Network interdiction problems are a natural way to study the sensitivity of a network optimization problem with respect to the removal of a limited set of edges or vertices. One of the oldest and best-studied interdiction problems is minimum spanning tree (MST) interdiction. Here, an undirected multigraph with nonnegative edge weights and positive interdiction costs on its edges is given, together with a positive budget B. The goal is to find a subset of edges R, whose total interdiction cost does not exceed B, such that removing R leads to a graph where the weight of an MST is as large as possible. Frederickson and Solis-Oba (SODA 1996) presented an O(log m)-approximation for MST interdiction, where m is the number of edges. Since then, no further progress has been made regarding approximations, and the question whether MST interdiction admits an O(1)-approximation remained open. We answer this question in the affirmative, by presenting a 14-approximation that overcomes two main hurdles that hindered further progress so far. Moreover, based on a well-known 2-approximation for the metric traveling salesman problem (TSP), we show that our O(1)-approximation for MST interdiction implies an O(1)-approximation for a natural interdiction version of metric TSP.
Facilities and their services can be lost due to natural disasters as well as to intentional strikes, either by terrorism or an army. An intentional strike against a system is called interdiction. The geographical distribution of facilities in a supply or service system may be particularly vulnerable to interdiction, and the resulting impacts of the loss of one or more facilities may be substantial. Critical infrastructure can be defined as those elements of infrastructure that, if lost, could pose a significant threat to needed supplies (e.g., food, energy, medicines), services (e.g., police, fire, and EMS), and communication or a significant loss of service coverage or efficiency. In this article we introduce two new spatial optimization models called the r-interdiction median problem and the r-interdiction covering problem. Both models identify for a given service supply system, that set of facilities that, if lost, would affect service delivery the most, depending upon the type of service protocol. These models can then be used to identify the most critical facility assets in a service supply system. Results of both models applied to spatial data are also presented. Several solutions derived from these two interdiction models are presented in greater detail and demonstrate the degree to which the loss of one or more facilities disrupts system efficiencies or coverage. Recommendations for further research are also made.   Let N = (V, A) be a directed, arc weighted network with node set V and arc set A. Associated with each arc [email protected] is a non-negative distance d(e) and a non-negative cost c(e) of removing it from N. Let B be the total amount available to spend on removing arcs. The most vital arcs problem (MVAP) is the problem of finding a subset K of arcs such that @S"e"@e"Kc(e) =< B and whose removal from N results in the greatest increase in the shortest distance between two specified nodes s and t in V. We show this problem to be NP-hard. In addition, we show that a closely related problem, whose solution provides a lower bound on the optimal solution value of MVAP, is polynomially solvable.  Interest in network interdiction has been rekindled because of attempts to reduce the flow of drugs and precursor chemicals moving through river and road networks in South America. This paper considers a problem in which an enemy attempts to maximize flow through a capacitated network while an interdictor tries to minimize this maximum flow by interdicting (stopping flow on) network arcs using limited resources. This problem is shown to be NP-complete even when the interdiction of an arc requires exactly one unit of resource. New, flexible, integer programming models are developed for the problem and its variations and valid inequalities and a reformulation are derived to tighten the LP relaxations of some of these models. A small computational example from the literature illustrates a hybrid (partly directed and partly undirected) model and the usefulness of the valid inequalities and the reformulation.   Given an undirected graph with weights on its vertices, the k most vital nodes independent set (k most vital nodes vertex cover) problem consists of determining a set of k vertices whose removal results in the greatest decrease in the maximum weight of independent sets (minimum weight of vertex covers, respectively). We also consider the complementary problems, minimum node blocker independent set (minimum node blocker vertex cover) that consists of removing a subset of vertices of minimum size such that the maximum weight of independent sets (minimum weight of vertex covers, respectively) in the remaining graph is at most a specified value. We show that these problems are NP-hard on bipartite graphs but polynomial-time solvable on unweighted bipartite graphs. Furthermore, these problems are polynomial also on cographs and graphs of bounded treewidth. Results on the non-existence of ptas are presented, too. Given a directed graph G=(V,A) with a non-negative weight (length) function on its arcs w:A→ℝ+ and two terminals s,t∈V, our goal is to destroy all short directed paths from s to t in G by eliminating some arcs of A. This is known as the short paths interdiction problem. We consider several versions of it, and in each case analyze two subcases: total limited interdiction, when a fixed number k of arcs can be removed, and node-wise limited interdiction, when for each node v∈V a fixed number k(v) of out-going arcs can be removed. Our results indicate that the latter subcase is always easier than the former one. In particular, we show that the short paths node-wise interdiction problem can be efficiently solved by an extension of Dijkstra’s algorithm. In contrast, the short paths total interdiction problem is known to be NP-hard. We strengthen this hardness result by deriving the following inapproximability bounds: Given k, it is NP-hard to approximate within a factor c<2 the maximum s–t distance d(s,t) obtainable by removing (at most) k arcs from G. Furthermore, given d, it is NP-hard to approximate within a factor @math the minimum number of arcs which has to be removed to guarantee d(s,t)≥d. Finally, we also show that the same inapproximability bounds hold for undirected graphs and or node elimination. In the Packing Interdiction problem we are given a packing LP together with a separate interdiction cost for each LP variable and a global interdiction budget. Our goal is to harm the LP: which variables should we forbid the LP from using (subject to forbidding variables of total interdiction cost at most the budget) in order to minimize the value of the resulting LP? Interdiction problems on graphs (interdicting the maximum flow, the shortest path, the minimum spanning tree, etc.) have been considered before; here we initiate a study of interdicting packing linear programs. Zenklusen showed that matching interdiction, a special case, is NP-hard and gave a 4-approximation for unit edge weights. We obtain an constant-factor approximation to the matching interdiction problem without the unit weight assumption. This is a corollary of our main result, an O(logq · min q, logk )-approximation to Packing Interdiction where q is the row-sparsity of the packing LP and k is the column-sparsity.