# A Characterization of Undirected Graphs Admitting Optimal Cost Shares

## TL;DR

This paper characterizes undirected graphs that allow optimal Steiner forests to be implemented as Nash equilibria through separable cost sharing protocols, based on forbidden subgraph conditions.

## Contribution

It provides a complete characterization of efficient undirected graphs for two-player network design games using forbidden subgraph criteria.

## Key findings

- Efficient graphs are characterized by the absence of specific forbidden subgraphs.
- Generalized series-parallel, fan, wheel, and small cycle graphs are efficient.
- The characterization applies to implementing low-cost Steiner forests as Nash equilibria.

## Abstract

In a seminal paper, Chen, Roughgarden and Valiant studied cost sharing protocols for network design with the objective to implement a low-cost Steiner forest as a Nash equilibrium of an induced cost-sharing game. One of the most intriguing open problems to date is to understand the power of budget-balanced and separable cost sharing protocols in order to induce low-cost Steiner forests. In this work, we focus on undirected networks and analyze topological properties of the underlying graph so that an optimal Steiner forest can be implemented as a Nash equilibrium (by some separable cost sharing protocol) independent of the edge costs. We term a graph efficient if the above stated property holds. As our main result, we give a complete characterization of efficient undirected graphs for two-player network design games: an undirected graph is efficient if and only if it does not contain (at least) one out of few forbidden subgraphs. Our characterization implies that several graph classes are efficient: generalized series-parallel graphs, fan and wheel graphs and graphs with small cycles.