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.
@InProceedings{fischer_et_al:LIPIcs.IPEC.2021.19, author = {Fischer, Dennis and Hartmann, Tim A. and Lendl, Stefan and Woeginger, Gerhard J.}, title = {{An Investigation of the Recoverable Robust Assignment Problem}}, booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)}, pages = {19:1--19:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-216-7}, ISSN = {1868-8969}, year = {2021}, volume = {214}, editor = {Golovach, Petr A. and Zehavi, Meirav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.19}, URN = {urn:nbn:de:0030-drops-154025}, doi = {10.4230/LIPIcs.IPEC.2021.19}, annote = {Keywords: assignment problem, matchings, exact matching, robust optimization, fixed paramter tractablity, RNC} }
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