Efficient determination of the k most vital edges for the minimum spanning tree problem

Abstract

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.

An O(1)-Approximation for Minimum Spanning Tree Interdiction

“…However, this does not lead to improvements for currently fastest deterministic algorithms for the most vital edge problem because no deterministic method is known to find an MST faster than in O(m · α(m, n)) time. Several exponential-time algorithms have been suggested for the k most vital edges problem for general k, including parallel algorithms [27,26,4]. The problem has also been considered under the aspect of parameterized complexity [17].…”

“…Notice that(4) can also be interpreted as a Lagrangean dual of min{− val(R) | R ⊆ E, c(R) ≤ B}. We focus on the Pareto front interpretation since it is natural for properties we want to highlight later.…”

Minimum edge blocker dominating set problem

“…The minimum vertex/edge blocker and the most vital vertices/ edges problems have been studied in literature with respect to different graph properties, such as connectivity (Addis, Di Summa, & Grosso, 2013;Arulselvan, Commander, Elefteriadou, & Pardalos, 2009;Di Summa, Grosso, & Locatelli, 2011;Shen, Smith, & Goli, 2012;Veremyev, Prokopyev, & Pasiliao, 2014), shortest path (Bar-Noy et al, 1995;Israeli & Wood, 2002;Khachiyan et al 2008), maximum flow (Altner, Ergun, & Uhan, 2010;Ghare, Montgomery, & Turner, 1971;Wollmer, 1964;Wood, 1993), spanning tree (Bazgan, Toubaline, & Vanderpooten, 2012;2013;Frederickson & Solis-Oba, 1996), assignment (Bazgan, Toubaline, & Vanderpooten, 2010b), 1-median (Bazgan et al, 2010a), 1-center (Bazgan et al, 2010a), matching (Ries et al, 2010;Zenklusen, 2010;Zenklusen et al, 2009), independent sets (Bazgan et al, 2011), vertex covers (Bazgan et al, 2011), andcliques (Mahdavi Pajouh, Boginski, &.…”

A dynamic reformulation heuristic for Generalized Interdiction Problems

“…When the follower problem in an interdiction problem can be formulated as linear program, duality-based reformulations give exact algorithms. This idea has been used, e.g., in [17] for the maximum flow problem, in [16] for multi-commodity flow problems, and in [20] for the spanning tree problems. The shortest-path variant for a problem where interdiction does not forbid the use of an item but causes a penalty in the follower objective is studied in [15], where a duality-based exact algorithm is given.…”