A Little Charity Guarantees Fair Connected Graph Partitioning

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