Let X = {x1, ..., xn} be a set, and let P = {P1, ..., Pm} be a set of subsets of X. The goal is to find the smallest sub-collection of sets R ⊆ P whose union covers all of X. More concretely, we can say that each xi, 1 ≤ i ≤ n represents a “skill”, and each Pj, 1 ≤ j ≤ m represents a “person”. In this case, xi ∈ Pj if person Pj has the skill xi. Then, as an employer, the goal is to find the min...

The front cover shows ABB serving the cement manufacturing industry, which it has done since 1961 with turnkey installations, electrical equipment and automation systems. In 1990, ABB recognised the MES (Manufacturing Execution Systems) business area as a natural extension to its technology portfolio. Since then, ABB has provided added value solutions that are independent from the installed pro...

A subset S of vertices of a graph G is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. Denote by ψk(G) the minimum cardinality of a k-path vertex cover in G. In this article a lower and an upper bound for ψk of the rooted product graphs are presented. Two characterizations are given when those bounds are attained. Moreover ψ2 and ψ3 are exactly de...

No part of this book may be reproduced, in any form or by any means, without permission in writing from the author. Cover picture: Three dimensional schematic representation of the steps that lead to controller design. The fish is cloned for scientific purposes only, without permission of the Dutch government.

We show that for each integer h ≥ 2, there exists a least positive integer ch so that if P is a poset having a planar cover graph and the height of P is h, then the dimension of P is at most ch. Trivially, c1 = 2. Also, Felsner, Li and Trotter showed that c2 exists and is 4, but their proof techniques do not seem to apply when h ≥ 3. We focus on establishing the existence of ch, although we sus...

We consider capacitated vertex cover with hard capacity constraints (VC-HC) on hypergraphs. In this problem we are given a hypergraph G = (V,E) with a maximum edge size f . Each edge is associated with a demand and each vertex is associated with a weight (cost), a capacity, and an available multiplicity. The objective is to find a minimum-weight vertex multiset such that the demands of the edge...

Let G = (X, Y; E) be a balanced bipartite graph of order 2n. The path-cover number pc(H) of a graph H is the minimum number of vertex-disjoint paths that use up all the vertices of H. S ⊆ V(G) is called a balanced set of G if |S ∩ X| = |S ∩ Y|. In this paper, we will give some sufficient conditions for a balanced bipartite graph G satisfying that for every balanced set S, there is a bi-cycle of...

This article studies crime rates’ response to economic conditions. Using a longitudinal data set covering about 20 countries over the period 1970-2010, we investigate whether crime rates respond asymmetrically to increases in unemployment and recoveries in economic conditions. We find a positive response of crime rates on variation in unemployment rates, but we do not find compelling evidence o...