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43
On the Difficulty of Some Shortest Path Problems
, 2003
"... We prove superlinear lower bounds for some shortest path problems in directed graphs, where no such bounds were previously known. The central problem in our study is the replacement paths problem: Given a directed graph G with nonnegative edge weights, and a shortest path P = {e_1, e_2, ..., e_p} ..."
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Cited by 34 (8 self)
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We prove superlinear lower bounds for some shortest path problems in directed graphs, where no such bounds were previously known. The central problem in our study is the replacement paths problem: Given a directed graph G with nonnegative edge weights, and a shortest path P = {e_1, e_2, ..., e_p} between two nodes s and t, compute the shortest path distances from s to t in each of the p graphs obtained from G by deleting one of the edges e_i. We show that the replacement paths problem requires &Omega;(m&radic;n) time in the worst case whenever m = O(n&radic;n). This also establishes a similar...
Increasing the Weight of Minimum Spanning Trees
, 1996
"... The problems of computing the maximum increase in the weight of the minimum spanning trees of a graph caused by the removal of a given number of edges, or by finite increases in the weights of the edges, are investigated. For the case of edge removals, the problem is shown to be NPhard and an \Omeg ..."
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Cited by 28 (1 self)
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The problems of computing the maximum increase in the weight of the minimum spanning trees of a graph caused by the removal of a given number of edges, or by finite increases in the weights of the edges, are investigated. For the case of edge removals, the problem is shown to be NPhard and an \Omega\Gamma/ = log k)approximation algorithm is presented for it, where k is the number of edges to be removed. The second problem is studied assuming that the increase in the weight of an edge has an associated cost proportional to the magnitude of the change. An O(n 3 m 2 log(n 2 =m)) time algorithm is presented to solve it. 1 Introduction Consider a communication network in which information is broadcast over a minimum spanning tree. There are applications for which it is important to determine the maximum degradation in the performance of the broadcasting protocol that can be expected as a result of traffic fluctuations and link failures [25]. Also, there are several combinatorial op...
The Complexity of Finding Most Vital Arcs and Nodes
, 1995
"... Let G(V; E) be a graph (either directed or undirected) with a nonnegative length `(e) associated with each arc e in E. For two specified nodes s and t in V , the k most vital arcs (or nodes) are those k arcs (nodes) whose removal maximizes the increase in the length of the shortest path from s to ..."
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Cited by 27 (0 self)
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Let G(V; E) be a graph (either directed or undirected) with a nonnegative length `(e) associated with each arc e in E. For two specified nodes s and t in V , the k most vital arcs (or nodes) are those k arcs (nodes) whose removal maximizes the increase in the length of the shortest path from s to t. We prove that finding the k most vital arcs (or nodes) is NPhard, even when all arcs have unit length. We also correct some errors in an earlier paper by Malik, Mittal and Gupta [ORL 8:223227, 1989]. Keywords: networks, graphs, NPComplete, vital arcs. 1. Introduction The Most Vital Arcs Problem (MVAP) is defined as follows. Input: A graph G = (V; E) (either directed or undirected) with a nonnegative length `(e) associated with each arc e in E, two specified nodes s and t in V , and a positive integer k. Research supported by NSF Research Initiation Award CCR9307462 and an NSF CAREER Award CCR9501355. Output: A set of k arcs whose removal maximizes the increase in the length ...
Finding the most vital node of a shortest path
 In Proc. COCOON
, 2001
"... Abstract. In an undirected, 2node connected graph G =(V,E) with positive real edge lengths, the distance between anytwo nodes r and s is the length of a shortest path between r and s in G. The removal of a node and its incident edges from G mayincrease the distance from r to s. Amost vital node of ..."
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Cited by 18 (1 self)
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Abstract. In an undirected, 2node connected graph G =(V,E) with positive real edge lengths, the distance between anytwo nodes r and s is the length of a shortest path between r and s in G. The removal of a node and its incident edges from G mayincrease the distance from r to s. Amost vital node of a given shortest path from r to s is a node (other than r and s) whose removal from G results in the largest increase of the distance from r to s. In the past, the problem of finding a most vital node of a given shortest path has been studied because of its implications in network management, where it is important to know in advance which component failure will affect network efficiencythe most. In this paper, we show that this problem can be solved in O(m + n log n) time and O(m) space, where m and n denote the number of edges and the number of nodes in G. 1
Finding the k Most Vital Edges with Respect to Minimum Spanning Tree
 Acta Informatica
, 1995
"... For a connected, undirected and weighted graph G = (V; E), the problem of finding the k most vital edges of G with respect to minimumspanning tree is to find k edges in G whose removal will cause greatest weight increase in the minimum spanning tree of the remaining graph. This problem is known to ..."
