D. P. Williamson, M. X. Goemans, M. Mihail, and V. V. Vazirani. A primal-dual approximation algorithm for generalized Steiner network problems. In Theory of Computing, pages 708 -- 717, San Diego, CA, May 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....each pair of vertices v, w has min rv , rw edge disjoint paths. This problem has applications to the design of fiber optic telecommunication networks [13] and a more complete discussion of the problem appears in Grotschel et al. 16] Williamson, Goemans, Mihail and Vazirani (WGMV, for short) [23] gave a combinatorial algorithm for this problem with an approximation factor of 2 maxv rv 1. Actually, this algorithm also works for the non uniform case, i.e. when each pair of vertices has an associated demand for a certain number of edge disjoint paths. However, our results will apply ....

.... of the WGMV algorithm, determining active sets is more involved and therefore, the above results do not generalize to these iterations (though [3] does show how to do this for one more iteration with the same time bounds) For subsequent iterations of the WGMV algorithm, the implementation of [23] took O(maxv rv ) total time over all these iterations, and the implementation due to Gabow, Goemans and Williamson [13] took O(maxv rv maxv rv n # m log log n) total time. Thus, no subquadratic (in n) implementation was known for the WGMV algorithm prior to this work. Further, all ....

D. Williamson, M. Goemans, M. Mihail, V. Vazirani. A primal-dual approximation algorithm for generalized steiner network problems, Combinatorica, 15, pp. 435--454, 1995.

.... problem contains the Set Cover problem as a special case, and hence no polynomial time algorithm can achieve an approximation better than O(log n) unless P=NP [10] In the last 10 years there has been significant progress in designing approximation algorithm for undirected network design problems [1, 5, 12, 4, 8], the analog of this problem where the graph G is undirected. General techniques have been developed for the undirected case, e.g. primal dual algorithms [1, 5, 12, 4] Recently, Kamal Jain gave a 2 approximation algorithm for the undirected case when the function f is weakly supermodular. The ....

....last 10 years there has been significant progress in designing approximation algorithm for undirected network design problems [1, 5, 12, 4, 8] the analog of this problem where the graph G is undirected. General techniques have been developed for the undirected case, e.g. primal dual algorithms [1, 5, 12, 4]. Recently, Kamal Jain gave a 2 approximation algorithm for the undirected case when the function f is weakly supermodular. The algorithm is based on a new technique: Jain proved that in any basic solution to the linear programming relaxation of the problem there is a variable whose value is at ....

D. P. Williamson, M. X. Goemans, M. Mihail, and V. V. Vazirani. A primal-dual approximation algorithm for generalized Steiner network problems. Combinatorica, 15:435--454, 1995.

....algorithm for any requirement function which is crossing supermodular. In undirected graphs, weakly supermodular functions have been widely studied as they model a broad class of network design problems, including for instance, the generalized Steiner tree problem. Following a long line of work ([1, 10, 11, 20]) Jain [13] devised an ingenious 2 approximation algorithm for weakly supermodular functions. He proved that every basic feasible solution to the linear programming (LP) relaxation of the problem contains a variable of value at least a half. Jain s algorithm finds and rounds such a large ....

D. Williamson, M. Goemans, M. Mihail, and V. Vazirani, A Primal-dual approximation algorithm for generalized steiner network problems, Combinatorica, 15 (1995), pp. 435-- 454.

....graph, approximation algorithm, strong connectivity, local improvement. 1. Introduction. Connectivity is fundamental to the study of graphs and graph algorithms. Recently, many approximation algorithms for finding minimumsubgraphs that meet given connectivity requirements have been developed [1, 9, 11, 15, 16, 24]. These results provide practical approximation algorithms for NP hard network design problems via an increased understanding of connectivity properties. Until now, the techniques developed have been applicable only to undirected graphs. We consider a basic network design problem in directed ....

D. P. Williamson, M. X. Goemans, M. Mihail and V. V. Vazirani, A primal-dual approximation algorithm for generalized Steiner network problems, Proc. 25th ACM Symposium on Theory of Computing, pp. 708--717, (1993). 15

....(see Garey and Johnson [GJ79] However, as pointed out by Winter [Win87] and Hwang and Richards [HR92] it is polynomially solvable in series parallel networks, Hallin networks, k planar networks with all nodes on the boundary and strongly chordal networks. Moreover, Williamson et al. [WGMV93] propose a primal dual heuristic for a generalization of the problem whose restriction to the Steiner Tree problem has a constant worst case bound of 2. Let us now present one directed and one undirected formulation of the Steiner Tree problem. The directed formulation is proposed by Wong [Won84] ....

D.P. Williamson, M.X. Goemans, M. Mihail, and V.V. Vazirani. A primal-dual approximation algorithm for generalized steiner network problems. Proc. of 25th ACM Symp. on the Theory of Computing, 1993.

....NP complete since the Steiner tree problem is a special case. The set X k can be represented in several ways with x k (Hwang et al. HRW92] A natural formulation that has been used recently in approximation algorithms for generalizations of the Steiner tree problem (e.g. in Williamson et al. [WGMV95], and in Goemans and Williamson [GW97] is based on cut constraints: X e2(S) x ke 1; 8S V; S T k 6= V n S) T k 6= 4.3a) x ke 2 f0; 1g; e 2 E (4.3b) Here (S) is the cut induced by S V , i.e. the set of edges with one end vertex in S and one in its complement V n S. Constraints ....

D. P. Williamsom, M. X. Goemans, M. Mihail, and V. V. Vazirani. A primal-dual approximation algorithm for generalized Steiner network problems. Combinatorica, 15, 435-454, 1995.

