When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2018.35
URN: urn:nbn:de:0030-drops-96171
URL: https://drops.dagstuhl.de/opus/volltexte/2018/9617/
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### Spanning-Tree Games

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### Abstract

We introduce and study a game variant of the classical spanning-tree problem. Our spanning-tree game is played between two players, min and max, who alternate turns in jointly constructing a spanning tree of a given connected weighted graph G. Starting with the empty graph, in each turn a player chooses an edge that does not close a cycle in the forest that has been generated so far and adds it to that forest. The game ends when the chosen edges form a spanning tree in G. The goal of min is to minimize the weight of the resulting spanning tree and the goal of max is to maximize it. A strategy for a player is a function that maps each forest in G to an edge that is not yet in the forest and does not close a cycle.
We show that while in the classical setting a greedy approach is optimal, the game setting is more complicated: greedy strategies, namely ones that choose in each turn the lightest (min) or heaviest (max) legal edge, are not necessarily optimal, and calculating their values is NP-hard. We study the approximation ratio of greedy strategies. We show that while a greedy strategy for min guarantees nothing, the performance of a greedy strategy for max is satisfactory: it guarantees that the weight of the generated spanning tree is at least w(MST(G))/2, where w(MST(G)) is the weight of a maximum spanning tree in G, and its approximation ratio with respect to an optimal strategy for max is 1.5+1/w(MST(G)), assuming weights in [0,1]. We also show that these bounds are tight. Moreover, in a stochastic setting, where weights for the complete graph K_n are chosen at random from [0,1], the expected performance of greedy strategies is asymptotically optimal. Finally, we study some variants of the game and study an extension of our results to games on general matroids.

### BibTeX - Entry

```@InProceedings{hefetz_et_al:LIPIcs:2018:9617,
author =	{Dan Hefetz and Orna Kupferman and Amir Lellouche and Gal Vardi},
title =	{{Spanning-Tree Games}},
booktitle =	{43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018)},
pages =	{35:1--35:16},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-086-6},
ISSN =	{1868-8969},
year =	{2018},
volume =	{117},
editor =	{Igor Potapov and Paul Spirakis and James Worrell},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},