We consider a two-player interdiction problem staged over a graph where the attacker’s objective is to minimize the cost of removing edges from the graph so that the defender’s objective, that is, the weight of a minimum spanning tree in the residual graph, is increased up to a predefined level r. Standard approaches for graph interdiction frame this type of problems as bilevel formulations, which are commonly solved by replacing the inner problem by its dual to produce a single-level reformulation. In this paper, we study an alternative integer program derived directly from the attacker’s solution space and show that this formulation yields a stronger linear relaxation than the bilevel counterpart. Furthermore, we analyze the convex hull of the feasible solutions of the problem and identify several families of facet-defining inequalities that can be used to strengthen this integer program. We then proceed by introducing a different formulation defined by a set of so-called supervalid inequalities that may exclude feasible solutions, albeit solutions whose objective value is not better than that of an edge cut of minimum cost. We discuss several computational aspects required for an efficient implementation of the proposed approaches. Finally, we perform an extensive set of computational experiments to test the quality of these formulations, analyzing and comparing the benefits of each model, as well as identifying further enhancements. Summary of Contribution: Network interdiction has received significant attention over the last couple of decades, with a notable peak of interest in recent years. This paper provides an interesting balance between the theoretical and computational aspects of solving a challenging network interdiction problem via integer programming. We present several technical developments, including a detailed study of the problem's solution space, multiple formulations, and a polyhedral analysis of the convex hull of feasible solutions. We then analyze the results of an extensive set of computational experiments that were used to validate the effectiveness of the different methods we developed in this paper.

Abstract In this paper, we study several statistical properties regarding the distance between events that take place on random locations along the edges of a given network. We derive analytical expressions for the arbitrary moments of such a distance, its probability density function, its cumulative distribution function, as well as their conditional counterparts for the cases in which the position of one event is known in advance. As part of this study, we implement our developments as a callable library for the Python language, to provide potential users with a computational engine able to calculate and visualize these statistics for any given network. We test our implementation on several networks of different sizes and topologies, analyze some of interesting properties we observed in our experiments, and discuss several applications for our proposed methodology. In particular, we focus our discussion on applications aimed to help with the optimal design of emergency response systems on infrastructure networks.

Abstract We consider the problem of detecting a collection of critical node structures of a graph whose deletion results in the maximum deterioration of the graph's connectivity. The proposed approach is aimed to generalize other existing models whose scope is restricted to removing individual and unrelated nodes. We consider two common metrics to quantify the connectivity of the residual graph: the total number of connected node pairs and the size of the largest connected component. We first discuss the computational complexity of the problem and then introduce a general mixed‐integer linear formulation, which depending on the kind of node structures, may have an exponentially large number of variables and constraints. To solve this potentially large model, we develop a branch‐price‐and‐cut framework, along with some valid inequalities and preprocessing algorithms to strengthen the formulation and reduce the overall execution time. We use the proposed approach to solve the problem for the cases, where the node structures form cliques or stars and provide further directions on how to extend the framework for detecting other kinds of critical structures as well. Finally, we test the quality of our approach by solving a collection of real‐life and randomly generated instances with various configurations, analyze the benefits of our model, and propose further enhancements.

The graph bipartitioning problem consists of dividing a graph into two disjoint subgraphs, such that each node is highly similar to others in the same subgraph, but also different from members of the other subgraph, according to some homogeneity criterion. This problem has received significant attention over the last few years because of its applicability in areas as diverse as data classification, image segmentation, and social network analysis. In this article we study a variation of the graph bipartitioning problem in which, in addition to considering homogeneity criteria for generating the partition, we also ensure that one of the subgraphs satisfies a set of predefined structural properties—that is, such a subgraph is required to induce a given motif. We focus our attention on imposing structural constraints that force one of the subgraphs to induce stars, cliques, and clique relaxations (quasi‐cliques) and discuss some specific applications for such particular cases. We tackle this problem by modeling it as a general fractional programming optimization problem and study several solution approaches. Moreover, we discuss additional algorithmic enhancements to tackle some of the aforementioned cases, and provide two greedy algorithms for the specific cases of induced cliques and stars, showing the approximation ratio for induced stars. Finally, we test the quality of our approach by solving a collection of several real‐life and randomly generated instances with various configurations, analyzing the benefits of the proposed models, as well as possible further extensions. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 71(4), 432–450 2018