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Bicriteria network design problems
 In Proc. 22nd Int. Colloquium on Automata, Languages and Programming
, 1995
"... We study a general class of bicriteria network design problems. A generic problem in this class is as follows: Given an undirected graph and two minimization objectives (under different cost functions), with a budget specified on the first, find a ¡subgraph from a given subgraphclass that minimizes ..."
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Cited by 75 (13 self)
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We study a general class of bicriteria network design problems. A generic problem in this class is as follows: Given an undirected graph and two minimization objectives (under different cost functions), with a budget specified on the first, find a ¡subgraph from a given subgraphclass that minimizes the second objective subject to the budget on the first. We consider three different criteria the total edge cost, the diameter and the maximum degree of the network. Here, we present the first polynomialtime approximation algorithms for a large class of bicriteria network design problems for the above mentioned criteria. The following general types of results are presented. First, we develop a framework for bicriteria problems and their approximations. Second, when the two criteria are the same we present a “black box ” parametric search technique. This black box takes in as input an (approximation) algorithm for the unicriterion situation and generates an approximation algorithm for the bicriteria case with only a constant factor loss in the performance guarantee. Third, when the two criteria are the diameter and the total edge costs we use a clusterbased approach to devise a approximation algorithms — the solutions output violate
Using Sparsification for Parametric Minimum Spanning Tree Problems
 Nordic J. Computing
, 1996
"... Two applications of sparsification to parametric computing are given. The first is a fast algorithm for enumerating all distinct minimum spanning trees in a graph whose edge weights vary linearly with a parameter. The second is an asymptotically optimal algorithm for the minimum ratio spanning t ..."
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Cited by 7 (2 self)
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Two applications of sparsification to parametric computing are given. The first is a fast algorithm for enumerating all distinct minimum spanning trees in a graph whose edge weights vary linearly with a parameter. The second is an asymptotically optimal algorithm for the minimum ratio spanning tree problem, as well as other search problems, on dense graphs. 1 Introduction In the parametric minimum spanning tree problem, one is given an nnode, medge undirected graph G where each edge e has a linear weight function w e (#)=a e +#b e . Let Z(#) denote the weight of the minimum spanning tree relative to the weights w e (#). It can be shown that Z(#) is a piecewise linear concave function of # [Gus80]; the points at which the slope of Z changes are called breakpoints. We shall present two results regarding parametric minimum spanning trees. First, we show that Z(#) can be constructed in O(min{nm log n, TMST (2n, n) # Department of Computer Science, Iowa State University, Ames, IA...
LinearTime Algorithms for Parametric Minimum Spanning Tree Problems on Planar Graphs
, 1995
"... A lineartime algorithm for the minimumratio spanning tree problem on planar graphs is presented. The algorithm is based on a new planar minimum spanning tree algorithm. The approach extends to other parametric minimum spanning tree problems on planar graphs and to other families of graphs having s ..."
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Cited by 3 (2 self)
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A lineartime algorithm for the minimumratio spanning tree problem on planar graphs is presented. The algorithm is based on a new planar minimum spanning tree algorithm. The approach extends to other parametric minimum spanning tree problems on planar graphs and to other families of graphs having small separators. 1 Introduction Suppose we are given an undirected graph G where each edge e has two weights a e and b e ; the b e 's are assumed to be either all negative or all positive. The minimum ratio spanning tree problem (MRST) [Cha77] is to find a spanning tree T of G such that the ratio P e2T a e = P e2T b e is minimized. One application of MRST arises in the design of communication networks. The number a e represents the cost of building link e, while b e represents the time required to build that link. The goal is to find a tree that minimizes the ratio of total cost over construction time. Other applications of MRST are given elsewhere [CMV89, Meg83]. The main result of thi...
An Analysis of Dinkelbach's Algorithm for 01 Fractional Programming Problems
, 1992
"... : The 01 fractional programming problem minimizes the fractional objective function (c 1 x 1 + c 2 x 2 + 1 1 1 + c n x n )=(d 1 x 1 + d 2 x 2 + 1 1 1 + d n x n ) = cx=dx under the condition that x = (x 1 ; 1 1 1 ; xn ) 2\Omega ` f0; 1g n ; where\Omega is the set of feasible solutions. For a fra ..."
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Cited by 1 (1 self)
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: The 01 fractional programming problem minimizes the fractional objective function (c 1 x 1 + c 2 x 2 + 1 1 1 + c n x n )=(d 1 x 1 + d 2 x 2 + 1 1 1 + d n x n ) = cx=dx under the condition that x = (x 1 ; 1 1 1 ; xn ) 2\Omega ` f0; 1g n ; where\Omega is the set of feasible solutions. For a fractional programming problem, Dinkelbach developed an algorithm which obtains an optimal solution of the given problem by solving a sequence of subproblems Q(), in which the linear objective function cx 0 dx is minimized under the same condition x 2\Omega : In this paper, we show that Dinkelbach 's algorithm solves at most O(log(nM)) subproblems in the worst case, where M = maxf max i=1;2;111;n jc i j; max i=1;2;111;n jd i j; 1g: 1 01 Fractional Programming Problems The problem of maximizing or minimizing the ratio of two linear functions is called a fractional programming problem and/or a hyperbolic programming problem. Fractional programming problems are found in various fields [2...