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The automated real time control of an electrical network is achieved through the estimation of its state using phasor measurement units. Given an undirected graph representing the network, we study the problem of finding the minimum number of phasor measurement units to place on the edges such that the graph is fully observed. This problem is also known as the Power Edge Set problem, a variant of the Power Dominating Set problem. It is naturally modeled using an iteration-indexed binary linear program, whose size turns out to be too large for practical purposes. We use a fixed-point argument to remove the iteration indices and obtain a more compact bilevel formulation. We then reformulate the latter to a single-level mixed-integer linear program, which performs better than the natural formulation. Lastly, we provide an algorithm that solves the bilevel program directly and much faster than a commercial solver can solve the previous models. We also discuss robust variants and extensions of the problem.

In his 2018 Stockholm prize winner lecture, Goldstein highlighted the need for problem-oriented policing (POP) to be not only effective but also fair. Contributing to the development of POP, this study examines how a wider perspective on problem-solving generally, and scoping in particular, can be adopted to address some of the growing challenges in 21st century policing. We demonstrate that the concept of ‘problem’ was too narrowly defined and that, as a result, many problem-solving models found in criminology are ill-structured to minimize the negative side-effects of interventions and deliver broader benefits. Problem-solving concepts and models are compared across disciplines and recommendations are made to improve POP, drawing on examples in architecture, conservation science, industrial ecology and ethics.

Abstract. We consider the k most vital edges (nodes) and min edge (node) blocker versions of the 1-median and 1-center location problems. Given a weighted connected graph with distances on edges and weights on nodes, the k most vital edges (nodes) 1-median (respectively 1-center) problem consists of finding a subset of k edges (nodes) whose removal from the graph leads to an optimal solution for the 1-median (respectively 1-center) problem with the largest total weighted distance (respectively maximum weighted distance). The complementary problem, min edge (node) blocker 1-median (respectively 1-center), consists of removing a subset of edges (nodes) of minimum cardinality such that an optimal solution for the 1-median (respectively 1-center) problem has a total weighted distance (respectively a maximum weighted distance) at least as large as a specified threshold. We show that k most vital edges 1-median and k most vital edges 1-center are NP -hard to approximate within a factor − ǫ respectively, for any ǫ > 0, while k most vital nodes 1-median and k most vital nodes 1-center are NP -hard to approximate within a factor 3 2 − ǫ, for any ǫ > 0. We also show that the complementary versions of these four problems are NP -hard to approximate within a factor 1.36.

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Efficient determination of the k most vital edges for the minimum spanning tree problem.Abstract. We study in this paper the problem of finding in a graph a subset of k edges whose deletion causes the largest increase in the weight of a minimum spanning tree. We propose for this problem an explicit enumeration algorithm whose complexity, when compared to the current best algorithm, is better for general k but very slightly worse for fixed k. More interestingly, unlike in the previous algorithms, we can easily adapt our algorithm so as to transform it into an implicit exploration algorithm based on a branch and bound scheme. We also propose a mixed integer programming formulation for this problem. Computational results show a clear superiority of the implicit enumeration algorithm both over the explicit enumeration algorithm and the mixed integer program.

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