The jump number of a partial order P is the minimum number of incomparable adjacent pairs in some linear extension of P. The jump number problem is known to be NP-hard in general. However some particular classes of po...
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The jump number of a partial order P is the minimum number of incomparable adjacent pairs in some linear extension of P. The jump number problem is known to be NP-hard in general. However some particular classes of posets admit easy calculation of the jump number.
The complexity status for interval orders still remains unknown. Here we present a heuristic that, given an interval order P, generates a linear extension Λ, whose jump number is less than 3/2 times the jump number of P.
Mehlhorn (1988) has presented an improved implementation of the Kou, Markowsky and Berman Steiner tree approximation algorithm (1981). By replacing one step of the original algorithm the complexity reduces from O(\S\....
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Mehlhorn (1988) has presented an improved implementation of the Kou, Markowsky and Berman Steiner tree approximation algorithm (1981). By replacing one step of the original algorithm the complexity reduces from O(\S\.(\E\ + \V\.log\V\) to O(\E\ + \V\.log\V\), where S is the set of terminals, E the set of all edges and V the set of all vertices in the graph. In this paper we will show that due to the improvement two steps of the algorithm may be omitted. This does not reduce the complexity of the algorithm but it makes it simpler.
We consider the problem of partitioning a finite sequence of points in Euclidean space into a given number of clusters (subsequences) minimizing the sum over all clusters of intracluster sums of squared distances of e...
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We consider the problem of partitioning a finite sequence of points in Euclidean space into a given number of clusters (subsequences) minimizing the sum over all clusters of intracluster sums of squared distances of elements of the clusters to their centers. It is assumed that the center of one of the desired clusters is the origin, while the centers of the other clusters are unknown and are defined as the mean values of cluster elements. Additionally, there are a few structural constraints on the elements of the sequence that enter the clusters with unknown centers: (1) the concatenation of indices of elements of these clusters is an increasing sequence, (2) the difference between two consequent indices is lower and upper bounded by prescribed constants, and (3) the total number of elements in these clusters is given as an input. It is shown that the problem is strongly NP-hard. A 2-approximation algorithm that is polynomial for a fixed number of clusters is proposed for this problem.
We consider the scheduling problem in which jobs with release times and delivery times are to be scheduled on one machine. We present a 3/2-approximation algorithm which runs in O(n log n) time and a new robust lower ...
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We consider the scheduling problem in which jobs with release times and delivery times are to be scheduled on one machine. We present a 3/2-approximation algorithm which runs in O(n log n) time and a new robust lower bound for this problem. The known until the present 3/2-approximation algorithm has O(n (2)log n) computational complexity.
Genome rearrangement algorithms are powerful tools to analyze gene orders in molecular evolution. Analysis of genomes evolving by reversals and transpositions leads to a combinatorial problem of sorting by reversals a...
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Genome rearrangement algorithms are powerful tools to analyze gene orders in molecular evolution. Analysis of genomes evolving by reversals and transpositions leads to a combinatorial problem of sorting by reversals and transpositions, the problem of finding a shortest sequence of reversals and transpositions that sorts one genome into the other. In this paper we present a 2k-approximation algorithm for sorting by reversals and transpositions for unsigned permutations where k is the approximation ratio of the algorithm used for cycle decomposition. For the best known value of k our approximation ratio becomes 2.8386 + delta for any delta > 0. We also derive a lower bound on reversal and transposition distance of an unsigned permutation. (C) 2007 Elsevier B.V. All rights reserved.
Given a directed graph G and an arc weight function w : E(G) --> R+, the maximum directed cut problem (MAX DICUT) is that of finding a directed cut delta (X) with maximum total weight. In this paper we consider a v...
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Given a directed graph G and an arc weight function w : E(G) --> R+, the maximum directed cut problem (MAX DICUT) is that of finding a directed cut delta (X) with maximum total weight. In this paper we consider a version of MAX DICUT-MAX DICUT with given sizes of parts or MAX DICUT WITH GSP-whose instance is that of MAX DICUT plus a positive integer p, and it is required to nd a directed cut delta (X) having maximum weight over all cuts delta (X) with |X| = p. Our main result is a 0.5-approximation algorithm for solving the problem. The algorithm is based on a tricky application of the pipage rounding technique developed in some earlier papers by two of the authors and a remarkable structural property of basic solutions to a linear relaxation. The property is that each component of any basic solution is an element of a set {0,delta ,1/2,1-delta ,1}, where delta is a constant that satisfies 0 < < 1/2 and is the same for all components.
