## 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^{))}, where w(MST(G)) 2 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+ _{w}(_{MST} ^{1} (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.

Original language | American English |
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Title of host publication | 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018 |

Editors | Igor Potapov, James Worrell, Paul Spirakis |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Print) | 9783959770866 |

DOIs | |

State | Published - 1 Aug 2018 |

Event | 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018 - Liverpool, United Kingdom Duration: 27 Aug 2018 → 31 Aug 2018 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 117 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018 |
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Country/Territory | United Kingdom |

City | Liverpool |

Period | 27/08/18 → 31/08/18 |

### Bibliographical note

Funding Information:The research leading to this paper was done when the author was visiting the Hebrew University. 2 The research leading to this paper has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013).

Publisher Copyright:

© Dan Hefetz, Orna Kupferman, Amir Lellouche, and Gal Vardi.

## Keywords

- Algorithms
- Games
- Greedy algorithms
- Minimum/maximum spanning tree