Rough set is a theory of data analysis and a mathematical tool for dealing with vagueness, incompleteness, and granularity. Matroid, as a branch of mathematics, is a structure that generalizes linear independence in vector spaces. In this paper, we establish the relationships between partition matroids and rough sets through k-rank matroids. On the one hand, k-rank matroids are proposed to repr...

This paper surveys some of the work that was inspired by Wagner’s general technique to prove completeness in the levels of the boolean hierarchy over NP and some related results. In particular, we show that it is DP-complete to decide whether or not a given graph can be colored with exactly four colors, where DP is the second level of the boolean hierarchy. This result solves a question raised ...

Let D be a finite and simple digraph with the vertex set V (D), and let f : V (D) → {−1, 1} be a two-valued function. If∑ x∈N[v] f(x) ≥ 1 for each v ∈ V (D), where N[v] consists of v and all vertices of D from which arcs go into v, then f is a signed dominating function on D. The sum f(V (D)) is called the weight w(f) of f . The minimum of weights w(f), taken over all signed dominating function...

upon specializing q = qτ := e 2πiτ , τ ∈ H the upper complex half-plane, where η(τ) := q1/24 ∏ n≥1(1 − qn) is Dedekind’s η-function, a weight 1/2 modular form. More recently, Bringmann and Ono [8] studied the generating function for partition ranks, where the rank of a partition, after Dyson, is defined to be the largest part of the partition minus the number of parts. For example, the rank of ...

We investigate hypergraphic LP relaxations for the Steiner tree problem, primarily the partition LP relaxation introduced by Könemann et al. [Math. Programming, 2009]. Specifically, we are interested in proving upper bounds on the integrality gap of this LP, and studying its relation to other linear relaxations. First, we show uncrossing techniques apply to the LP. This implies structural prope...

Matrix partition problems generalize a number of natural graph partition problems, and have been studied for several standard graph classes. We prove that each matrix partition problem has only finitely many minimal obstructions for split graphs. Previously such a result was only known for the class of cographs. (In particular, there are matrix partition problems which have infinitely many mini...

Abstract Methods for improving upper and lower bounds various coverings of planar sets are proposed. New numbers partition constituents presented, suggestions the generalization presented methods offered.

We propose an algorithm for computing the number of ordered integer partitions with upper bounds. This problem’s task is to compute the number of distributions of z balls into n urns with constrained capacities i1, . . . , in (see [10]). It has applications in the theory of database preferences as described in [3] and [9]. The running time of our algorithm depends only on the number of urns and...

We discuss matrix partition problems for graphs that admit a partition into k independent sets and ` cliques. We show that when k + ` 6 2, any matrix M has finitely many (k, `) minimal obstructions and hence all of these problems are polynomial time solvable. We provide upper bounds for the size of any (k, `) minimal obstruction when k = ` = 1 (split graphs), when k = 2, ` = 0 (bipartite graphs...