Guaranteeing MMS for All but One Agent When Allocating Indivisible Chores

TL;DR

This work proposes a notion called $\alpha$-approximate all-but-one maximin share ($\alpha$-AMMS) which is a stronger version of $\alpha$-approximate MMS.

Abstract

We study the problem of allocating $m$ indivisible chores to $n$ agents with additive cost functions under the fairness notion of maximin share (MMS). In this work, we propose a notion called $α$-approximate all-but-one maximin share ($α$-AMMS) which is a stronger version of $α$-approximate MMS. An allocation is called $α$-AMMS if $n-1$ agents are guaranteed their MMS values and the remaining agent is guaranteed $α$-approximation of her MMS value. We show that there exist $α$-AMMS allocations, with $α= 9/8$ for three agents; $α= 4/3$ for four agents; and $α= (n+1)^2/4n$ for $n\geq 5$ agents.

Figures (7)

• Figure 1: An example of an MMS-feasibility graph with $n=4$ and there exists a perfect matching between agent $\{4\}$ and bundles $\{P_3,P_4\}$.
• Figure 2: MMS-feasibility graph with $|S^*| = 2, |L(S^*)| = 1$.
• Figure 3: MMS feasibility graphs with $|S^*| = 3, |L(S^*)| \leq 2$.
• Figure 4: Each atomic bundle is an intersection of a bundle in $B$ (an MMS partition of agent $1$) and a bundle in $P$ (an MMS partition of agent $4$). We remark that $P_1$ is the bundle that agent $1$ likes while she dislikes all other bundles $P_2, P_3, P_4$.
• Figure 5: Illustrations for subgraph $G'$ and $S$ with $L(S)$.

Theorems & Definitions (41)

• Definition 2.1: Proportionality
• Definition 2.2
• Definition 2.3: $\alpha$-MMS
• Definition 2.4: $\alpha$-AMMS
• Definition 2.5: MMS-feasibility Graph
• proof
• Definition 2.8: Reduced Instance
• Lemma 3.1
• proof
• Theorem 3.2