The reachability problem asks to decide if there exists a path from one vertex to another in a digraph. In a grid digraph, the vertices are the points of a two-dimensional square grid, and an edge can occur between a ...
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The reachability problem asks to decide if there exists a path from one vertex to another in a digraph. In a grid digraph, the vertices are the points of a two-dimensional square grid, and an edge can occur between a vertex and its immediate horizontal and vertical neighbors *** and Doerr (CCCG'11) presented the first simultaneous time-space bound for reachability in grid digraphs by solving the problem in polynomial time and O(n(1/2+E)) space. In 2018, the space complexity was improved to O(n(1/3)) by Ashida and Nakagawa (SoCG'18).In this paper, we show that there exists a polynomial-time algorithm that uses O(n(1/4+E)) space to solve the reachability problem in a grid digraph containing n vertices. We define and construct a new separator-like device called pseudoseparator to develop a divide-and-conquer algorithm. This algorithm works in a space-efficient manner to solve reachability.
Let N be a planar undirected network with distinguished vertices s, t, a total of n vertices, and each edge labeled with a positive real (the edge’s cost) from a set L. This paper presents an algorithm for computing ...
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Let N be a planar undirected network with distinguished vertices s, t, a total of n vertices, and each edge labeled with a positive real (the edge’s cost) from a set L. This paper presents an algorithm for computing a minimum (cost) s-t cut of N. For general L, this algorithm runs in time O(nlog
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We consider the reconstruction of shared secrets in communication networks, which are modelled by graphs whose components are subject to possible failure. The reconstruction probability can be approximated using minim...
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We consider the reconstruction of shared secrets in communication networks, which are modelled by graphs whose components are subject to possible failure. The reconstruction probability can be approximated using minimal cuts, if the failure probabilities of vertices and edges are close to zero. As the main contribution of this paper, node separators are used to design a heuristic for the near-optimal placement of secrets sets on the vertices of the graph.
This paper studies methods for finding minimum spanning trees in graphs. Results include 1. several algorithms with $O(m\log \log n)$ worst-case running times, where n is the number vertices and m is the number of edg...
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This paper studies methods for finding minimum spanning trees in graphs. Results include 1. several algorithms with $O(m\log \log n)$ worst-case running times, where n is the number vertices and m is the number of edges in the problem graph; 2. an $O(m)$ worst-case algorithm for dense graphs (those for which m is $\Omega (n^{1 + \varepsilon } )$ for some positive constant $\varepsilon $); 3. an $O(n)$ worst-case algorithm for planar graphs; 4. relationships with other problems which might lead general lower bound for the complexity of the minimum spanning tree problem.
We consider a generalization of the rooted triplet distance between two phylogenetic trees to two phylogenetic networks. We show that if each of the two given phylogenetic networks is a so-called galled tree with n le...
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We consider a generalization of the rooted triplet distance between two phylogenetic trees to two phylogenetic networks. We show that if each of the two given phylogenetic networks is a so-called galled tree with n leaves then the rooted triplet distance can be computed in O(n(2.687)) time. Our upper bound is obtained by reducing the problem of computing the rooted triplet distance between two galled trees to that of counting monochromatic and almost-monochromatic triangles in an undirected, edge-colored graph. To count different types of colored triangles in a graph efficiently, we extend an existing technique based on matrix multiplication and obtain several new algorithmic results that may be of independent interest: (i) the number of triangles in a connected, undirected, uncolored graph with m edges can be computed in o(m(1.408)), time;(ii) if G is a connected, undirected, edge-colored graph with n vertices and C is a subset of the set of edge colors then the number of monochromatic triangles of G with colors in C can be computed in o(n(2.687)) time;and (iii) if G is a connected, undirected, edge-colored graph with n vertices and R is a binary relation on the colors that is computable in O(1) time then the number of R-chromatic triangles in G can be computed in o(n(2.687)) time. (C) 2013 Elsevier B.V. All rights reserved.
A path-based support of a hypergraph H is a graph with the same vertex set as H in which each hyperedge induces a Hamiltonian subgraph. While it is NP-hard to decide whether a path-based support has a monotone drawing...
