Pseudomatroids

Generalized skew bisubmodularity: A characterization and a min–max theorem

“…The concept of bisubmodularity was introduced in the study of ∆-matroids by Bouchet [3] and independently by Chandrasekaran-Kabadi [5] (also see [6,1]). Examples of ∆-matroids include the base family of a matroid as well as the family of matchable vertex sets in a graph, and bisubmodularity plays an important rôle in combinatorial optimization for establishing the common generalization of matroid theory and matching theory from the optimization view point (see, e.g., [4]).…”

Δ-matroid and jump system

“…For any discrete system (E, ) and any Definition 3.1 [7,10]. The rank function r : 3 E → Z + of a ∆-matroid (E, ) is defined as…”

“…Theorem 3.2 [7,10]. A function r : 3 E → Z + is the rank function of a ∆-matroid if and only if it satisfies the following:…”

“…In [7,10], the concept of pseudomatroid was developed as a proper generalization of the concept of matroid. The same concept was independently developed as ∆-matroid in [4,5].…”

Polynomial combinatorial algorithms for skew-bisubmodular function minimization

“…The main body of the algorithm will also be used in the strongly polynomial time algorithm given in the next section, and hence we shall refer to it as REFINE. An iteration of the while-loop in REFINE (i.e., lines [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] corresponds to a scaling phase with a scaling parameter δ discussed above. …”