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How to Secure Matchings Against Edge Failures

Authors Felix Hommelsheim, Moritz Mühlenthaler, Oliver Schaudt

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Author Details

Felix Hommelsheim
  • Department of Mathematics, TU Dortmund University
Moritz Mühlenthaler
  • Department of Mathematics, TU Dortmund University
Oliver Schaudt
  • Department of Mathematics, RWTH Aachen University

Funding

  • Hommelsheim, Felix: Research partially supported by the German Research Foundation (DFG), RTG 1855.
  • Mühlenthaler, Moritz: Research partially supported by the German Research Foundation (DFG), RTG 1855.

Acknowledgements

We would like to thank Viktor Bindewald for the fruitful discussions about some of the results in this paper.

Cite As Get BibTex

Felix Hommelsheim, Moritz Mühlenthaler, and Oliver Schaudt. How to Secure Matchings Against Edge Failures. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.STACS.2019.38

Abstract

Suppose we are given a bipartite graph that admits a perfect matching and an adversary may delete any edge from the graph with the intention of destroying all perfect matchings. We consider the task of adding a minimum cost edge-set to the graph, such that the adversary never wins. We show that this problem is equivalent to covering a digraph with non-trivial strongly connected components at minimal cost. We provide efficient exact and approximation algorithms for this task. In particular, for the unit-cost problem, we give a log_2 n-factor approximation algorithm and a polynomial-time algorithm for chordal-bipartite graphs. Furthermore, we give a fixed parameter algorithm for the problem parameterized by the treewidth of the input graph. For general non-negative weights we give tight upper and lower approximation bounds relative to the Directed Steiner Forest problem. Additionally we prove a dichotomy theorem characterizing minor-closed graph classes which allow for a polynomial-time algorithm. To obtain our results, we exploit a close relation to the classical Strong Connectivity Augmentation problem as well as directed Steiner problems.

Subject Classification

ACM Subject Classification
  • Hardware → Robustness
  • Mathematics of computing → Matchings and factors
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Approximation algorithms
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Mathematical optimization
Keywords
  • Matchings
  • Robustness
  • Connectivity Augmentation
  • Graph Algorithms
  • Treewidth

