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21
When trees collide: An approximation algorithm for the generalized Steiner problem on networks
, 1994
"... We give the first approximation algorithm for the generalized network Steiner problem, a problem in network design. An instance consists of a network with linkcosts and, for each pair fi; jg of nodes, an edgeconnectivity requirement r ij . The goal is to find a minimumcost network using the a ..."
Abstract

Cited by 212 (31 self)
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We give the first approximation algorithm for the generalized network Steiner problem, a problem in network design. An instance consists of a network with linkcosts and, for each pair fi; jg of nodes, an edgeconnectivity requirement r ij . The goal is to find a minimumcost network using the available links and satisfying the requirements. Our algorithm outputs a solution whose cost is within 2dlog 2 (r + 1)e of optimal, where r is the highest requirement value. In the course of proving the performance guarantee, we prove a combinatorial minmax approximate equality relating minimumcost networks to maximum packings of certain kinds of cuts. As a consequence of the proof of this theorem, we obtain an approximation algorithm for optimally packing these cuts; we show that this algorithm has application to estimating the reliability of a probabilistic network.
Sublinear Time Algorithms for Metric Space Problems
"... In this paper we give approximation algorithms for the following problems on metric spaces: Furthest Pair, k median, Minimum Routing Cost Spanning Tree, Multiple Sequence Alignment, Maximum Traveling Salesman Problem, Maximum Spanning Tree and Average Distance. The key property of our algorithms i ..."
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Cited by 79 (2 self)
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In this paper we give approximation algorithms for the following problems on metric spaces: Furthest Pair, k median, Minimum Routing Cost Spanning Tree, Multiple Sequence Alignment, Maximum Traveling Salesman Problem, Maximum Spanning Tree and Average Distance. The key property of our algorithms is that their running time is linear in the number of metric space points. As the full specification o`f an npoint metric space is of size \Theta(n 2 ), the complexity of our algorithms is sublinear with respect to the input size. All previous algorithms (exact or approximate) for the problems we consider have running time\Omega\Gamma n 2 ). We believe that our techniques can be applied to get similar bounds for other problems. 1 Introduction In recent years there has been a dramatic growth of interest in algorithms operating on massive data sets. This poses new challenges for algorithm design, as algorithms quite efficient on small inputs (for example, having quadratic running time) ...
A Polynomial Time Approximation Scheme for Minimum Routing Cost Spanning Trees
, 1998
"... Given an undirected graph with nonnegative costs on the edges, the routing cost of any of its spanning trees is the sum over all pairs of vertices of the cost of the path between the pair in the tree. Finding a spanning tree of minimum routing cost is NPhard, even when the costs obey the triangle i ..."
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Cited by 42 (6 self)
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Given an undirected graph with nonnegative costs on the edges, the routing cost of any of its spanning trees is the sum over all pairs of vertices of the cost of the path between the pair in the tree. Finding a spanning tree of minimum routing cost is NPhard, even when the costs obey the triangle inequality. We show that the general case is in fact reducible to the metric case and present a polynomialtime approximation scheme valid for both versions of the problem. In particular, we show how to build a spanning tree of an nvertex weighted graph with routing cost within (1 + ffl) from the minimum in time O(n O( 1 ffl ) ). Besides the obvious connection to network design, trees with small routing cost also find application in the construction of good multiple sequence alignments in computational biology. The communication cost spanning tree problem is a generalization of the minimum routing cost tree problem where the routing costs of different pairs are weighted by different r...
Light graphs with small routing cost
 Networks
"... edge weights wand let aij be the nonnegative requirement between vertices iand j. For any spanning subgraphHofG,theweightofHisthetotalweightofits edges and the routing cost of His �� � i<j aijdH(i, j), where dH(i, j) is the distance between iand jin H. In this paper, we investigated two special c ..."
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Cited by 4 (1 self)
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edge weights wand let aij be the nonnegative requirement between vertices iand j. For any spanning subgraphHofG,theweightofHisthetotalweightofits edges and the routing cost of His �� � i<j aijdH(i, j), where dH(i, j) is the distance between iand jin H. In this paper, we investigated two special cases of the problem of finding aspanning subgraph with small weight and small routing cost. For the case where all the distances in Gare 1, we show that the problem is NPcomplete, and give asimple approximation algorithm for it. Furthermore, we define some sufficient conditions for the problem to be polynomialtime solvable. For the case where all the requirements are 1, we develop an algorithm for finding aspanning tree with small weight and small routing cost. The algorithm provides tradeoffs
A tight upper bound on the probabilistic embedding of seriesparallel graphs
 In Proceedings of Symp. on Discr. Algorithms, SODA’06
, 2006
"... In [EEST05] it is shown that every graph can be probabilistically embedded into a distribution over its spanning trees with expected distortion O(log 2 n log log n), narrowing the gap left by [AKPW95], where a lower bound of Ω(log n) is established. This lower bound holds even for the class of serie ..."
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Cited by 3 (1 self)
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In [EEST05] it is shown that every graph can be probabilistically embedded into a distribution over its spanning trees with expected distortion O(log 2 n log log n), narrowing the gap left by [AKPW95], where a lower bound of Ω(log n) is established. This lower bound holds even for the class of seriesparallel graphs as proved in [GNRS99]. In this paper we close this gap for seriesparallel graphs, namely, we prove that every nvertex seriesparallel graph can be probabilistically embedded into a distribution over its spanning trees with expected stretch O(log n) for every two vertices. We gain our upper bound by presenting a polynomial time probabilistic algorithm that constructs spanning trees with low expected stretch. This probabilistic algorithm can be derandomized to yield a deterministic polynomial time algorithm for constructing a spanning tree of a given seriesparallel graph G, whose communication cost is at most O(log n) times larger than that of G.
