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Spanning-Tree Games

Authors Dan Hefetz, Orna Kupferman, Amir Lellouche, Gal Vardi

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ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • Algorithms
  • Games
  • Minimum/maximum spanning tree
  • Greedy algorithms

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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.

Cite As Get BibTex

Dan Hefetz, Orna Kupferman, Amir Lellouche, and Gal Vardi. Spanning-Tree Games. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 35:1-35:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.MFCS.2018.35

Author Details

Dan Hefetz
  • Department of Computer Science, Ariel University, Israel
Orna Kupferman
  • School of Computer Science and Engineering, The Hebrew University, Israel
Amir Lellouche
  • Department of Computer Science, Weizmann Institute of Science, Israel
Gal Vardi
  • School of Computer Science and Engineering, The Hebrew University, Israel

Funding

  • Hefetz, Dan: The research leading to this paper was done when the author was visiting the Hebrew University.
  • Kupferman, Orna: 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).

References

  1. T. Bartnicki, J. A. Grytczuk, H. A. Kierstead, and X. Zhu. The map coloring game. American Mathematical Monthly, 114:793-803, 2007. Google Scholar
  2. C.G. Bird. On cost allocation for a spanning tree: a game theoretic approach. Networks, 6(4):335-350, 1976. Google Scholar
  3. J. Bruno and L. Weinberg. A constructive graph-theoretic solution of the shannon switching game. IEEE Transactions on Circuit Theory, 17(1):74-81, 1970. Google Scholar
  4. J. Cardinal, E.D. Demaine, S. Fiorini, G. Joret, S. Langerman, I. Newman, and O. Weimann. The stackelberg minimum spanning tree game. Algorithmica, 59(2):129-144, 2011. Google Scholar
  5. J. Cardinal, E.D. Demaine, S. Fiorini, G. Joret, I. Newman, and O. Weimann. The stackelberg minimum spanning tree game on planar and bounded-treewidth graphs. Journal of combinatorial optimization, 25(1):19-46, 2013. Google Scholar
  6. A.K. Chandra, D.C. Kozen, and L.J. Stockmeyer. Alternation. Journal of the Association for Computing Machinery, 28(1):114-133, 1981. Google Scholar
  7. A. Claus and D. J. Kleitman. Cost allocation in networks: The bulk supplier problem. Networks, 4(1):1-17, 1974. Google Scholar
  8. S.A. Cook. Path systems and language recognition. In Proc. 2nd ACM Symp. on Theory of Computing, pages 70-72, 1970. Google Scholar
  9. C. Cooper, A. Frieze, N. Ince, S. Janson, and J. Spencer. On the length of a random minimum spanning tree. Combinatorics, Probability and Computing, 25:89-107, 2016. Google Scholar
  10. T.H. Cormen, C.E. Leiserson, and R.L. Rivest. Introduction to Algorithms. MIT Press and McGraw-Hill, 1990. Google Scholar
  11. B. Dutta and A. Kar. Cost monotonicity, consistency and minimum cost spanning tree games. Games and Economic Behavior, 48(2):223-248, 2004. Google Scholar
  12. A. M. Frieze. On the value of a random minimum spanning tree problem. Discrete Applied Mathematics, 10:47-56, 1985. Google Scholar
  13. Z. Füredi, D. Reimer, and A. Seress. Triangle-free game and extremal graph problems. Congressus Numerantium, 82:123-128, 1991. Google Scholar
  14. R.L Graham and P. Hell. On the history of the minimum spanning tree problem. Annals of the History of Computing, 7(1):43-57, 1985. Google Scholar
  15. D. Granot and G. Huberman. Minimum cost spanning tree games. Mathematical programming, 21(1):1-18, 1981. Google Scholar
  16. D. Granot and G. Huberman. On the core and nucleolus of minimum cost spanning tree games. Mathematical programming, 29(3):323-347, 1984. Google Scholar
  17. D. Hefetz, M. Krivelevich, A. Naor, and M. Stojaković. On saturation games. European Journal of Combinatorics, 41:315-335, 2016. Google Scholar
  18. A. Kar. Axiomatization of the shapley value on minimum cost spanning tree games. Games and Economic Behavior, 38(2):265-277, 2002. Google Scholar
  19. J.B. Kruskal. On the shortest spanning subtree of a graph and the traveling salesman problem. Proceedings of the American Mathematical society, 7(1):48-50, 1956. Google Scholar
  20. O. Kupferman. Examining classical graph-theory problems from the viewpoint of formal-verification methods. In Proc. 49th ACM Symp. on Theory of Computing, page 6, 2017. Google Scholar
  21. O. Kupferman, G. Vardi, and M.Y. Vardi. Flow games. In Proc. 37th Conf. on Foundations of Software Technology and Theoretical Computer Science, 2017, to appear. Google Scholar
  22. A. Lehman. A solution of the shannon switching game. Journal of the Society for Industrial and Applied Mathematics, 12(4):687-725, 1964. Google Scholar
  23. O. Lichtenstein and A. Pnueli. Checking that finite state concurrent programs satisfy their linear specification. In Proc. 12th ACM Symp. on Principles of Programming Languages, pages 97-107, 1985. Google Scholar
  24. N. Nisan, T. Roughgarden, E. Tardos, and V.V. Vazirani. Algorithmic Game Theory. Cambridge University Press, 2007. Google Scholar
  25. J. Oxley. Matroid Theory, 2nd edition. Oxford University Press, 2011. Google Scholar
  26. R.C. Prim. Shortest connection networks and some generalizations. Bell Labs Technical Journal, 36(6):1389-1401, 1957. Google Scholar
  27. L.J. Stockmeyer. On the combinational complexity of certain symmetric boolean functions. Mathematical Systems Theory, 10:323-336, 1977. Google Scholar
  28. R.E. Tarjan. Data structures and network algorithms. SIAM, 1983. Google Scholar
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