Matroid bases with cardinality constraints on the intersection.

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 the 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 polynomial-time algorithm was left as an open question in Hradovich et al. (J Comb Optim 34(2):554-573, 2017). 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. We also present a strongly polynomial, primal-dual algorithm that computes a minimum cost solution for every feasible size of the intersection k in one run with asymptotic running time equal to one run of Frank's matroid intersection algorithm. Additionally, we discuss generalizations of the problems from matroids to polymatroids, and from two to three or more matroids. We obtain a strongly polynomial time algorithm for the recoverable robust polymatroid base problem with interval uncertainties.

Long-term consequences of non-intentional flows of substances: modelling non-intentional flows of lead in the Dutch economic system and evaluating their environmental consequences.

Substances may enter the economy and the environment through both intentional and non-intentional flows. These non-intentional flows, including the occurrence of substances as pollutants in mixed primary resources (metal ores, phosphate ores and fossil fuels) and their presence in re-used waste streams from intentional use may have environmental and economic consequences in terms of pollution and resource availability. On the one hand, these non-intentional flows may cause pollution problems. On the other hand, these flows have the potential to be a secondary source of substances. This article aims to quantify and model the non-intentional flows of lead, to evaluate their long-term environmental consequences, and compare these consequences to those of the intentional flows of lead. To meet this goal, the model combines all the sources of non-intentional flows of lead within one model, which also includes the intentional flows. Application of the model shows that the non-intentional flows of lead related to waste streams associated with intentional use are decreasing over time, due to the increased attention given to waste management. However, as contaminants in mixed primary resources application, lead flows are increasing as demand for these applications is increasing.