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References

  1. David Adjiashvili. Beating Approximation Factor Two for Weighted Tree Augmentation with Bounded Costs. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2384-2399. Society for Industrial and Applied Mathematics, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.157.
  2. David Adjiashvili, Viktor Bindewald, and Dennis Michaels. Robust Assignments via Ear Decompositions and Randomized Rounding. In 43rd International Colloquium on Automata, Languages, and Programming, volume 55, pages 71:1-71:14, Dagstuhl, Germany, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.71.
  3. David Adjiashvili, Sebastian Stiller, and Rico Zenklusen. Bulk-robust combinatorial optimization. Mathematical Programming, 149(1-2):361-390, 2015. URL: http://dx.doi.org/10.1007/s10107-014-0760-6.
  4. David Adjiashvili and Rico Zenklusen. An s-t connection problem with adaptability. Discrete Applied Mathematics, 159(8):695-705, 2011. URL: http://dx.doi.org/10.1016/j.dam.2010.12.018.
  5. Kristóf Bérczi, Satoru Iwata, Jun Kato, and Yutaro Yamaguchi. Making Bipartite Graphs DM-Irreducible. SIAM Journal on Discrete Mathematics, 32:560-590, 2018. URL: http://dx.doi.org/10.1137/16M1106717.
  6. Piotr Berman, Arnab Bhattacharyya, Konstantin Makarychev, Sofya Raskhodnikova, and Grigory Yaroslavtsev. Approximation algorithms for spanner problems and Directed Steiner Forest. Information and Computation, 222:93-107, 2013. 38th International Colloquium on Automata, Languages and Programming (ICALP 2011). URL: http://dx.doi.org/10.1016/j.ic.2012.10.007.
  7. Hans L. Bodlaender. A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth. SIAM Journal on Computing, 25(6):1305-1317, 1996. URL: http://dx.doi.org/10.1137/S0097539793251219.
  8. Hans L. Bodlaender, Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Information and Computation, 243:86-111, 2015. URL: http://dx.doi.org/10.1016/j.ic.2014.12.008.
  9. Robert C. Brigham, Frank Harary, Elizabeth C. Violin, and Jay Yellen. Perfect-matching Preclusion. Congressus Numerantium, 174:185-192, 2005. Google Scholar
  10. Chandra Chekuri, Guy Even, Anupam Gupta, and Danny Segev. Set Connectivity Problems in Undirected Graphs and the Directed Steiner Network Problem. In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 532-541. Society for Industrial and Applied Mathematics, 2008. Google Scholar
  11. Eddie Cheng and László Lipták. Matching preclusion for some interconnection networks. Networks, 50(2):173-180, 2007. URL: http://dx.doi.org/10.1002/net.20187.
  12. Joseph Cheriyan, András Sebő, and Zoltán Szigeti. Improving on the 1.5-Approximation of a Smallest 2-Edge Connected Spanning Subgraph. SIAM Journal on Discrete Mathematics, 14(2):170-180, 2001. URL: http://dx.doi.org/10.1137/S0895480199362071.
  13. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  14. Mitre C. Dourado, Dirk Meierling, Lucia D. Penso, Dieter Rautenbach, Fabio Protti, and Aline Ribeiro de Almeida. Robust recoverable perfect matchings. Networks, 66(3):210-213, 2015. URL: http://dx.doi.org/10.1002/net.21624.
  15. Andrew L Dulmage and Nathan S Mendelsohn. Coverings of bipartite graphs. Canadian Journal of Mathematics, 10(4):516-534, 1958. Google Scholar
  16. Kapali P. Eswaran and Robert E. Tarjan. Augmentation problems. SIAM Journal on Computing, 5(4):653-665, 1976. URL: http://dx.doi.org/10.1137/0205044.
  17. Jon Feldman and Matthias Ruhl. The Directed Steiner Network Problem is Tractable for a Constant Number of Terminals. SIAM Journal on Computing, 36(2):543-561, 2006. URL: http://dx.doi.org/10.1137/S0097539704441241.
  18. Moran Feldman, Guy Kortsarz, and Zeev Nutov. Improved Approximating Algorithms for Directed Steiner Forest. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 922-931. Society for Industrial and Applied Mathematics, 2009. Google Scholar
  19. Samuel Fiorini, Martin Groß, Jochen Könemann, and Laura Sanità. Approximating Weighted Tree Augmentation via Chvátal-Gomory Cuts. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 817-831, 2018. URL: http://dx.doi.org/10.1137/1.9781611975031.53.
  20. András Frank and Tibor Jordán. Graph Connectivity Augmentation. In Handbook of Graph Theory, Combinatorial Optimization, and Algorithms, chapter 14, pages 313-346. CRC Press, 2015. Google Scholar
  21. Greg N Frederickson and Joseph Ja’Ja’. Approximation Algorithms for Several Graph Augmentation Problems. SIAM Journal on Computing, 10(2):270-283, 1981. URL: http://dx.doi.org/10.1137/0210019.
  22. Harold N. Gabow, Michel X. Goemans, Éva Tardos, and David P. Williamson. Approximating the smallest k-edge connected spanning subgraph by LP-rounding. Networks, 53(4):345-357, 2009. URL: http://dx.doi.org/10.1002/net.20289.
  23. Eran Halperin and Robert Krauthgamer. Polylogarithmic inapproximability. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing, pages 585-594, 2003. URL: http://dx.doi.org/10.1145/780542.780628.
  24. Alan J. Hoffman, Anthonius W. J. Kolen, and Michel Sakarovitch. Totally-Balanced and Greedy Matrices. SIAM Journal on Algebraic Discrete Methods, 6(4):721-730, 1985. URL: http://dx.doi.org/10.1137/0606070.
  25. Mathieu Lacroix, A. Ridha Mahjoub, Sébastien Martin, and Christophe Picouleau. On the NP-completeness of the perfect matching free subgraph problem. Theoretical Computer Science, 423:25-29, 2012. URL: http://dx.doi.org/10.1016/j.tcs.2011.12.065.
  26. Laurence A. Wolsey and George L. Nemhauser. Integer and Combinatorial Optimization. Wiley Series in Discrete Mathematics and Optimization. Wiley, 1999. Google Scholar
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