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1.
The aim of this paper is to give a complete picture of approximability for two tree consensus problems which are of particular interest in computational biology: Maximum Agreement SubTree (MAST) and Maximum Compatible Tree (MCT). Both problems take as input a label set and a collection of trees whose leaf sets are each bijectively labeled with the label set. Define the size of a tree as the number of its leaves. The well-known MAST problem consists of finding a maximum-sized tree that is topologically embedded in each input tree, under label-preserving embeddings. Its variant MCT is less stringent, as it allows the input trees to be arbitrarily refined. Our results are as follows. We show that MCT is -hard to approximate within bound n1−? on rooted trees, where n denotes the size of each input tree; the same approximation lower bound was already known for MAST [J. Jansson, Consensus algorithms for trees and strings, Ph. D. Thesis, Lund University, 2003]. Furthermore, we prove that MCT on two rooted trees is not approximable within bound 2log1−?n, unless all problems in are solvable in quasi-polynomial time; the same result was previously established for MAST on three rooted trees [J. Hein, T. Jiang, L. Wang, K. Zhang, On the complexity of comparing evolutionary trees, Discrete Applied Mathematics 71 (1-3) (1996) 153-169] (note that MAST on two trees is solvable in polynomial time [M.A. Steel, T.J. Warnow, Kaikoura tree theorems: Computing the maximum agreement subtree, Information Processing Letters 48 (2) (1993) 77-82]). Let CMAST, resp. CMCT, denote the complement version of MAST, resp. MCT: CMAST, resp. CMCT, consists of finding a tree that is a feasible solution of MAST, resp. MCT, and whose leaf label set excludes a smallest subset of the input labels. The approximation threshold for CMAST, resp. CMCT, on rooted trees is shown to be the same as the approximation threshold for CMAST, resp. CMCT, on unrooted trees; it was already known that both CMAST and CMCT are approximable within ratio three on rooted and unrooted trees [V. Berry, F. Nicolas, Maximum agreement and compatible supertrees, in: S.C. Sahinalp, S. Muthukrishnan, U. Dogrusoz (Eds.), Proceedings of the 15th Annual Symposium on Combinatorial Pattern Matching, CPM’04, in: LNCS, vol. 3109, Springer-Verlag, 2004, pp. 205-219; G. Ganapathy, T.J. Warnow, Approximating the complement of the maximum compatible subset of leaves of k trees, in: K. Jansen, S. Leonardi, V. V. Vazirani (Eds.), Proceedings of the 5th International Workshop on Approximation Algorithms for Combinatorial Optimization, APPROX’02, in: LNCS, vol. 2462, Springer-Verlag, 2002, pp. 122-134]. The latter results are completed by showing that CMAST is -hard on three rooted trees and that CMCT is -hard on two rooted trees.  相似文献   

2.
A method that utilizes the polynomially solvable critical independent set problem for solving the maximum independent set problem on graphs with a nonempty critical independent set is developed. The effectiveness of the proposed approach on large graphs with large independence number is demonstrated through extensive numerical experiments.  相似文献   

3.
In this paper we analyze several approaches to the Maximum Independent Set (MIS) problem in hypergraphs with degree bounded by a parameter Δ. Since independent sets in hypergraphs can be strong and weak, we denote by MIS (MSIS) the problem of finding a maximum weak (strong) independent set in hypergraphs, respectively. We propose a general technique that reduces the worst case analysis of certain algorithms on hypergraphs to their analysis on ordinary graphs. This technique allows us to show that the greedy algorithm for MIS that corresponds to the classical greedy set cover algorithm has a performance ratio of (Δ+1)/2. It also allows us to apply results on local search algorithms on graphs to obtain a (Δ+1)/2 approximation for the weighted MIS and (Δ+3)/5−? approximation for the unweighted case. We improve the bound in the weighted case to ⌈(Δ+1)/3⌉ using a simple partitioning algorithm. We also consider another natural greedy algorithm for MIS that adds vertices of minimum degree and achieves only a ratio of Δ−1, significantly worse than on ordinary graphs. For MSIS, we give two variations of the basic greedy algorithm and describe a family of hypergraphs where both algorithms approach the bound of Δ.  相似文献   

