An essential step towards gaining a deeper insight into intricate mechanisms underlying the formation and functioning of complex networks is extracting and understanding their building blocks encoded in the clustering...
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An essential step towards gaining a deeper insight into intricate mechanisms underlying the formation and functioning of complex networks is extracting and understanding their building blocks encoded in the clustering structure. At its core, the problem of partitioning vertices into clusters may be regarded as a dual problem to vertex colouring and, as such, permitted us to leverage the Petford-Welsh colouring algorithm to devise a highly scalable decentralised heuristic approach to cluster detection. As long as the graph under scrutiny admits a fairly well-defined clustering structure per se, the modified Petford-Welsh algorithm tends to perform on a par with or even surpasses existing techniques.
In this paper, we investigate the computational complexity of subgraph reconfiguration problems in directed graphs. More specifically, we focus on the problem of reconfiguring arborescences in a digraph, where an arbo...
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In this paper, we investigate the computational complexity of subgraph reconfiguration problems in directed graphs. More specifically, we focus on the problem of reconfiguring arborescences in a digraph, where an arborescence is a directed graph such that its underlying undirected graph forms a tree and all vertices have in-degree at most 1. Given two arborescences in a digraph, the goal of the problem is to determine whether there is a (reconfiguration) sequence of arborescences between the given arborescences such that each arborescence in the sequence can be obtained from the previous one by removing an arc and then adding another arc. We show that this problem can be solved in polynomial time, whereas the problem is PSPACE-complete when we restrict arborescences in a reconfiguration sequence to directed paths or relax to directed acyclic graphs. We also show that there is a polynomial-time algorithm for finding a shortest reconfiguration sequence between two spanning arborescences. (c) 2022 Elsevier B.V. All rights reserved.
The problem of finding the minimum cut of an undirected unweighted graph is studied on the external memory model. First, a lower bound of Omega((E/V)Sort(V)) on the number of I/Os is shown for the problem, where V is ...
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The problem of finding the minimum cut of an undirected unweighted graph is studied on the external memory model. First, a lower bound of Omega((E/V)Sort(V)) on the number of I/Os is shown for the problem, where V is the number of vertices and E is the number of edges. Then the following are presented, (1) a deterministic minimum cut algorithm that uses O(min{c(7) (log(3+is an element of) E)(MST(V, E)), c(log E)(MST(V, E)) + (V /root M) Sort(V)}) I/Os;here is an element of > 0 is a small constant, MST(V, E) is the number of I/Os needed to compute a minimum spanning tree of the graph, and c is the value of the minimum cut. The algorithm performs better on dense graphs than the algorithm of [1], which requires O(E + c(2)V log(V/c)) I/Os, when executed on the external memory model. For a delta-fat graph (for delta > 0, the maximum tree packing of the graph is at least (I + delta)c/2), our algorithm computes a minimum cut in O(c(log E)MST(V, E)) I/Os. (2) A randomized algorithm that computes minimum cut with high probability in O(c(log E).MST(V, E) Sort(E) log(2) V + V/B Sort(V) log V) I/Os. (3) A (2 + is an element of)-minimum cut algorithm that requires O((E/V) MST(V, E)) I/Os. (C) 2014 Elsevier B.V. All rights reserved.
This paper is concerned with applying bandwidth and profile reduction reordering algorithms prior to computing an incomplete Cholesky factorization and using this as a preconditioner for the conjugate gradient method....
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This paper is concerned with applying bandwidth and profile reduction reordering algorithms prior to computing an incomplete Cholesky factorization and using this as a preconditioner for the conjugate gradient method. Hundreds of reordering algorithms have been proposed to solve the problems of bandwidth and profile reductions since the mid-1960s. In previous publications, a large range of heuristics for bandwidth and/or profile reductions was reviewed. Based on this experience, 13 heuristics were selected as the most promising methods. These are evaluated in this paper along with a variant of the breadth-first search procedure that is proposed. Numerical results confirm the effectiveness of this modified reordering algorithm for linear systems derived from specific application areas. Moreover, the most promising heuristics for several application areas are identified when reducing the computational cost of the incomplete Cholesky-conjugate gradient method.
