On the Price of Stability for Undirected Network Design

network design problem.
We provide new bounds for the price of stability for network design with fair cost allocation for undirected graphs. We consider the most general case, for which the best known upper bound is the Harmonic number $H_n$, where $n$ is the number of agents, and the best previously known lower bound is $12/7\approx1.778$.

We present a nontrivial lower bound of $42/23\approx1.8261$.
Furthermore, we show that for two players, the price of stability is exactly $4/3$, while for three players it is at least $74/48\approx 1.542$ and at most $1.65$.
These are the first improvements on the bound of $H_n$ for general networks. In particular, this demonstrates a separation between the price of stability on undirected graphs and that on directed graphs,
where $H_n$ is tight.
Previously, such a gap was only known for the cases where all players
have a shared source, and for weighted players.