4.
In a recent paper Glover (J. Heuristics 9:175–227, 2003) discussed a variety of surrogate constraint-based heuristics for solving optimization problems in graphs. The key ideas put forth in the paper were illustrated by giving specializations designed for certain covering and coloring problems. In particular, a family of methods designed for the maximum cardinality independent set problem was presented. In this paper we report on the efficiency and effectiveness of these methods based on considerable computational testing carried out on test problems from the literature as well as some new test problems.  相似文献   

5.
Two continuous formulations of the maximum independent set problem on a graph G=(V,E) are considered. Both cases involve the maximization of an n-variable polynomial over the n-dimensional hypercube, where n is the number of nodes in G. Two (polynomial) objective functions F(x) and H(x) are considered. Given any solution to x0 in the hypercube, we propose two polynomial-time algorithms based on these formulations, for finding maximal independent sets with cardinality greater than or equal to F(x0) and H(x0), respectively. A relation between the two approaches is studied and a more general statement for dominating sets is proved. Results of preliminary computational experiments for some of the DIMACS clique benchmark graphs are presented.  相似文献   

6.
In this paper we characterize the local maxima of a continuous global optimization formulation for finding the independence number of a graph. Classical Karush-Kuhn-Tucker conditions and simple combinatorial arguments are found sufficient to deduce several interesting properties of the local and global maxima. These properties can be utilized in developing new approaches to the maximum independent set problem.  相似文献   

7.
    
《Discrete Mathematics》2019,342(4):934-942
Fricke, Hedetniemi, Hedetniemi, and Hutson asked whether every tree with domination number γ has at most 2γ minimum dominating sets. Bień gave a counterexample, which allows us to construct forests with domination number γ and 2.0598γ minimum dominating sets. We show that every forest with domination number γ has at most 2.4606γ minimum dominating sets, and that every tree with independence number α has at most 2α1+1 maximum independent sets.  相似文献   

8.
    
In this note, we consider several polynomial optimization formulations of the maximum independent set problem and the use of the Lasserre hierarchy with these different formulations. We demonstrate using computational experiments that the choice of formulation may have a significant impact on the resulting bounds. We also provide theoretical justifications for the observed behavior.  相似文献   

9.
We study the on-line version of the maximum independent set problem, for the case of disk graphs which are graphs resulting from intersections of disks on the plane. In particular, we investigate whether randomization can be used to break known lower bounds for deterministic on-line independent set algorithms and present new upper and lower bounds.  相似文献   

10.
We study various optimization problems in t-subtree graphs, the intersection graphs of t-subtrees, where a t-subtree is the union of t disjoint subtrees of some tree. This graph class generalizes both the class of chordal graphs and the class of t-interval graphs, a generalization of interval graphs that has recently been studied from a combinatorial optimization point of view. We present approximation algorithms for the Maximum Independent Set, Minimum Coloring, Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique problems.  相似文献   

11.
For a subgraph X of G, let αG3(X) be the maximum number of vertices of X that are pairwise distance at least three in G. In this paper, we prove three theorems. Let n be a positive integer, and let H be a subgraph of an n-connected claw-free graph G. We prove that if n2, then either H can be covered by a cycle in G, or there exists a cycle C in G such that αG3(H?V(C))αG3(H)?n. This result generalizes the result of Broersma and Lu that G has a cycle covering all the vertices of H if αG3(H)n. We also prove that if n1, then either H can be covered by a path in G, or there exists a path P in G such that αG3(H?V(P))αG3(H)?n?1. By using the second result, we prove the third result. For a tree T, a vertex of T with degree one is called a leaf of T. For an integer k2, a tree which has at most k leaves is called a k-ended tree. We prove that if αG3(H)n+k?1, then G has a k-ended tree covering all the vertices of H. This result gives a positive answer to the conjecture proposed by Kano et al. (2012).  相似文献   

