Critical edges/nodes for the minimum spanning tree problem: complexity and approximation

Bazgan, Cristina; Toubaline, Sonia; Vanderpooten, Daniel

doi:10.1007/s10878-011-9449-4

Cited by 21 publications

(8 citation statements)

References 15 publications

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“…The smaller distance sum is, the more important the node may be. Figure 5 (c) also shows that the in-degree of the nodes 11,12,13,14,15,16,17,18 and 19 is equal to zero, but out-degree of these nodes is greater than 0. There are also some node with out-degree 0 in Fig.…”

Section: Case Studymentioning

confidence: 92%

“…The concept of critical nodes also was regarded as middlemen and was extended to directed networks [17]. Modeling the network as a weighted connected graph, an applicable method of identifying critical nodes and edges was proposed to find a subset in which nodes and edges are removed to cause the largest cost [18]. Wehmuth et al [19] proposed a methodology to locate the most critical nodes in terms of network robustness in a fully distributed way.…”

Section: Introductionmentioning

confidence: 99%

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Critical Nodes Identification of Power Grids Based on Network Efficiency

Kang

Zhu

Zhang

et al. 2018

IEICE Trans. Inf. & Syst.

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Critical nodes identification is of great significance in protecting power grids. Network efficiency can be used as an evaluation index to identify the critical nodes and is an indicator to quantify how efficiently a network exchanges information and transmits energy. Since power grid is a heterogeneous network and can be decomposed into small functionallyindependent grids, the concept of the Giant Component does not apply to power grids. In this paper, we first model the power grid as the directed graph and define the Giant Efficiency sub-Graph (GEsG). The GEsG is the functionally-independent unit of the network where electric energy can be transmitted from a generation node (i.e., power plants) to some demand nodes (i.e., transmission stations and distribution stations) via the shortest path. Secondly, we propose an algorithm to evaluate the importance of nodes by calculating their critical degree, results of which can be used to identify critical nodes in heterogeneous networks. Thirdly, we define node efficiency loss to verify the accuracy of critical nodes identification (CNI) algorithm and compare the results that GEsG and Giant Component are separately used as assessment criteria for computing the node efficiency loss. Experiments prove the accuracy and efficiency of our CNI algorithm and show that the GEsG can better reflect heterogeneous characteristics and power transmission of power grids than the Giant Component. Our investigation leads to a counterintuitive finding that the most important critical nodes may not be the generation nodes but some demand nodes.

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Section: Case Studymentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Critical Nodes Identification of Power Grids Based on Network Efficiency

Kang

Zhu

Zhang

et al. 2018

IEICE Trans. Inf. & Syst.

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show abstract

“…Due to the immediate practical relevance, there are numerous studies on "most vital edges (and vertices)" and related problems. We focus on shortest paths here, while there are also studies for problems such as Minimum Spanning Tree [3,4,11,15,20] or Maximum Flow [15,25,27], to mention only two. With respect to shortest path computation, the following is known.…”

Section: Shortest Path Most Vital Edges (Sp-mve)mentioning

confidence: 99%

A Refined Complexity Analysis of Finding the Most Vital Edges for Undirected Shortest Paths

Bazgan

Nichterlein

Niedermeier

2015

Lecture Notes in Computer Science

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Abstract. We study the NP-hard Shortest Path Most Vital Edges problem arising in the context of analyzing network robustness. For an undirected graph with positive integer edge lengths and two designated vertices s and t, the goal is to delete as few edges as possible in order to increase the length of the (new) shortest st-path as much as possible. This scenario has been mostly studied from the viewpoint of approximation algorithms and heuristics, while we particularly introduce a parameterized and multivariate point of view. We derive refined tractability as well as hardness results, and identify numerous directions for future research. Among other things, we show that increasing the shortest path length by at least one is much easier than to increase it by at least two.

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“…To complete this overview of the various tracks allowing relevant results on the subject, we have also been interested in variations of the problem, inspired by the work on the minimum spanning tree and other classical combinatorial optimization problems [2,3,4]. These variants aim to detect critical subsets of edges or nodes in the graph, which can be used to detect a skeleton that we do not further question, and decrease the time consumption of an exact search on remaining edges.…”

Section: Introductionmentioning

confidence: 99%

Scaffolding Problems Revisited: Complexity, Approximation and Fixed Parameter Tractable Algorithms, and Some Special Cases

et al. 2018

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This paper is devoted to new results about the scaffolding problem, an integral problem of genome inference in bioinformatics. The problem consists in finding a collection of disjoint cycles and paths covering a particular graph called the "scaffold graph". We examine the difficulty and the approximability of the scaffolding problem in special classes of graphs, either close to trees, or very dense. We propose negative and positive results, exploring the frontier between difficulty and tractability of computing and/or approximating a solution to the problem. Also, we explore a new direction through related problems consisting in finding a family of edges having a strong effect on solution weight. Corollary 2. Let G be a class of graphs such that, for each bipartite graph G there is a supergraph of G in G. Scaffolding is N P-complete on G, even if σ p = 0 and σ c = 1 and ω max = 1. Construction 1 also implies subexponential lower bounds for our problems based on the widely believed complexitytheoretic hypothesis known as the "Exponential-Time Hypothesis" 1 (ETH, see [18, 24]). In fact, the lower bound is established directly from the fact that (planar) Directed Hamiltonian Cycle does not admit a O(2 o(|E(G)|))-time algorithm [19, Theorem 3.5] and that Construction 1 only linearly blows up the instance size. Corollary 3. Let G be a class of graphs such that, for each bipartite graph G there is a supergraph of G in G. Assuming ETH, there is no 2 o(|E(G)|)-time algorithm for Scaffolding on G, even if σ p = 0 and σ c = 1 and ω max = 1. 3.1.3 Weighted cases in sparse graphs The hardness of Scaffolding for dense graphs proved by Theorem 2 motivates the search for tractable cases among classes of sparse graphs. It is known that Scaffolding is polynomial-time solvable on graphs that are close to being a forest (constant treewidth) [22], so we consider a different sparsity measure here. We investigate whether Scaffolding becomes polynomial-time solvable if the result of removing the given perfect matching M * from G forms a forest. We call this class of graphs "quasi forests". Remark that real scaffold graphs are not always quasi-forest, however this is a first step towards their structure. We start off by modifying Construction 1 to make the resulting graph a quasi tree (see Construction 3 and Figure 3). Unfortunately, this requires fixing the length of the sought Hamiltonian cycle. To circumvent this, we present another construction, reducing the N P-complete Weighted 2-SAT to Scaffolding, that does not require fixing the lengths.

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Critical edges/nodes for the minimum spanning tree problem: complexity and approximation

Cited by 21 publications

References 15 publications

Critical Nodes Identification of Power Grids Based on Network Efficiency

Critical Nodes Identification of Power Grids Based on Network Efficiency

A Refined Complexity Analysis of Finding the Most Vital Edges for Undirected Shortest Paths

Scaffolding Problems Revisited: Complexity, Approximation and Fixed Parameter Tractable Algorithms, and Some Special Cases

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