The k most vital arcs in the shortest path problem

Malik, Kavindra; Mittal, Ashok; Gupta, Sanjeev

doi:10.1016/0167-6377(89)90065-5

Cited by 134 publications

(50 citation statements)

References 5 publications

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“…The detour path is defined as the shortest path from v i to t with the removal of edge e(v i , v j ). Nardelli et al (1998) redefined the vital edge of a shortest path with respect to the longest detour path and gave an algorithm for the new problem with the same time and space complexity as Malik et al (1989). Nardelli et al (2001) also improved the result to O(m · α(m, n)).…”

Section: Introductionmentioning

confidence: 85%

“…e i on shortest path Nardelli et al (1998). It is proved that R P(i, j) = min Malik et al (1989), where A i, j ← I ≤i, j≤J For any P S i , there exists no P S i satisfying the following conditions: R i = R i , De 1 (i, i + 1) < De 1 (i, i + 1) and De 2 (i, i + 1) = De 2 (i, i + 1), and the corresponding path P i passing node v a (as in Fig. 6).…”

Section: Lemmamentioning

confidence: 98%

“…The replacement path is the shortest path between the source s and the destination t after some edge is removed. Malik et al (1989) presented an O(m + n log n)-time algorithm to find the most vital edge whose removal results in the longest replacement path. An O(m · α(m, n))-time algorithm was given for the problem by Nardelli et al (2001), where α(m, n) is the inverse Ackermann function.…”

Section: Introductionmentioning

confidence: 99%

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Optimal shortest path set problem in undirected graphs

Zhang

Wen

2014

J Comb Optim

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This paper proposes the optimal shortest path set problem in an undirected graph G = (V, E), in which some vehicles have to go from a source node s to a destination node t. However, at most k edges in the graph may be blocked during the traveling. The goal is to find a minimum collection of paths for the vehicles before they start off to assure the fastest arrival of at least one vehicle, no matter which l (0 ≤ l ≤ k) edges are blocked. We consider two scenarios for this problem. In the first scenario with k = 1, we propose the concept of common replacement path and design the Least-Overlap Algorithm to find the common replacement path. Based on this, an algorithm to compute the optimal shortest path set is presented and its time complexity is proved to be O(n 2 ). In the second scenario with k > 1, we consider the case where the blocked edges are consecutive ones on a shortest path from s to t and the vertices connecting two blocked edges are also blocked (i.e., routes passing through these vertices are not allowed), and an algorithm is presented to compute the optimal shortest path set in this scenario with time complexity O(mn + k 2 n 2 log n).

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Section: Introductionmentioning

confidence: 85%

Section: Lemmamentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Optimal shortest path set problem in undirected graphs

Zhang

Wen

2014

J Comb Optim

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“…The problem of finding all the replacement shortest paths, one for each edge of P G (s, z), has been efficiently solved in O(m + n log n) time on a pointer machine [11], and in O(m α(m, n)) time on a word RAM [12], respectively. Both algorithms are based on a pre-computation of the SPTs S G (s) and S G (z).…”

Section: The Single-edge Casementioning

confidence: 99%

“…As soon as, quite naturally, one assumes that the agent's valuation function is proportional to the owned edge length (obviously once that the edge is part of the solution), then this problem enjoys the property of being utilitarian, in that the quality of any feasible output can be measured by simply summing up all the agents' valuations. For utilitarian problems, there exists a well-known class of truthful mechanisms, i.e., the Vickrey-Clarke-Groves (VCG) mechanisms [5,8,21], and therefore the shortest path problem can be solved efficiently (in terms of runtime for computing the output specification and the payments to the agents) in O(m + n log n) time [10,11]. In contrast with VCG-mechanisms, which handle arbitrary valuation functions but only utilitarian problems, Archer and Tardos [1] defined another class of truthful mechanisms, named one-parameter mechanisms, allowing to solve general (i.e., non-utilitarian) problems, but with the restriction that the data input held by each agent must be expressed by a single parameter.…”

Section: Introductionmentioning

confidence: 99%

Exact and Approximate Truthful Mechanisms for the Shortest Paths Tree Problem

Gualà

Proietti

2007

Algorithmica

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Let a communication network be modeled by an undirected graph G=(V,E) of n nodes and m edges, and assume that edges are controlled by selfish agents. In this paper we analyze the problem of designing a truthful mechanism for computing one of the most popular structures in communication networks, i.e., the single-source shortest paths tree.\ud \ud More precisely, we will study several realistic scenarios, in which each agent can own either a single or multiple edges of G. In particular, for the single-edge case, we will show that: (i) in the classic utilitarian case, the problem can be solved efficiently in O(mnlogα(m,n)) time, where ±(m,n) is the inverse of the Ackermanns function; (ii) in a meaningful non-utilitarian case, namely that in which agents valuation functions only depend on the edge lengths, the problem can be solved in O(m+nlogn) time. Conversely, for the multiple-edges case, we will show in the utilitarian case an O(mP+nPlogn) time truthful mechanism, where P=O(n) denotes the number of agents participating in the solution, while in the same non-utilitarian case we will prove a general lower bound to the approximation ratio that can be achieved by any truthful mechanism, by showing that no c-approximate mechanism can exist, for any fixed c < 5+√13 / 3+√13

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Stochastic Network Interdiction

Morton

2011

Wiley Encyclopedia of Operations Research and Management Science

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We describe discrete stochastic network interdiction of shortest‐path, and maximum‐reliability‐path, networks. A nested “max–min” model is reformulated using duality and, if necessary, an equivalent penalty‐based model. The resulting mixed‐integer program is then amenable to solution by standard solvers. Special‐purpose decomposition algorithms, and accompanying valid inequalities, are also developed, which permit solution of larger scale problem instances. Throughout the article, we point to important models that arise when key assumptions in our formulations are modified.

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The k most vital arcs in the shortest path problem

Cited by 134 publications

References 5 publications

Optimal shortest path set problem in undirected graphs

Optimal shortest path set problem in undirected graphs

Exact and Approximate Truthful Mechanisms for the Shortest Paths Tree Problem

Stochastic Network Interdiction

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