On Approximate MMS Allocations on Restricted Graph Classes

TL;DR

This paper investigates the existence of approximate maximin share allocations of indivisible goods with connectivity constraints across various restricted graph classes, expanding understanding of fair division under complex graph structures.

Contribution

It proves the existence of approximate fair allocations for several specific graph classes, advancing the theory of fair division with connectivity constraints.

Findings

Approximate MMS allocations exist for block graphs and cacti.

Such allocations are also possible for complete multipartite and split graphs.

The results extend the classes of graphs where fair division guarantees are known.

Abstract

We study the problem of fair division of a set of indivisible goods with connectivity constraints. Specifically, we assume that the goods are represented as vertices of a connected graph, and sets of goods allocated to the agents are connected subgraphs of this graph. We focus on the widely-studied maximin share criterion of fairness. It has been shown that an allocation satisfying this criterion may not exist even without connectivity constraints, i.e., if the graph of goods is complete. In view of this, it is natural to seek approximate allocations that guarantee each agent a connected bundle of goods with value at least a constant fraction of the maximin share value to the agent. It is known that for some classes of graphs, such as complete graphs, cycles, and $d$-claw-free graphs for any fixed $d$, such approximate allocations indeed exist. However, it is an open problem whether they exist for the class of all graphs. In this paper, we continue the systematic study of the existence of approximate allocations on restricted graph classes. In particular, we show that such allocations exist for several well-studied classes, including block graphs, cacti, complete multipartite graphs, and split graphs.