M. X. Goemans and R. Ravi, \The constrained minimum spanning tree problem," Proceedings of SWAT 96, LNCS 1097, 66-75, 1996. 10  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... with n elements and rank k, the number of different bases occurring as the solutions to a parametric matroid optimization problem is O(n min(k, n k) 1 2 ) 18, 27] Various authors have studied related questions of computing an optimal value for the time parameter in this sequence of bases [3, 5, 12,15,16,31,33, 35]. Katoh (personal communication) noticed already one connection between parametric matroid optimization and a classical problem of computational geometry: k sets. Given a set of n points in the plane, in how many combinatorially distinct ways can subsets of exactly k points be covered by ....

R. Ravi and M. X. Goemans. The constrained minimum spanning tree problem. Proc. 5th Scand. Worksh. Algorithm Theory, pp. 66--75. Springer-Verlag, Lecture Notes in Computer Science 1097, 1996.

.... ) max # Z(#) Problem (P1) is a standard problem in sensitivity analysis [Gus83] P2) arises in the solution of the minimum ratio spanning tree problem (MRST) Cha77, Meg83] and (P3) arises in Lagrangian relaxation, in particular, when dealing with spanning tree problems with side constraints [CMV89, RaGo96]. Our result should be contrasted with the fastest known algorithm for the above problems, due to Cole [Cole87] which solves them in O(TMST (m, n) log n) O(m log #(m, n) log n) time. Thus, our algorithm is faster than Cole s for all su#ciently dense graphs i.e. m = n 2 o(log n) and ....

R. Ravi and M.X. Goemans. The constrained minimum spanning tree problem. In Proc. 5th Scandinavian Workshop on Algorithm Theory, pp. 66--75, LNCS 1097, Springer-Verlag, 1996.

.... the minimum spanning tree problem in which there are two unrelated costs per edge, say weight and length: given a budget L on length, the algorithm finds a spanning tree whose length is at most (1 ffl)L and whose weight is no more than the minimum weight of a spanning tree having length at most L (Ravi and Goemans, 1996). There are similar bait and switch algorithms for pairs of cost measures in spanning trees: degree total cost, degree bottleneck cost, and degree diameter (Ravi et al. 1993; Ravi, 1994) 10 Hard to approximate problems For some optimization problems, worst case performance guarantees are ....

Ravi, R. and Goemans, M. X. (1996). The constrained minimum spanning tree problem. In Proc. 5th Scand. Worksh. Algorithm Theory, number 1097 in Lecture Notes in Computer Science, pages 66--75. Springer-Verlag.

....many fundamental network problems. In this paper we initiate this study for the facility location problem and leave open a host of other problems. For example it is natural to study the problem of finding a spanning tree of the network whose maximum weight over time is minimum. Ravi and Goemans [19] studied this problem in the context of bicriterion approximation and their results imply a 1 ffl approximation for this problem when there are only two time slots . Similar bicriterion approximation results are known for the shortest path problem [25, 7] Some of the other fundamental ....

R. Ravi, M. X. Goemans, "The constrained minimum spanning tree problem", SWAT 1996, 66-75.

....many fundamental network problems. In this paper we initiate this study for the facility location problem and leave open a host of other problems. For example it is natural to study the problem of finding a spanning tree of the network whose maximum weight over time is minimum. Ravi et al. [18] studied this problem in the context of bicriterion approximation and their results imply a 1 ffl approximation for this problem when there are only two time slots . Similar bicriterion approximation results are known for the shortest path problem [6] Some of the other fundamental problems, ....

R. Ravi, M. X. Goemans, "The constrained minimum spanning tree problem", SWAT 1996, 66-75.

....many fundamental network problems. In this paper we initiate this study for the facility location problem and leave open a host of other problems. For example it is natural to study the problem of finding a spanning tree of the network whose maximum weight over time is minimum. Ravi et al. [18] studied this problem in the context of bicriterion approximation and their results imply a 1 ffl approximation for this problem when there are only two time slots . Similar bicriterion approximation results are known for the shortest path problem [6] Some of the other fundamental problems, ....

R. Ravi, M. X. Goemans, "The constrained minimum spanning tree problem", SWAT 1996, 66-75.

....c(T) is at most B and the total cost l(T) is a minimum among all spanning trees that obey the budget constraint. In the framework of bicriteria problems, the above problem can be expressed as (lTOTAL WEIGHT, c TOTAL WEIGHT, SPANNING TREE) This problem has been addressed by Ravi and Goemans [RG96] who obtained the following result. Theorem 8 For all 0, there is a polynomial time approximation algorithm for the Two Cost Spanning Tree problem with a performance of (1; 1 ) The running time of the algorithm is in O(n 1= m log 2 n n log 3 n) 2 6 We now explain in more ....

....iteration are twofold. On the one hand, we can show how to decompose an optimal solution into a set of marked claws one of which provides a good upgrading set at the current stage (see Lemma 14) On the other hand, we are able to find a good claw in each iteration by using the algorithm from [RG96] to solve a couple of auxiliary instances of the Two Cost Spanning Tree Problem (see Lemma 15) In each of these instances of Two Cost Spanning Tree Problem, we add edges derived from one particular marked claw to the current MST to obtain an auxiliary graph H. Each edge from the claw is added ....

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R. Ravi and M. X. Goemans, The constrained minimum spanning tree problem, Proceedings Scandinavian Workshop on Algorithmic Theory (SWAT'96), Reykjavik, Iceland, July 1996. 17

....c(T) is at most B and the total cost l(T) is a minimum among all spanning trees that obey the budget constraint. In the framework of bicriteria problems, the above problem can be expressed as (lTOTAL WEIGHT, c TOTAL WEIGHT, SPANNING TREE) This problem has been addressed by Ravi and Goemans [RG96] who obtained the following result. Theorem 8 For all 0, there is a polynomial time approximation algorithm for the Two Cost Spanning Tree problem with a performance of (1; 1 ) 2 4.2 Algorithm and Performance Guarantee The remainder of Section 4 is devoted to a proof of the following ....

R. Ravi and M. X. Goemans, The constrained minimum spanning tree problem, Proceedings Scandinavian Workshop on Algorithmic Theory (SWAT'96), Reykjavik, Iceland, Lecture Notes in Computer Science, vol. 1097, July 1996, pp. 66--75. 16

....that the total cost c(T ) is at most B and the total cost l(T ) is a minimum among all spanning trees that obey the budget constraint. The above problem can be expressed as the bicriteria problem (c Total Weight, l Total Weight, Spanning Tree) This problem has been addressed by Ravi and Goemans [11] who obtained the following result. Theorem10. For all 0, there is a polynomial time approximation algorithm for the Two Cost Spanning Tree problem with a performance of (1 ; 1) 2.2 Algorithm and Performance Guarantee The steps of our algorithm are shown in Figure 1. This algorithm ....

R. Ravi and M. X. Goemans, "The Constrained Minimum Spanning Tree Problem ", Proc. Scandinavian Workshop on Algorithmic Theory, Reykjavik, July 1996. This article was processed using the L A T E X macro package with LLNCS style

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M. X. Goemans and R. Ravi, \The constrained minimum spanning tree problem," Proceedings of SWAT 96, LNCS 1097, 66-75, 1996. 10

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R. Ravi and M. X. Goemans, The constrained minimum spanning tree problem. Proceedings Scandinavian Workshop on Algorithmic Theory (SWAT'96), Reykjavik, Iceland, July 1996.

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