Given a collection of rooted phylogenetic trees with overlapping sets of leaves, a compatible supertree S is a single tree whose set of leaves is the union of the input sets of leaves and such that S agrees with each ...
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Given a collection of rooted phylogenetic trees with overlapping sets of leaves, a compatible supertree S is a single tree whose set of leaves is the union of the input sets of leaves and such that S agrees with each input tree when restricted to the leaves of the input tree. Typically, with trees from real data, no compatible supertree exists, and various methods may be utilized to reconcile the incompatibilities in the input trees. This paper focuses on a measure of robustness of a supertree method called its "radius" R. The larger the value of R is, the further the data set can be from a natural correct tree T, and yet, the method will still output T. It is shown that the maximal possible radius for a method is R=1/2. Many familiar methods, both for supertrees and consensus trees, are shown to have R=0, indicating that they need not output a tree T that would seem to be the natural correct answer. A polynomial-time method, Normalized Triplet Supertree ( NTS), with the maximal possible R=1/2 is defined. A geometric interpretation is given, and NTS is shown to solve an optimization problem. Additional properties of NTS are described.
We show that, for fixed k, there is a polynomial-time algorithm that finds a maximum (or maximum-weight) stable set in any graph that belongs to the class of k-colorable P-5-free graphs, or, more generally. to the Cla...
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We show that, for fixed k, there is a polynomial-time algorithm that finds a maximum (or maximum-weight) stable set in any graph that belongs to the class of k-colorable P-5-free graphs, or, more generally. to the Class Of P5-free graphs that contain no clique of size k + 1. This is based on the following structural result: every connected k-colorable P5-free graph has a vertex whose non-neighbors induce a (k - 1)-colorable subgraph. (C) 2009 Elsevier B.V. All rights reserved.
We prove that maximum weight branchings in directed graphs can be approximated in time O(m) tip to a factor of 1 - epsilon. where epsilon > 0 is an arbitrary constant. (C) 2008 Elsevier B.V. All rights reserved.
We prove that maximum weight branchings in directed graphs can be approximated in time O(m) tip to a factor of 1 - epsilon. where epsilon > 0 is an arbitrary constant. (C) 2008 Elsevier B.V. All rights reserved.
Let G = (V, E) be a simple undirected graph with a set V of vertices and a set E of edges. Each vertex v is an element of V has a demand d(v) is an element of Z(+), and a cost c(v) is an element of R+, where Z(+) and ...
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Let G = (V, E) be a simple undirected graph with a set V of vertices and a set E of edges. Each vertex v is an element of V has a demand d(v) is an element of Z(+), and a cost c(v) is an element of R+, where Z(+) and R+ denote the set of nonnegative integers and the set of nonnegative reals, respectively. The source location problem with vertex-connectivity requirements in a given graph G asks to find a set S of vertices minimizing Sigma(v is an element of S) c(v) such that there are at least d(v) pairwise vertex-disjoint paths from S to v for each vertex v is an element of V-S. It is known that the problem is not approximable within a ratio of O(ln Sigma(v is an element of V) d(v)), unless NP has an O(N-loglogN)-time deterministic algorithm. Also, it is known that even if every vertex has a uniform cost and d* = 4 holds, then the problem is NP-hard, where d* = max{d(v) vertical bar v is an element of V}. In this paper, we consider the problem in the case where every vertex has uniform cost. We propose a simple greedy algorithm for providing a max{d*, 2d* - 6}-approximate solution to the problem in O(min{d*, root vertical bar V vertical bar d*vertical bar V vertical bar(2)) time, while we also show that there exists an instance for which it provides no better than a (d* - 1)-approximate solution. Especially, in the case of d* <= 4, we give a tight analysis to show that it achieves an approximation ratio of 3. We also show the APX-hardness of the problem even restricted to d* <= 4. (C) 2009 Elsevier B.V. All rights reserved.
Reuse of proven process models increases modeling efficiency;and ensure the quality if process models. To provide a better support for reuse, the retrieval mechanisms of process repositories should be able to propose ...
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ISBN:
(纸本)9781424438518
Reuse of proven process models increases modeling efficiency;and ensure the quality if process models. To provide a better support for reuse, the retrieval mechanisms of process repositories should be able to propose similar process models that ranked according to their similarity degrees to the query request. As a process model and a query request on process structure can both be viewed as rooted, directed, and acyclic graphs, the problem of querying structural in formation of BPEL processes can be reduced to a graph matchmaking problem. In this paper we present a novel and efficient graph-based algorithm for querying structural information of BPEL processes based on an incomplete matchmaking semantics. Our algorithm performs in the worst case in polynomial time in the orders of the query graph and the process graph.
The minimum spanning tree problem originated in the 1920s when O. Borůvka identified and solved the problem during the electrification of Moravia. This graph theory problem and its numerous applications have inspired ...
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An L(2, 1)-labeling of a graph G is an assignment f from the vertex set V(G) to the set of nonnegative integers such that [f (x) - f (y)] >= 2 if x and y are adjacent and [f (x) - f (y)] >= 1 if x and y are at d...
