# An Investigation of the Recoverable Robust Assignment Problem

We investigate the so-called recoverable robust assignment problem on balanced bipartite graphs with 2n vertices, a mainstream problem in robust optimization: For two given linear cost functions c_1 and c_2 on the edges and a given integer k, the goal is to find two perfect matchings M_1 and M_2 that minimize the objective value c_1(M_1)+c_2(M_2), subject to the constraint that M_1 and M_2 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 the parameter k, and also with respect to the recoverability parameter k'=n-k. This hardness result holds even in the highly restricted special case where both cost functions c_1 and c_2 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 M_1 is frozen, and where the optimization goal is to compute the best corresponding matching M_2, the second stage recoverable assignment problem. We show that this problem variant is contained in the randomized parallel complexity class RNC_2, and that it is at least as hard as the infamous problem Exact Matching in Red-Blue Bipartite Graphs whose computational complexity is a long-standing open problem

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