Spanning-tree games

Dan Hefetz, Orna Kupferman, Amir Lellouche, Gal Vardi

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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 Kn 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 languageEnglish
Title of host publication43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018
EditorsIgor Potapov, James Worrell, Paul Spirakis
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Print)9783959770866
DOIs
StatePublished - 1 Aug 2018
Event43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018 - Liverpool, United Kingdom
Duration: 27 Aug 201831 Aug 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume117
ISSN (Print)1868-8969

Conference

Conference43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018
Country/TerritoryUnited Kingdom
CityLiverpool
Period27/08/1831/08/18

Keywords

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

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