For a graph G = (V, E) with no isolated vertices, a set D subset of V is called a semipaired dominating set of G if (i) D is a dominating set of G, and (ii) D can be partitioned into two element subsets such that the ...
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For a graph G = (V, E) with no isolated vertices, a set D subset of V is called a semipaired dominating set of G if (i) D is a dominating set of G, and (ii) D can be partitioned into two element subsets such that the vertices in each two element set are at distance at most two. The minimum cardinality of a semipaired dominating set of G is called the semipaired domination number of G, and is denoted by gamma(pr2)(G). TheMINIMUM SEMIPAIRED DOMINATION problem is to find a semipaired dominating set of G of cardinality gamma(pr2)(G). In this paper, we initiate the algorithmic study of the MINIMUM SEMIPAIRED DOMINATION problem. We show that the decision version of the MINIMUM SEMIPAIRED DOMINATION problem is NP-complete for bipartite graphs and chordal graphs. On the positive side, we present a linear-time algorithm to compute a minimum cardinality semipaired dominating set of interval graphs. We also propose a 1 + ln(2 Delta + 2)-approximation algorithm for the MINIMUM SEMIPAIRED DOMINATION problem, where Delta denotes the maximum degree of the graph and show that the MINIMUM SEMIPAIRED DOMINATION problem cannot be approximated within (1 - epsilon) ln vertical bar V vertical bar for any epsilon > 0 unless P=NP.
The linear-width of a graph G is defined to be the smallest integer k such that the edges of G can be arranged in a linear ordering (e(1),...,e(r)) in such a way that for every i = 1,..., r - 1, there are at most k ve...
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The linear-width of a graph G is defined to be the smallest integer k such that the edges of G can be arranged in a linear ordering (e(1),...,e(r)) in such a way that for every i = 1,..., r - 1, there are at most k vertices incident to edges that belong both to {e(1),...,e(i)} and to {e(i+1),...,e(r)}. In this paper, we give a set of 57 graphs and prove that it is the set of the minimal forbidden miners for the class of graphs with linear-width at most two. Our proof also gives a linear time algorithm that either reports that a given graph has linear-width more than two or outputs an edge ordering of minimum linear-width. We further prove a structural connection between linear-width and the mixed search number which enables us to determine, for any k greater than or equal to 1, the set of acyclic forbidden miners for the class of graphs with linear-width less than or equal to k. Moreover, due to this connection, our algorithm can be transfered to two linear time algorithms that check whether a graph has mixed search or edge search number at most two and, if so, construct the corresponding sequences of search moves. (C) 2000 Elsevier Science B.V. All rights reserved.
As graphs grow in size, many real-world graphs are difficult to load into the primary memory of a computer. Thus, computing depth-first search (DFS) results (i.e., depth-first order or DFS-Tree) on the semi-external m...
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As graphs grow in size, many real-world graphs are difficult to load into the primary memory of a computer. Thus, computing depth-first search (DFS) results (i.e., depth-first order or DFS-Tree) on the semi-external memory model is important to investigate. Semi external algorithms assume that the primary memory can at least hold a spanning tree T of a graph G and gradually restructure T into a DFS-Tree, which is nontrivial. In this paper, we present a comprehensive study for the semi-external DFS problem. Based on a theoretical analysis of this problem, we introduce a new semi-external DFS algorithm called EPDFS with a lightweight index N thorn -index. Unlike traditional algorithms, we focus on addressing such a complex problem efficiently with fewer I/Os, simpler CPU calculations (implementation-friendly), and less random I/O accesses (key-to-efficiency). Extensive experimental evaluations are performed on both synthetic and real graphs, and experimental results confirm that the proposed EP-DFS algorithm markedly outperforms existing algorithms. (c) 2022 Elsevier Inc. All rights reserved.