12.
Using ideas and results from polynomial time approximation and exact computation we design approximation algorithms for several NP-hard combinatorial problems achieving ratios that cannot be achieved in polynomial time (unless a very unlikely complexity conjecture is confirmed) with worst-case complexity much lower (though super-polynomial) than that of an exact computation. We study in particular two paradigmatic problems, max independent set and min vertex cover.  相似文献   

13.
On the Simple Connectivity of Fatou Components   总被引:1,自引:0,他引:1  
We shall show that for certain holomorphic maps, all Fatou components are simply connected. We also discuss the relation between wandering domains and singularities for certain meromorphic maps. Received May 8, 2002, Accepted May 24, 2002  相似文献   

14.
A subset A of a group G is sum-free if a + b does not belong to A for any a, bA. Asymptotics of the number of sum-free sets in groups of prime order are proved.  相似文献   

15.
Polynomial-time approximation schemes for packing and piercing fat objects   总被引:1,自引:0,他引:1  
We consider two problems: given a collection of n fat objects in a fixed dimension, (1) ( packing) find the maximum subcollection of pairwise disjoint objects, and (2) ( piercing) find the minimum point set that intersects every object. Recently, Erlebach, Jansen, and Seidel gave a polynomial-time approximation scheme (PTAS) for the packing problem, based on a shifted hierarchical subdivision method. Using shifted quadtrees, we describe a similar algorithm for packing but with a smaller time bound. Erlebach et al.'s algorithm requires polynomial space. We describe a different algorithm, based on geometric separators, that requires only linear space. This algorithm can also be applied to piercing, yielding the first PTAS for that problem.  相似文献   

16.
    

In this paper we show that the Julia set of a finitely generated rational semigroup is connected if the union of the Julia sets of generators is contained in a subcontinuum of . Under a nonseparating condition, we prove that the Julia set of a finitely generated polynomial semigroup is connected if its postcritical set is bounded.

  相似文献   


17.
Let G be a graph. Then the hamiltonian index h(G) of G is the smallest number of iterations of line graph operator that yield a hamiltonian graph. In this paper we show that for every 2-connected simple graph G that is not isomorphic to the graph obtained from a dipole with three parallel edges by replacing every edge by a path of length l≥3. We also show that for any two 2-connected nonhamiltonian graphs G and with at least 74 vertices. The upper bounds are all sharp.  相似文献   

18.
This paper is devoted to the lower bounds on the maximum genus of graphs. A simple statement of our results in this paper can be expressed in the following form:

Let G be a k-edge-connected graph with minimum degree δ, for each positive integer k(3), there exists a non-decreasing function f(δ) such that the maximum genus γM(G) of G satisfies the relation γM(G)f(δ)β(G), and furthermore that limδ→∞f(δ)=1/2, where β(G)=|E(G)|-|V(G)|+1 is the cycle rank of G.

The result shows that lower bounds of the maximum genus of graphs with any given connectivity become larger and larger as their minimum degree increases, and complements recent results of several authors.  相似文献   


19.
A discrete optimization problem of assigning linearly ordered character-states to the hypothetical ancestors of an evolutionary tree under the principle of maximum parsimony has been discussed. Under the transformation relation of linearly ordered character-states, Farris (1970) and Swofford and Maddison (1987) have dealt with the problem on completely bifurcating phylogenetic trees and presented a solution. Hanazawa et al. (1995) have mathematically formulated the problem with its generalization to any tree and called it the MPR (most-parsimonious reconstruction) problem. Then they have presented clear algorithms for the MPR problem and the related problems. We present a more efficient algorithm for one of the problems, the problem of obtaining the MPR sets. The complexity of the previous algorithm for this problem is O(n2) for the number n of nodes in a given tree, but that of the new algorithm is O(n).  相似文献   

20.
Entringer, Goddard, and Henning studied graphs in which every vertex belongs to both an (m + 1)‐clique and an independent (n + 1)‐set; they proved that there is such a graph of order p if and only if . We give an alternative and slightly easier proof of this fact, relating it to combinatorial matrix theory. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 173–175, 2000  相似文献   

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