach, we show the existence of $1/2$-approximate MMS allocations, improving on the $11/30$ guarantee of Li and Vetta. Moreover, we prove that ($2/3 + ε$)-approximate MMS allocations do not always exist in this model for every $ε> 0$, an improvement from the recent $3/4 + ε$ result of Li and Deng. Our existence proof is constructive, but does not directly yield a polynomial-time approximation algorithm. However, we show that a $2/5$-approximate MMS allocation can be found in polynomial time, given valuation oracles. Finally, we show that our existence and approximation results transfer to a variety of problems within constrained fair allocation, improving on existing results in some of these settings.

Figures (1)

• Figure 1: An instance with no ($2/3 + \epsilon$)-approximate MMS allocation for any $\epsilon > 0$. The independent sets of the hereditary set system are given by the triples marked by the filled and outlined gray areas, and all their subsets. One agent assigns a value of 1 to each item in the three upper rows. The remaining $n - 1$ agents assign a value of 1 to each item in the three lower rows.

Theorems & Definitions (36)

• Lemma 2.1
• proof
• Lemma 2.2: Corollary 1.1 in Aigner-Horev:22
• Theorem 2.3: Theorem 4.1 in Aigner-Horev:22
• Theorem 2.4: Theorem 1.2 in Aigner-Horev:22
• Theorem 3.1
• proof : Proof of \ref{['thr:existence']}
• Corollary 3.2
• proof
• Theorem 3.3