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How to Secure Matchings Against Edge Failures

Authors Felix Hommelsheim, Moritz Mühlenthaler, Oliver Schaudt

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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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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.

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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

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.

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