Matroid bases with cardinality constraints on the intersection

Given two matroids $$\mathcal {M}_{1} = (E, \mathcal {B}_{1})$$ M 1 = ( E , B 1 ) and $$\mathcal {M}_{2} = (E, \mathcal {B}_{2})$$ M 2 = ( E , B 2 ) on a common ground set E with base sets $$\mathcal {B}_1$$ B 1 and $$\mathcal {B}_2$$ B 2 , some integer $$k \in \mathbb {N}$$ k ∈ N , and two cost functions $$c_{1}, c_{2} :E \rightarrow \mathbb {R}$$ c 1 , c 2 : E → R , we consider the optimization problem to find a basis $$X \in \mathcal {B}_{1}$$ X ∈ B 1 and a basis $$Y \in \mathcal {B}_{2}$$ Y ∈ B 2 minimizing the cost $$\sum _{e\in X} c_1(e)+\sum _{e\in Y} c_2(e)$$ ∑ e ∈ X c 1 ( e ) + ∑ e ∈ Y c 2 ( e ) subject to either a lower bound constraint $$|X \cap Y| \le k$$ | X ∩ Y | ≤ k , an upper bound constraint $$|X \cap Y| \ge k$$ | X ∩ Y | ≥ k , or an equality constraint $$|X \cap Y| = k$$ | 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.

Efficient algorithm for Minimum cost flow problem with partial lane reversals

In this paper, we investigate the minimum cost flow problem in two terminal series parallel network. We present modified minimum cost flow algorithm that computes the maximum dynamic and the earliest arrival flows in strongly polynomial time and also preserves all unused arc capacities. We also present strongly polynomial time minimum cost partial contraflow algorithm that solves both problems with partial reversals of arc capacities on two terminal series parallel networks.

Generalized Littlewood-Richardson coefficients for branching rules of GL(n) and extremal weight crystals

This thesis is devoted to the combinatorial and geometric study of certain multiplicities, which we call generalized Littlewood-Richardson coefficients. These are sums of products of single Littlewood-Richardson coefficients, and the specific ones we study describe the branching rules for the direct sum and diagonal embeddings of GL(n) as well as the decompositions of extremal weight crystals of type A+. By representing these multiplicities as dimensions of weight spaces of quiver semi-invariants, we use quiver theory to prove their saturation and describe necessary and sufficient conditions for them to be nonzero, culminating in statements similar to Horn's classical conjecture. We then use these conditions to prove various combinatorial properties, including how these multiplicities can be factored and that these numbers in certain cases satisfy the same conjectures as single Littlewood-Richardson coefficients. Finally, we provide a polytopal description of these multiplicities and prove that their positivity can be computed in strongly polynomial time.

Inverse min-max spanning r-arborescence problem under the weighted sum-type Hamming distance

The inverse min-max spanning [Formula: see text]-arborescence problem under the weighted sum-type Hamming distance on graphs is to modify the edge cost vector with respect to given modification bounds such that a given spanning [Formula: see text]-arborescence becomes a min-max spanning [Formula: see text]-arborescence and the total modification cost under the sum-type Hamming distance for all edges is minimized. It is shown that the problem is solvable in strongly polynomial time.