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An Investigation of the Recoverable Robust Assignment Problem

Authors Dennis Fischer, Tim A. Hartmann, Stefan Lendl, Gerhard J. Woeginger

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Fixed parameter tractability
Keywords
  • assignment problem
  • matchings
  • exact matching
  • robust optimization
  • fixed paramter tractablity
  • RNC

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Abstract

We investigate the so-called recoverable robust assignment problem on complete bipartite graphs, a mainstream problem in robust optimization: For two given linear cost functions c₁ and c₂ on the edges and a given integer k, the goal is to find two perfect matchings M₁ and M₂ that minimize the objective value c₁(M₁)+c₂(M₂), subject to the constraint that M₁ and M₂ have at least k edges in common.
We derive a variety of results on this problem. First, we show that the problem is W[1]-hard with respect to parameter k, and also with respect to the complementary parameter k' = n/2-k. This hardness result holds even in the highly restricted special case where both cost functions c₁ and c₂ only take the values 0 and 1. (On the other hand, containment of the problem in XP is straightforward to see.) Next, as a positive result we construct a polynomial time algorithm for the special case where one cost function is Monge, whereas the other one is Anti-Monge. Finally, we study the variant where matching M₁ is frozen, and where the optimization goal is to compute the best corresponding matching M₂. This problem variant is known to be contained in the randomized parallel complexity class RNC², and we show that it is at least as hard as the infamous problem Exact Red-Blue Matching in Bipartite Graphs whose computational complexity is a long-standing open problem.

Cite As Get BibTex

Dennis Fischer, Tim A. Hartmann, Stefan Lendl, and Gerhard J. Woeginger. An Investigation of the Recoverable Robust Assignment Problem. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.IPEC.2021.19

Author Details

Dennis Fischer
  • Department of Computer Science, RWTH Aachen, Germany
Tim A. Hartmann
  • Department of Computer Science, RWTH Aachen, Germany
Stefan Lendl
  • Institute of Operations und Information Systems, Universität Graz, Austria
Gerhard J. Woeginger
  • Department of Computer Science, RWTH Aachen, Germany

Funding

Stefan Lendl acknowledges the support of the Austrian Science Fund (FWF): W1230. Dennis Fischer and Gerhard J. Woeginger have been supported by the DFG RTG 2236 "UnRAVeL".

Acknowledgements

We thank Bettina Klinz for several helpful discussions about the topic.

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