Steiner Trees and Beyond: Approximation Algorithms for Network Design
, 1993
"... We present approximation algorithms for several NPhard optimization problems arising in network design. Almost all of our algorithms run in polynomial time and output solutions with a worstcase performance guarantee on the quality of the output solution. A typical problem that we consider can be s ..."
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Cited by 3 (1 self)
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We present approximation algorithms for several NPhard optimization problems arising in network design. Almost all of our algorithms run in polynomial time and output solutions with a worstcase performance guarantee on the quality of the output solution. A typical problem that we consider can be stated as follows: given an undirected graph and certain connectivity requirements, find a subgraph that satisfies these requirements and has minimum cost. In this thesis, we address many different connectivity requirements such as spanning trees, Steiner trees, generalized Steiner forests, and twoconnected networks. The cost criteria that we consider range from the total cost of the edges in the network, the total cost of the nodes in the network, the maximum degree of any node in the network, the maximum cost of any edge in the network to some combination of these. We also address the maximumleaf spanning tree problem and provide the first approximation algorithms for this problem. In t...
Approximation algorithms for minimizing average distortion
 Proceedings of the 21st Annual Symposium on Theoretical Aspects of Computer Science
, 2004
"... Abstract This paper considers embeddings f of arbitrary finite metrics into the line metric! so thatnone of the distances is shrunk by the embedding f; the quantity of interest is the factor by whichthe average distance in the metric is stretched. We call this quantity the average distortion of the ..."
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Cited by 2 (1 self)
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Abstract This paper considers embeddings f of arbitrary finite metrics into the line metric! so thatnone of the distances is shrunk by the embedding f; the quantity of interest is the factor by whichthe average distance in the metric is stretched. We call this quantity the average distortion of the noncontracting map f.We prove that finding the best embedding of even a tree metric into a line metric so as to minimize the average distortion is NPhard, and hence focus on approximating the average distortion of thebest possible embedding for the given input metric. We give a constantfactor approximation for the problem of embedding general metrics into the line metric. For the case of npoint tree metrics,we provide a quasipolynomial time approximation scheme (QPTAS) which outputs an embedding with distortion at most (1 + ffl) times the optimum in time nO(log n/ffl 2). Finally, when the average
Optimization Problems in Computational Molecular Biology
, 1997
"... 1 An important everyday task for geneticists and molecular biologists is that of isolating and analyzing some particular DNA regions (markers), each drawn from a limited and known set of possible values (alleles). This procedure is called genotyping and is based on gel electrophoresis. In order to ..."
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Cited by 2 (1 self)
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1 An important everyday task for geneticists and molecular biologists is that of isolating and analyzing some particular DNA regions (markers), each drawn from a limited and known set of possible values (alleles). This procedure is called genotyping and is based on gel electrophoresis. In order to increase the throughput of genotyping, recently a new experiment has been proposed which tries to analyze many different markers at once. We study the mathematical problem corresponding to this model and give a branch and bound algorithm for its solution. We show that by using the techniques described in this chapter, genotyping of pooled markers can be computed effectively, thus potentially achieving a considerable reduction in time and expense. 2.1 Introduction In this chapter we investigate the possibility of using combinatorial optimization techniques to increase the rate at which genotyping is currently performed on individuals. A brief simplified description of the situation is as fo...
Connecting Two Trees with Optimal Routing Cost ∗
"... We study the problem of finding the optimal connection between two disconnected vertexweighted trees. We are given a distance function on the vertices and seek to minimize the routing cost of the tree resulting from adding one single edge between the two trees. The routing cost is defined as the su ..."
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We study the problem of finding the optimal connection between two disconnected vertexweighted trees. We are given a distance function on the vertices and seek to minimize the routing cost of the tree resulting from adding one single edge between the two trees. The routing cost is defined as the sum of the weighted distances between all pairs of vertices in the induced treemetric. The problem arises when augmenting and/or repairing communication networks or infrastructure networks. We present an asymptotically optimal quadratictime algorithm for the general case and show that the problem can be solved more efficiently for the Euclidean metric, when vertices are mapped to points in the plane, as well as for compactly representable graph metrics. 1
On the Optimum Requirement Graph Problem
"... Proc. of the 19th Workshop on Combinatorial Mathematics and Computation Theory Given a requirement aij for each pair of nodes i,j ∈{1..n}, the optimum requirement graph problem looks for a kedge graph H such that the routing cost ∑ i<j aijdH(i, j) is minimum, where dH(i, j) is the number of edge ..."
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Proc. of the 19th Workshop on Combinatorial Mathematics and Computation Theory Given a requirement aij for each pair of nodes i,j ∈{1..n}, the optimum requirement graph problem looks for a kedge graph H such that the routing cost ∑ i<j aijdH(i, j) is minimum, where dH(i, j) is the number of edges of the shortest path between i and j on H. In this paper, we show that the problem is NPhard and give a constant ratio approximation algorithm. Furthermore, we define a sufficient condition for the problem to be polynomialtime solvable. 1