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Cited by 17 (0 self)
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For a connected, undirected and weighted graph G = (V; E), the problem of finding the k most vital edges of G with respect to minimumspanning tree is to find k edges in G whose removal will cause greatest weight increase in the minimum spanning tree of the remaining graph. This problem is known to be NP hard for arbitrary k. In this paper, we first describe a simple exact algorithm for this problem, based on the approach of edge replacement in the minimum spanning tree of G. Next we present polynomialtime randomized algorithms that produce optimal and approximate solutions to this problem. For jV j = n and jEj = m, our algorithm producing optimal solution has a time complexity of O(mn) with probability of success at least e \Gamma k 2 2(m\Gamman\Gamma1) \Gamma 2 log c k k\Gamma4 , c = 1 + 1 2 k=2 , and the algorithm producing approximate solution runs in time O(mn + nk 2 log k) and yields results within factor 2 to the optimal one. Finally we show that both of our randomize...
On short paths interdiction problems : total and nodewise limited interdiction
 Theory of Computing Systems
"... nodewise limited interdiction. 1 by ..."
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Finding All the Best Swaps of a Minimum Diameter Spanning Tree Under Transient Edge Failures
 Journal of Graph Algorithms and Applications
, 1998
"... Abstract. In network communication systems, frequently messages are routed along a minimum diameter spanning tree (MDST) of the network, to minimize the maximum delay in delivering a message. When a transient edge failure occurs, it is important to choose a temporary replacement edge which minimizes ..."
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Cited by 14 (6 self)
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Abstract. In network communication systems, frequently messages are routed along a minimum diameter spanning tree (MDST) of the network, to minimize the maximum delay in delivering a message. When a transient edge failure occurs, it is important to choose a temporary replacement edge which minimizes the diameter of the new spanning tree. Such an optimal replacement is called the best swap. As a natural extension, the allbestswaps (ABS) problem is the problem of finding the best swap for every edge of the MDST. Given a weighted graph G =(V, E), where V  = n and E  = m,wesolvetheABSprobleminO(n √ m)time and O(m + n) space, thus improving previous bounds for m = o(n 2). 1
A nearly optimal algorithm for approximating replacement paths and k shortest simple paths in general graphs
 In Proc. SODA
, 2010
"... Let G = (V, E) be a directed graph with positive edge weights, let s, t be two specified vertices in this graph, and let π(s, t) be the shortest path between them. In the replacement paths problem we want to compute, for every edge e on π(s, t), the shortest path from s to t that avoids e. The naive ..."
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Let G = (V, E) be a directed graph with positive edge weights, let s, t be two specified vertices in this graph, and let π(s, t) be the shortest path between them. In the replacement paths problem we want to compute, for every edge e on π(s, t), the shortest path from s to t that avoids e. The naive solution to this problem would be to remove each edge e, one at a time, and compute the shortest s − t path each time; this yields a running time of O(mn + n 2 log n). Gotthilf and Lewenstein [8] recently improved this to O(mn+n 2 log log n), but no o(mn) algorithms are known. We present the first approximation algorithm for replacement paths in directed graphs with positive edge weights. Given any ɛ ∈ [0, 1), our algorithm returns (1 + ɛ)approximate replacement paths in O(ɛ −1 log 2 n log(nC/c)(m+n log n)) = Õ(m log(nC/c)/ɛ) time, where C is the largest edge weight in the graph and c is the smallest weight. We also present an even faster (1 + ɛ) approximate algorithm for the simpler problem of approximating the k shortest simple s − t paths in a directed graph with positive edge weights. That is, our algorithm outputs k different simple s−t paths, where the kth path we output is a (1 + ɛ) approximation to the actual kth shortest simple s − t path. The running time of our algorithm is O(kɛ −1 log 2 n(m + n log n)) = Õ(km/ɛ). The fastest exact algorithm for this problem has a running time of O(k(mn+n 2 log log n)) = Õ(kmn) [8]. The previous best approximation algorithm was developed by Roditty [15]; it has a stretch of 3/2 and a running time of Õ(km√n) (it does not work for replacement paths). Note that all of our running times are nearly optimal except for the O(log(nC/c)) factor in the replacements paths algorithm. Also, our algorithm can solve the variant of approximate replacement paths where we avoid vertices instead of edges. 1
Dualfailure distance and connectivity oracles
 In Proc. of the 20th ACMSIAM Symposium On Discrete Algorithms (SODA
, 2009
"... Spontaneous failure is an unavoidable aspect of all networks, particularly those with a physical basis such as communications networks or road networks. Whether due to malicious coordinated attacks or other causes, failures temporarily change the topology of the network and, as a consequence, its co ..."
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Spontaneous failure is an unavoidable aspect of all networks, particularly those with a physical basis such as communications networks or road networks. Whether due to malicious coordinated attacks or other causes, failures temporarily change the topology of the network and, as a consequence, its connectivity and distance metric. In this paper we look at the problem of efficiently answering connectivity, distance, and shortest route queries in the presence of two node or link failures. Our data structure uses Õ(n2) space and answers queries in Õ(1) time, which is within a polylogarithmic factor of optimal and nearly matches the singlefailure distance oracles of Demestrescu et al. It may yet be possible to find distance/connectivity oracles capable of handling any fixed number of failures. However, the sheer complexity of our algorithm suggests that moving beyond dualfailures will require a fundamentally different approach to the problem. 1