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 $\text{RNC}_2$, and that it is at least as hard as the infamous problem \probl{Exact Matching in Red-Blue Bipartite Graphs} whose computational complexity is a long-standing open problem