Let P be a set of points on the plane, and d(p, q) be the distance between a pair of points p, q in P. For a point p. P and a subset S subset of P with vertical bar S vertical bar >= 3, the 2-dispersion cost, denot...
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Let P be a set of points on the plane, and d(p, q) be the distance between a pair of points p, q in P. For a point p. P and a subset S subset of P with vertical bar S vertical bar >= 3, the 2-dispersion cost, denoted by cost(2)( p, S), of p with respect to S is the sum of (1) the distance from p to the nearest point in S\{p} and (2) the distance from p to the second nearest point in S\{p}. The 2-dispersion cost cost(2)(S) of S subset of P with vertical bar S vertical bar >= 3 is min(p is an element of S) {cost(2)( p, S)}. Given a set P of n points and an integer k we wish to compute k point subset S of P with maximum cost(2)(S). In this paper we give a simple 1/(4 root 3) approximation algorithm for the problem.
approximation algorithms have so far mainly been studied for problems that are not known to have polynomial time algorithms for solving them exactly. Here we propose an approximation algorithm for the weighted matchin...
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approximation algorithms have so far mainly been studied for problems that are not known to have polynomial time algorithms for solving them exactly. Here we propose an approximation algorithm for the weighted matching problem in graphs which can be solved in polynomial time. The weighted matching problem is to find a matching in an edge weighted graph that has maximum weight. The first polynomial-time algorithm for this problem was given by Edmonds in 1965. The fastest known algorithm for the weighted matching problem has a running time of O(nm + n(2) log n). Many real world problems require graphs of such large size that this running time is too costly. Therefore, there is considerable need for faster approximation algorithms for the weighted matching problem. We present a linear-time approximation algorithm for the weighted matching problem with a performance ratio arbitrarily close to 2/3. This improves the previously best performance ratio of 1/2. Our algorithm is not only of theoretical interest, but because it is easy to implement and the constants involved are quite small it is also useful in practice.
An instance of the Network Steiner Problem consists of an undirected graph with edge lengths and a subset of vertices;the goal is to find a minimum cost Steiner tree of the given subset (i.e., minimum cost subset of e...
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An instance of the Network Steiner Problem consists of an undirected graph with edge lengths and a subset of vertices;the goal is to find a minimum cost Steiner tree of the given subset (i.e., minimum cost subset of edges which spans it). An 11/6-approximation algorithm for this problem is given. The approximate Steiner tree can be computed in the time O(Absolute value of V Absolute value of E + Absolute value of S4), where V is the vertex set, E is the edge set of the graph, and S is the given subset of vertices.
In the current study, an unrelated parallel machine scheduling problem with release dates is considered, which is to obtain a job assignment with minimal sum of weighted completion times. Although this problem is NP-h...
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In the current study, an unrelated parallel machine scheduling problem with release dates is considered, which is to obtain a job assignment with minimal sum of weighted completion times. Although this problem is NP-hard in the strong sense, which renders the optimality seeking a formidable task within polynomial time, a 4-approximation algorithm based on the constant worst-case bound is devised and proved in comparison with the existing 16/3-approximation (Hall et al. in Math Oper Res 22(3):513-544, 1997). In the newly proposed algorithm, the original scheduling problem is divided into several sub-problems based on release dates. For each sub-problem, a convex quadratic integer programming model is constructed in accordance with the specific problem structure. Then a semi-definite programming approach is implemented to produce a lower bound via the semi-definite relaxation of each sub-problem. Furthermore, by considering the binary constraint, a branch and bound based method and a local search strategy are applied separately to locate the optimal solution of each sub-problem. Then the solution of the original scheduling problem is derived by integrating the outcomes of the sub-problems via the proposed approximation algorithm. In the numerical analysis, it is discovered that the proposed methods could always obtain a scheduling result within 1% of the optimal solution. And an asymptotic trend could be observed where the quality of solutions improves even further as the scale of the scheduling problem increases.
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