A Little Charity Guarantees Fair Connected Graph Partitioning

Abstract

Motivated by fair division applications, we study a fair connected graph partitioning problem, in which an undirected graph with m nodes must be divided between n agents such that each agent receives a connected subgraph and the partition is fair. We study approximate versions of two fairness criteria: \alpha-proportionality requires that each agent receive a subgraph with at least (1/\alpha)*m/n nodes, and \alpha-balancedness requires that the ratio between the sizes of the largest and smallest subgraphs be at most \alpha. Unfortunately, there exist simple examples in which no partition is reasonably proportional or balanced. To circumvent this, we introduce the idea of charity. We show that by "donating" just n-1 nodes, we can guarantee the existence of 2-proportional and almost 2-balanced partitions (and find them in polynomial time), and that this result is almost tight. More generally, we chart the tradeoff between the size of charity and the approximation of proportionality or balancedness we can guarantee.

On the Price of Fairness in the Connected Discrete Cake Cutting Problem

“…In a recent couple of years, motivated by real-world applications, constraints have been investigated in the company with fair division; we refer the readers to the survey by Suksompong [37]. One of the most frequently studied constraints is connectivity, where the items are assumed to be distributed on a graph [13,14,19] and each agent should receive a connected subgraph. Line structure as we considered in this work is an important special case which admits some interesting positive results.…”