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Stabilizing Weighted Graphs

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Zhuan Khye Koh and Laura Sanità. Stabilizing Weighted Graphs. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 83:1-83:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ICALP.2018.83

Abstract

An edge-weighted graph G=(V,E) is called stable if the value of a maximum-weight matching equals the value of a maximum-weight fractional matching. Stable graphs play an important role in some interesting game theory problems, such as network bargaining games and cooperative matching games, because they characterize instances which admit stable outcomes. Motivated by this, in the last few years many researchers have investigated the algorithmic problem of turning a given graph into a stable one, via edge- and vertex-removal operations. However, all the algorithmic results developed in the literature so far only hold for unweighted instances, i.e., assuming unit weights on the edges of G. We give the first polynomial-time algorithm to find a minimum cardinality subset of vertices whose removal from G yields a stable graph, for any weighted graph G. The algorithm is combinatorial and exploits new structural properties of basic fractional matchings, which are of independent interest. In particular, one of the main ingredients of our result is the development of a polynomial-time algorithm to compute a basic maximum-weight fractional matching with minimum number of odd cycles in its support. This generalizes a fundamental and classical result on unweighted matchings given by Balas more than 30 years ago, which we expect to prove useful beyond this particular application. In contrast, we show that the problem of finding a minimum cardinality subset of edges whose removal from a weighted graph G yields a stable graph, does not admit any constant-factor approximation algorithm, unless P=NP. In this setting, we develop an O(Delta)-approximation algorithm for the problem, where Delta is the maximum degree of a node in G.

Subject Classification

ACM Subject Classification
• Mathematics of computing → Matchings and factors
• Mathematics of computing → Approximation algorithms
• Mathematics of computing → Graph algorithms
• Theory of computation → Discrete optimization
• Theory of computation → Algorithmic game theory
• Theory of computation → Network games
Keywords
• combinatorial optimization
• network bargaining
• cooperative game

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References

1. Sara Ahmadian, Hamideh Hosseinzadeh, and Laura Sanità. Stabilizing network bargaining games by blocking players. In Proceedings of the 18th International Conference on Integer Programming and Combinatorial Optimization (IPCO), pages 164-177, 2016.
2. Egon Balas. Integer and fractional matchings. North-Holland Mathematics Studies, 59:1-13, 1981.
3. Michel Balinski. On maximum matching, minimum covering and their connections. In Proceedings of the Princeton Symposium on Mathematical Programming, pages 303-312, 1970.
4. Péter Biró, Matthijs Bomhoff, Petr A. Golovach, Walter Kern, and Daniël Paulusma. Solutions for the stable roommates problem with payments. Theor. Comput. Sci., 540:53-61, 2014.
5. Péter Biró, Walter Kern, and Daniël Paulusma. Computing solutions for matching games. Int. J. Game Theory, 41(1):75-90, 2012.
6. Adrian Bock, Karthekeyan Chandrasekaran, Jochen Könemann, Britta Peis, and Laura Sanità. Finding small stabilizers for unstable graphs. Math. Program., 154(1-2):173-196, 2015.
7. Karthekeyan Chandrasekaran, Corinna Gottschalk, Jochen Könemann, Britta Peis, Daniel Schmand, and Andreas Wierz. Additive stabilizers for unstable graphs. arXiv e-prints, Aug 2016. URL: http://arxiv.org/abs/1608.06797.
8. Xiaotie Deng, Toshihide Ibaraki, and Hiroshi Nagamochi. Algorithmic aspects of the core of combinatorial optimization games. Math. Oper. Res., 24(3):751-766, 1999.
9. Jack Edmonds. Paths, trees, and flowers. Canad. J. Math., 17:449-467, 1965.
10. Takehiro Ito, Naonori Kakimura, Naoyuki Kamiyama, Yusuke Kobayashi, and Yoshio Okamoto. Efficient stabilization of cooperative matching games. Theor. Comput. Sci., 677:69-82, 2017.
11. Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC), pages 767-775, 2002.
12. Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2-ε. J. Comput. Syst. Sci., 74(3):335-349, 2008.
13. Jon M. Kleinberg and Éva Tardos. Balanced outcomes in social exchange networks. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC), pages 295-304, 2008.
14. Jochen Könemann, Kate Larson, and David Steiner. Network bargaining: Using approximate blocking sets to stabilize unstable instances. Theory Comput. Syst., 57(3):655-672, 2015.
15. Sounaka Mishra, Venkatesh Raman, Saket Saurabh, Somnath Sikdar, and C. R. Subramanian. The complexity of könig subgraph problems and above-guarantee vertex cover. Algorithmica, 61(4):857-881, 2011.
16. George L. Nemhauser and Leslie E. Trotter Jr. Vertex packings: Structural properties and algorithms. Math. Program., 8(1):232-248, 1975.
17. Lloyd S. Shapley and Martin Shubik. The assignment game I: The core". International Journal of Game Theory, 1(1):111-130, 1971.