WADS 2007: Algorithms and Data Structures pp 64-76

# The Stackelberg Minimum Spanning Tree Game

• Jean Cardinal
• Erik D. Demaine
• Samuel Fiorini
• Gwenaël Joret
• Stefan Langerman
• Ilan Newman
• Oren Weimann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4619)

## 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,3 + 2ln 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.

## Keywords

Span Tree Minimum Span Tree Linear Programming Relaxation Price Function Stackelberg Game
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• Jean Cardinal
• 1
• Erik D. Demaine
• 2
• Samuel Fiorini
• 3
• Gwenaël Joret
• 1
• Stefan Langerman
• 1
• Ilan Newman
• 4
• Oren Weimann
• 2
1. 1.Computer Science Department, Université Libre de Bruxelles, B-1050 BrusselsBelgium
2. 2.MIT Computer Science and Artificial Intelligence Laboratory, Cambridge, MA 02139USA
3. 3.Department of Mathematics, Université Libre de Bruxelles, B-1050 BrusselsBelgium
4. 4.Department of Computer Science, University of Haifa, Haifa 31905Israel