Graphical Resource Allocation with Matching-Induced Utilities

Zheng Chen, Bo Li, Minming Li, Guochuan Zhang

Keywords: Fair Division, Graphical Resources, Matching, Maximin Share, Envy-freeness.

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 the 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. We show that for homogeneous agents, there is an EF1 allocation that ensures at least a constant fraction of the maximum possible social welfare. However, the social welfare guarantee of EF1 allocations degrades to $1/n$ for heterogeneous agents, where $n$ is the number of agents. Fortunately, for two special yet typical cases, namely binary-weight and two-agent, we are able to design polynomial-time algorithms ensuring a constant fractions of the maximum social welfare.