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A path-based support of a hypergraph H is a graph with the same vertex set as H in which each hyperedge induces a Hamiltonian subgraph. While it is NP-hard to decide whether a path-based support has a monotone drawing, to determine a path-based support with the minimum number of edges, or to decide whether there is a planar path-based support, we show that a path-based tree support can be computed in polynomial time if it exists. (C) 2011 Elsevier B.V. All rights reserved.
The Gomory-Hu tree or cut tree [R. E. Gomory and T. C. Hu, J. Soc. Indust. Appl. Math., 9 (1961), pp. 55--570] is a classic data structure for reporting (s,t)-mincuts (and by duality, the values of (s, t)-maxflows) fo...
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The Gomory-Hu tree or cut tree [R. E. Gomory and T. C. Hu, J. Soc. Indust. Appl. Math., 9 (1961), pp. 55--570] is a classic data structure for reporting (s,t)-mincuts (and by duality, the values of (s, t)-maxflows) for all-pairs of vertices s and t in an undirected graph. Gomory and Hu showed that it can be computed using n - 1 exact maxflow computations. Surprisingly, this remains the best algorithm for Gomory-Hu trees more than 50 years later, even for approximate mincuts. In this paper, we break this longstanding barrier and give an algorithm for computing a (1+e)-approximate Gomory-Hu tree using polylog(n) maxflow computations. Specifically, we obtain the running time bounds we describe below. We obtain a randomized (Monte Carlo) algorithm for undirected, weighted graphs that runs in O(m + n(3/2)) time and returns a (1+e)-approximate Gomory-Hu tree with high probability (w.h.p.). Previously, the best running time known was O(n(5/2)), which is obtained by running Gomory and Hu's original algorithm on a cut sparsifier of the graph. Next, we obtain a randomized (Monte Carlo) algorithm for undirected, unweighted graphs that runs in m(4/3+o(1)) time and returns a (1 + e)-approximate Gomory-Hu tree w.h.p. This improves on our first result for sparse graphs, namely m = o(n(9/8)). Previously, the best running time known for unweighted graphs was O(mn) for an exact Gomory-Hu tree [A. Bhalgat et al., Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, CA, 2007, pp. 605-614];no better result is known if approximations are allowed. As a consequence of our Gomory-Hu tree algorithms, we also solve the (1+e)-approximate all-pairs mincut (APMC) and single-source mincut (SSMC) problems in the same time bounds. (These problems are simpler in that the goal is to only return the (s, t)-mincut values, and not the mincuts.) This improves on the recent algorithm for these problems in O(n(2)) time due to Abboud, Krauthgamer, and Trabelsi [2020 IEEE 61st Annu
We consider a problem for constructing a minimum cost r-edge-connected multigraph in which degree d(v) of each vertex v is an element of V is specified. In this paper, we propose a 3-approximation algorithm for this p...
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We consider a problem for constructing a minimum cost r-edge-connected multigraph in which degree d(v) of each vertex v is an element of V is specified. In this paper, we propose a 3-approximation algorithm for this problem under the assumption that edge cost is metric, r(u, v) is an element of {1, 2} for each u, v is an element of V, and d(v) >= 2 for each v is an element of V. This problem is a generalization of metric TSP. We also propose an approximation algorithm for the digraph version of the problem.
The concept of mutual-visibility in graphs has been recently introduced. If X is a subset of vertices of a graph G, then vertices u and v are X- visible if there exists a shortest u , v-path P such that V(P) ∩ X ⊆ { ...
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The concept of mutual-visibility in graphs has been recently introduced. If X is a subset of vertices of a graph G, then vertices u and v are X- visible if there exists a shortest u , v-path P such that V(P) ∩ X ⊆ { u , v}. If every two vertices from X are X- visible, then X is a mutual-visibility set. The mutual-visibility number of G is the cardinality of a largest mutual-visibility set of G . It is known that computing the mutual-visibility number of a graph is NP-complete, whereas it has been shown that there are exact formulas for special graph classes like paths, cycles, blocks, cographs, and grids. In this paper, we study the mutual-visibility in distance-hereditary graphs and show that the mutual-visibility number can be computed in linear time for this class.
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