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ISBN:
(纸本)3540699007
An L(2, 1)-labeling of a graph G is an assignment f from the vertex set V(G) to the set of nonnegative integers such that [f (x) - f (y)] >= 2 if x and y are adjacent and [f (x) - f (y)] >= 1 if x and y are at distance 2 for all x and y in V(G). A k-L(2, 1)-labeling is an L(2, 1)-labeling f : V(G) -> {0, ... , k}, and the L(2, 1)-labeling problem asks the minimum k, which we denote by lambda(G), among all possible L(2, 1)-labelings. It is known that this problem is NP-hard even for graphs of treewidth 2. Tree is one of a few classes for which the problem is polynomially solvable, but still only an O(Delta(4.5)n) time algorithm for a tree T has been known so far, where Delta is the maximum degree of T and n = vertical bar V(T)vertical bar. In this paper, we first show that an existent necessary condition for lambda(T) = Delta + 1 is also sufficient for a tree T with Delta = Omega(root n), which leads to a linear time algorithm for computing lambda(T) under this condition. We then show that lambda(T) can be computed in O(Delta(1.5)n) time for any tree T. Combining these, we finally obtain an O(n(1.75)) time algorithm, which substantially improves upon previously known results. (C) 2009 Elsevier B.V. All rights reserved.
Given a graph G = (V, E) and a requirement function r: W1 x W2 → R+ for two families W1, W2 ⊆ 2V - {θ}, we consider the problem (called area-to-area edge-connectivity augmentation problem) of augmenting G by a small...
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ISBN:
(纸本)9781920682750
Given a graph G = (V, E) and a requirement function r: W1 x W2 → R+ for two families W1, W2 ⊆ 2V - {θ}, we consider the problem (called area-to-area edge-connectivity augmentation problem) of augmenting G by a smallest number of new edges so that the resulting graph G satisfies δG (X) ≥ r(W1, W2) for all X ⊆ V, W1 ∈ W1, and W2 ∈ W2 with W1 ⊆ X ⊆ V - W2, where δG(X) denotes the degree of a vertex set X in G. This problem can be regarded as a natural generalization of the global, local, and node-to-area edge-connectivity augmentation *** this paper, we show that there exists a constant c such that the problem is inapproximable within a ratio of clog α(W1, W2), unless P=NP, even restricted to the directed global node-to-area edge-connectivity augmentation or undirected local node-to-area edge-connectivity augmentation, where α(W1, W2) denotes the number of pairs W1 ∈ W1 and W2 ∈ W2 with r(W1, W2) > 0. We also provide an O(log α (W1, W2))-approximation algorithm for the area-to-area edge-connectivity augmentation problem. This together with the negative result implies that the problem is Θ(log α (W1, W2))-approximable, unless P=NP, which solves open problems for node-to-area edge-connectivity augmentation (Ishii et al. 2008, Ishii and Hagiwara 2006, Miwa and Ito 2004).Furthermore, we characterize the node-to-area and area-to-area edge-connectivity augmentation problems as the augmentation problems with modulotone and extended modulotone functions.
The NP-complete CLOSEST 4-LEAF POWER problem asks, given an undirected graph, whether it can be modified by at most r edge insertions or deletions such that it becomes a 4-leaf power. Herein, a 4-leaf power is a graph...
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The NP-complete CLOSEST 4-LEAF POWER problem asks, given an undirected graph, whether it can be modified by at most r edge insertions or deletions such that it becomes a 4-leaf power. Herein, a 4-leaf power is a graph that can be constructed by considering an unrooted tree-the 4-leaf root-with leaves one-to-one labeled by the graph vertices, where we connect two graph vertices by an edge iff their corresponding leaves are at distance at most 4 in the tree. Complementing previous work on CLOSEST 2-LEAF POWER and CLOSEST 3-LEAF POWER, we give the first algorithmic result for CLOSEST 4-LEAF POWER, showing that CLOSEST 4-LEAF POWER is fixed-parameter tractable with respect to the parameter r. (C) 2008 Elsevier B.V. All rights reserved.
Let G = (V, E) be a connected graph such that each edge e is an element of E is weighted by nonnegative real w(e). Let s be a vertex designated as a source, k be a positive integer, and S subset of V be a set of termi...
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Let G = (V, E) be a connected graph such that each edge e is an element of E is weighted by nonnegative real w(e). Let s be a vertex designated as a source, k be a positive integer, and S subset of V be a set of terminals. The capacitated multicast tree routing problem (CMTR) asks to find a partition {Z(1), Z(2), . . . , Z(l)} of S and a set {T-1, T-2, . . . , T-l} of trees of G such that Z(i) consists of at most k terminals and each Ti spans Z(i) boolean OR {s}. The objective is to minimize Sigma(l)(i=1) w(T-i), where w(T-i) denotes the sum of weights of all edges in T-i. In this paper, we propose a (3/2 + (4/3)rho)-approximation algorithm to the CMTR, where rho is the best achievable approximation ratio for the Steiner tree problem. (C) 2007 Elsevier B.V. All rights reserved.
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