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.
Many interdiction problems beyond the minimum spanning tree setting have been studied. This includes interdiction versions of the maximum @math - @math flow problem @cite_7 @cite_3 @cite_14 (a setting often called ), the shortest path problem @cite_26 @cite_5 , the maximum matching problem @cite_21 @cite_31 , interdicting the connectivity of a graph @cite_37 , interdiction of packings @cite_31 , stable set interdiction @cite_24 , and variants of facility location @cite_38 . However, the theoretical understanding of most interdiction problems still seems rather limited. A good example for which a large gap remains between the best known hardness results and approximation algorithms is network flow interdiction. Network flow interdiction is a strongly NP-hard problem @cite_7 for which no approximation results are known, except for a pseudo-approximation @cite_22 which is allowed to violate the budget by a factor of @math .
@cite_38 @cite_37 @cite_14 @cite_26 @cite_22 @cite_7 @cite_21 @cite_3 @cite_24 @cite_5 @cite_31
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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. 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. 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. A network interdiction problem usually involves two players who compete in a min‐max or max‐min game. One player, the network owner, tries to optimize its objective over the network, for example, as measured by a shortest path, maximum flow, or minimum cost flow. The opposing player, called the interdictor, alters the owner’s network to maximally impair the owner’s objective (e.g., by destroying arcs that maximize the owner’s shortest path). This chapter