The Stackelberg Minimum Spanning Tree Game

TL;DR

This paper studies a network pricing game where a leader sets prices on blue edges to maximize revenue from a minimum spanning tree, analyzing its complexity, approximability, and providing an integer linear programming formulation.

Contribution

It proves APX-hardness for the problem with limited red costs, offers an approximation algorithm with a proven ratio, and presents an ILP formulation with matching integrality gap.

Findings

Problem is APX-hard with two red costs.

Provides an approximation algorithm with ratio at most min{k, 1+ln b, 1+ln W}.

ILP formulation's integrality gap matches the approximation guarantee.

Abstract

We consider a one-round two-player network pricing game, the Stackelberg Minimum Spanning Tree game or StackMST. The game is played on a graph (representing a network), whose edges are colored either red or blue, and where the red edges have a given fixed cost (representing the competitor's prices). The first player chooses an assignment of prices to the blue edges, and the second player then buys the cheapest possible minimum spanning tree, using any combination of red and blue edges. The goal of the first player is to maximize the total price of purchased blue edges. This game is the minimum spanning tree analog of the well-studied Stackelberg shortest-path game. We analyze the complexity and approximability of the first player's best strategy in StackMST. In particular, we prove that the problem is APX-hard even if there are only two different red costs, and give an approximation algorithm whose approximation ratio is at most $\min \{k,1+\ln b,1+\ln W\}$, where $k$ is the number of distinct red costs, $b$ is the number of blue edges, and $W$ is the maximum ratio between red costs. We also give a natural integer linear programming formulation of the problem, and show that the integrality gap of the fractional relaxation asymptotically matches the approximation guarantee of our algorithm.