Welfare Loss in Connected Resource Allocation

TL;DR

This paper analyzes the worst-case welfare loss in connected resource allocation problems, introducing the concept of price of connectivity and providing bounds for various graph classes and agent scenarios.

Contribution

It introduces the egalitarian and utilitarian price of connectivity, offering tight bounds for different graph classes and agent counts, advancing understanding of welfare loss due to connectivity constraints.

Findings

Tight bounds on price of connectivity for specific graph classes.

Analysis for two agents and arbitrary agents.

Insights into welfare loss due to connectivity constraints.

Abstract

We study the allocation of indivisible items that form an undirected graph and investigate the worst-case welfare loss when requiring that each agent must receive a connected subgraph. Our focus is on both egalitarian and utilitarian welfare. Specifically, we introduce the concept of egalitarian (resp., utilitarian) price of connectivity, which captures the worst-case ratio between the optimal egalitarian (resp., utilitarian) welfare among all allocations and that among connected allocations. We provide tight or asymptotically tight bounds on the price of connectivity for several large classes of graphs in the case of two agents -- including graphs with vertex connectivity $1$ or $2$ and complete bipartite graphs -- as well as for paths, stars, and cycles in the general case where the number of agents can be arbitrary.