An Investigation of the Recoverable Robust Assignment Problem

Abstract

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 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 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 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, the second stage recoverable assignment problem. We show that this problem variant is contained in the randomized parallel complexity class RNC2, 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.

Investigating the Recoverable Robust Single Machine Scheduling Problem Under Interval Uncertainty

“…As observed in Fischer et al (2020), it is straightforward to see that problem (RecSMSP) can be solved in polynominal time for constant value of ∆ by enumerating all possibilities for the intersection set |X ∩ Y |. Note that whilst this approach would require us to check n 2 ∆ many candidates for general cost functions, this number is reduced to n ∆ in our case.…”

“…Robust assignment problems were also considered in Deı et al (2006) using discrete scenarios and in Pereira and Averbakh (2011) under interval uncertainty in combination with the regret criterion. Most closely related to the work presented in this paper is Fischer et al (2020), where the complexity of the recoverable robust assignment problem with interval uncertainty is studied. Amongst other results, they show that this problem is W[1]-hard with respect to ∆ and n − ∆.…”

Benchmarking Problems for Robust Discrete Optimization

“…For this reason, min-max problems with interval uncertainty are excluded from the remainder of this study. Similarly, two-stage and recoverable robust problems with interval uncertainty are equivalent to problems with only a single scenario and are therefore not considered here, though they find some interest in the recent literature (see, e.g., [KZ17,FHLW20,BG21]). Furthermore, while some papers consider two-stage problems with discrete scenarios, they are most commonly not given by an explicit list, but described implicitly, see, for example, [HDJPR21,Sub21].…”

On the Recoverable Traveling Salesman Problem

“…The recoverable robust setting was also considered for the Assignment Problem in [6], where the authors show W [1]-hardness and present special cases that can be solved in polynomial time. In a related single-machine Scheduling Problem setting, [1] derive a 2-approximation algorithm.…”