Maximin Shares with Lower Quotas

TL;DR

The paper advances the study of MMS guarantees under per-agent lower and upper quotas for indivisible goods and chores with heterogeneous additive valuations. It develops a polynomial-time algorithm that guarantees a $\frac{2n}{3n-1}$-MMS allocation for single-category goods (and a $\frac{3n-1}{2n}$-MMS for chores), and extends to a multi-category setting achieving $\frac{n}{2n-1}$-MMS (goods) or $\frac{2n-1}{n}$-MMS (chores). The core techniques combine reducing to ordered instances, valid reductions via carefully constructed bundles, and a multi-bag, invariant-preserving bag-filling framework that iteratively assigns items while maintaining quota feasibility. The results generalize previous upper-quota-based MMS approximations, provide tightness results for the single-category goods case, and discuss multi-category extensions, related lower-quota constraints in matching, and potential avenues for improvement and broader applicability. Overall, the work broadens the applicability of MMS-based fairness to quota-constrained allocations with practical implications for resource and personnel distribution problems.

Abstract

We study the fair division of indivisible items among $n$ agents with heterogeneous additive valuations, subject to lower and upper quotas on the number of items allocated to each agent. Such constraints are crucial in various applications, ranging from personnel assignments to computing resource distribution. This paper focuses on the fairness criterion known as maximin shares (MMS) and its approximations. Under arbitrary lower and upper quotas, we show that a $\left(\frac{2n}{3n-1}\right)$-MMS allocation of goods exists and can be computed in polynomial time, while we also present a polynomial-time algorithm for finding a $\left(\frac{3n-1}{2n}\right)$-MMS allocation of chores. Furthermore, we consider the generalized scenario where items are partitioned into multiple categories, each with its own lower and upper quotas. In this setting, our algorithm computes an $\left(\frac{n}{2n-1}\right)$-MMS allocation of goods or a $\left(\frac{2n-1}{n}\right)$-MMS allocation of chores in polynomial time. These results extend previous work on the cardinality constraints, i.e., the special case where only upper quotas are imposed.

Figures (1)

• Figure 1: An illustration of key steps of \ref{['alg:single:goods:main-start']} in \ref{['alg:single:goods']}, where each item $g_j\in M$ is denoted by its index $j$. \ref{['fig:single:goods:init']} At Lines \ref{['alg:single:goods:B_k-init-for-start']} to \ref{['alg:single:goods:B_k-init-for-end']}, the item set $M$ is split into the initial bags $B^{(|N|)}_1,B^{(|N|)}_2,\ldots,B^{(|N|)}_{|N|}$ in a greedy manner. \ref{['fig:single:goods:move']} At Line \ref{['alg:single:goods:move']}, an item is moved from the bag $B$ to $B^{(t-1)}_k$; this continues until either the size of these bags are completely exchanged, or the bag value $v_i(B)$ falls below a threshold, $\frac{3}{2}\alpha\hat{\mu}_i$, for every remaining agent $i\in N^{(t)}$. \ref{['fig:single:goods:swap']} At Line \ref{['alg:single:goods:swap']}, where $|B| = |B^{(t)}_k|$, an item from $B$ is exchanged with another from $B^{(t-1)}_k$; this continues until either all items in these bags, except $g_k$ and $g_t$, are completely exchanged from their initial state, or $v_i(B)$ falls below $\frac{3}{2}\alpha\hat{\mu}_i$ for every $i\in N^{(t)}$.

Theorems & Definitions (53)

• Definition 1
• Lemma 1: Corollary of hummel2022maximin
• Lemma 2
• Corollary 1
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 3
• Lemma 4