Given two matroids M 1 = (E, B 1 ) and M 2 = (E, B 2 ) on a common ground set E with base sets B 1 and B 2 , some integer k ∈ N, and two cost functions c 1 , c 2 : E → R, we consider the optimization problem to find a basis X ∈ B 1 and a basis Y ∈ B 2 minimizing cost e∈X c 1 (e)+ e∈Y c 2 (e) subject to either a lower bound constraint |X ∩Y | ≤ k, an upper bound constraint |X ∩ Y | ≥ k, or an equality constraint |X ∩ Y | = k on the size of the intersection of the two bases X and Y . The problem with lower bound constraint turns out to be a generalization of the Recoverable Robust Matroid problem under interval uncertainty representation for which the question for a strongly polynomialtime algorithm was left as an open question in [6].We show that the two problems with lower and upper bound constraints on the size of the intersection can be reduced to weighted matroid intersection, and thus be solved with a strongly polynomial-time primal-dual algorithm. The question whether the problem with equality constraint can also be solved efficiently turned out to be a lot harder. As our main result, we present a strongly-polynomial, primal-dual algorithm for the problem with equality constraint on the size of the intersection.Additionally, we discuss generalizations of the problems from matroids to polymatroids, and from two to three or more matroids.

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Budgeted uncertainty sets have been established as a major influence on uncertainty modeling for robust optimization problems. A drawback of such sets is that the budget constraint only restricts the global amount of cost increase that can be distributed by an adversary. Local restrictions, while being important for many applications, cannot be modeled this way.We introduce a new variant of budgeted uncertainty sets, called locally budgeted uncertainty. In this setting, the uncertain parameters are partitioned, such that a classic budgeted uncertainty set applies to each part of the partition, called region.In a theoretical analysis, we show that the robust counterpart of such problems for a constant number of regions remains solvable in polynomial time, if the underlying nominal problem can be solved in polynomial time as well. If the number of regions is unbounded, we show that the robust selection problem remains solvable in polynomial time, while also providing hardness results for other combinatorial problems.In computational experiments using both random and real-world data, we show that using locally budgeted uncertainty sets can have considerable advantages over classic budgeted uncertainty sets.

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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.