Given a directed graph G and a pair of nodes s and t, an s-t bridge of G is an edge whose removal breaks all s-t paths of G (and thus appears in all s-t paths). Computing all s-t bridges of G is a basic graph problem,...
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Given a directed graph G and a pair of nodes s and t, an s-t bridge of G is an edge whose removal breaks all s-t paths of G (and thus appears in all s-t paths). Computing all s-t bridges of G is a basic graph problem, solvable in linear time. In this paper, we consider a natural generalisation of this problem, with the notion of "safety" from bioinformatics. We say that a walk W is safe with respect to a set W of s-t walks, if W is a subwalk of all walks in W. We start by considering the maximal safe walks when W consists of: all s-t paths, all s-t trails, or all s-t walks of G. We show that the solutions for the first two problems immediately follow from finding all s-t bridges after incorporating simple characterisations. However, solving the third problem requires non-trivial techniques for incorporating its characterisation. In particular, we show that there exists a compact representation computable in linear time, that allows outputting all maximal safe walks in time linear in their length. Our solutions also directly extend to multigraphs, except for the second problem, which requires a more involved approach. We further generalise these problems, by assuming that safety is defined only with respect to a subset of visible edges. Here we prove a dichotomy between the s-t paths and s-t trails cases, and the s-t walks case: the former two are NP-hard, while the latter is solvable with the same complexity as when all edges are visible. We also show that the same complexity results hold for the analogous generalisations of s-t articulation points (nodes appearing in all s-t paths). We thus obtain the best possible results for natural "safety"-generalisations of these two fundamental graph problems. Moreover, our algorithms are simple and do not employ any complex data structures, making them ideal for use in practice.
The Phylogenetic kth Root Problem (PRk) is the problem of finding a (phylogenetic) tree T from a given graph G = (V, E) such that (1) T has no degree-2 internal nodes, (2) the external nodes (i.e., leaves) of T are ex...
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The Phylogenetic kth Root Problem (PRk) is the problem of finding a (phylogenetic) tree T from a given graph G = (V, E) such that (1) T has no degree-2 internal nodes, (2) the external nodes (i.e., leaves) of T are exactly the elements of V, and (3) (u, v) is an element of E if and only if the distance between it and v in tree T is at most k, where k is some fixed threshold k. Such a tree T, if exists. is called a phylogenetic kth root of graph G. The computational complexity of PRk is open, except for k <= 4. Recently, Chen et al. investigated PRk under a natural restriction that the maximum degree of the phylogenetic root is bounded from above by a constant. They presented a linear-time algorithm that determines if a given connected G has such a phylogenetic kth root, and if so. demonstrates one. In this paper, we supplement their work by presenting a linear-time algorithm for disconnected graphs. (c) 2005 Elsevier Inc. All rights reserved.
In this brief, we study first- and second-order consensus algorithms for the scale-free small-world Koch network, where vertices are subject to white noise. We focus on three cases of consensus schemes: 1) first-order...
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In this brief, we study first- and second-order consensus algorithms for the scale-free small-world Koch network, where vertices are subject to white noise. We focus on three cases of consensus schemes: 1) first-order leaderless algorithm;2) first-order algorithm with a single leader;and 3) second-order leaderless algorithm. We are concerned with the coherence of the Koch network in the H-2 norm, which captures the level of agreement of vertices in face of stochastic disturbances. Based on the particular network construction, we derive explicit expressions of the coherence for all the three consensus algorithms, as well as their dependence on the network size. Particularly, for the first-order leader-follower model, we show that coherence relies on the shortest-path distance between the leader and the largest-degree vertices, as well as the degree of the leader. The asymptotic behaviors for coherence of the three consensus algorithms in Koch network behave differently from those associated with other networks lacking scale-free small-world features, indicating significant influences of the scale-free small-world topology on the performance of the consensus algorithms in noisy environments.
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