Fair Graphical Resource Allocation with Matching-Induced Utilities

TL;DR

This paper investigates fair allocation of graph-based resources with utilities based on maximum matchings, providing algorithms for certain cases and analyzing the trade-offs between fairness and social welfare.

Contribution

It introduces polynomial-time algorithms for fair allocation under MMS and EF1 criteria in specific scenarios, highlighting the limitations and trade-offs involved.

Findings

Constant-approximation algorithms for homogeneous agents

EF1 allocations have social welfare at most 1/n of optimal in general

Efficient algorithms for binary-weight, two-agent, and homogeneous-agent cases

Abstract

Motivated by real-world applications, we study the fair allocation of graphical resources, where the resources are the vertices in a graph. Upon receiving a set of resources, an agent's utility equals the weight of a maximum matching in the induced subgraph. We care about maximin share (MMS) fairness and envy-freeness up to one item (EF1). Regarding MMS fairness, the problem does not admit a finite approximation ratio for heterogeneous agents. For homogeneous agents, we design constant-approximation polynomial-time algorithms, and also note that significant amount of social welfare is sacrificed inevitably in order to ensure (approximate) MMS fairness. We then consider EF1 allocations whose existence is guaranteed. However, the social welfare guarantee of EF1 allocations cannot be better than $1/n$ for the general case, where $n$ is the number of agents.Fortunately, for three special cases, binary-weight, two-agents and homogeneous-agents, we are able to design polynomial-time algorithms that also ensure a constant fractions of the maximum social welfare.