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D. P. Williamson, M. X. Goemans, M. Mihail, and V. V. Vazirani. A primal-dual approximation algorithm for generalized Steiner network problems. In Proceedings of the 25th Annual ACM Symposium on Theory of Computing, pages 708--717, 1993. To appear in Combinatorica. 23

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D. P. Williamson, M. X. Goemans, M. Mihail, and V. V. Vazirani. A primal-dual approximation algorithm for generalized Steiner network problems. Combinatorica, 15:435--454, 1995. 12

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D. P. Williamson, M. X. Goemans, M. Mihail, and V. V. Vazirani, A primal-dual approximation algorithm for generalized Steiner network problems, in Proceedings of the 25th Annual ACM Symposium on Theory of Computing, 1993, pp. 708--717. To appear in Combinatorica.

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D.P. Williamson, M.X. Goemans, M. Mihail, and V.V. Vazirani, A primal-dual approximation algorithm for generalized Steiner network problems, Combinatorica 15 (1995), pp. 435--454.

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D.P. Williamson, M.X. Goemans, M. Mihail, and V.V. Vazirani. A primal--dual approximation algorithm for generalized Steiner network problems. Combinatorica, 15:435--454, 1995.

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D. P. Williamson, M. X. Goemans, M. Mihail, and V. V. Vazirani. A primal-dual approximation algorithm for generalized Steiner network problems. Combinatorica, 15:435-454, December 1995.

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D. P. Williamson, M. X. Goemans, M. Mihail, and V. V. Vazirani. A primal-dual approximation algorithm for generalized Steiner network problems. Combinatorica, 15:435--454, 1995.

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D. P. Williamson, M. X. Goemans, M. Mihail, and V. V. Vazirani. A primal-dual approximation algorithm for generalized Steiner network problems. Combinatorica, 15:435--454, 1995.

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D. P. Williamson, M. X. Goemans, M. Mihail, and V. V. Vazirani. A primal-dual approximation algorithm for generalized Steiner network problems. Combinatorica, 15:435-454, December 1995.

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D.P. Williamson, M.X. Goemans, M. Mihail, and V.V. Vazirani. A primal-dual approximation algorithm for generalized Steiner network problems. Combinatorica, 15:435--454, 1995. 138

....[ f1g stating an upper bound on the number of copies of edge e we are allowed to use; if u e = 1, then there is no bound on the number of copies of edge e. All LP duality based approximation algorithms for the metric Steiner tree problem and its generalizations work with the undirected relaxation [1, 9, 10, 20]. In order to give the integer programming formulation on which this relaxation is based, we will define a cut requirement function f : 2 . For S V , f(S) is defined to be the largest connectivity requirement separated by the cut (S; S) i.e. f(S) maxfr(u; v)ju 2 S and v 2 Sg. Let us ....

....limits the approximation factor that an algorithm using this relaxation can achieve. As a consequence of the factor 2 approximation algorithm for the Steiner network problem, we also get that the integrality gap of the undirected relaxation is 2. Previously, algorithms achieving guarantees of 2k [20] and 2H k [10] where k is the largest requirement, were obtained for this problem. 4 The bidirected cut relaxation The undirected relaxation has an integrality gap of 2 not only for as general a problem as the Steiner network problem, but also for the minimum spanning tree problem, a problem in ....

D. Williamson, M. Goemans, M. Mihail, and V. Vazirani. A primal-dual approximation algorithm for generalized steiner network problems. Combinatorica, 15, 1995.

....etc (these problems are defined later in Section 2. Subsequently, this technique has been at the core of several other approximation algorithms, e.g. the algorithm of Klein and Ravi [5] for the 2 Edge Connected Subgraph problem, and the algorithm of Williamson, Goemans, Mihail and Vazirani [6] for the Survivable Network Design problem. Before describing our results, we give a brief overview of the general clustering technique, without reference to any particular problem. The output of the clustering procedure will be a forest, constructed edge by edge over several rounds. In each ....

....of the clustering procedure. However, we show that a single run of the clustering procedure su#ces to obtain a 2 approximation running in O( n m) log n) time. Finally, we mention that Issue 1 is the bottleneck for implementing the algorithm of Williamson, Goemans, Mihail and Vazirani [6] for the Survivable Network Design Problem with higher connectivities in nearly linear time. The current fastest implementation of this algorithm is due to Gabow, Goemans and Williamson [3] which takes O(c cn # m log log n) time, where c is the maximum connectivity desired. However, as ....

D. Williamson, M. Goemans, M. Mihail, V. Vazirani. A primal-dual approximation algorithm for generalized steiner network problems, Combinatorica, 15, pp. 435--454, 1995.

....etc (these problems are de ned later in Section 2. Subsequently, this technique has been at the core of several other approximation algorithms, e.g. the algorithm of Klein and Ravi [6] for the 2 Edge Connected Subgraph problem, and the algorithm of Williamson, Goemans, Mihail and Vazirani [7] for the Survivable Courant Institute, NYU, New York. cole cs.nyu.edu. Work was supported in part by NSF grant CCR 9800085. y Indian Institute of Science, Bangalore. ramesh csa.iisc.ernet.in. Work done in part while visiting Courant Institute, NYU, and supported in part by NSF grant ....

....running in O( n m) log 2 n) time. The fact that the n iterations above can be reduced to 1 iteration has also been observed by Johnson, Minko and Phillips [4] Finally, we mention that Issue 1 is the bottleneck for implementing the algorithm of Williamson, Goemans, Mihail and Vazirani [7] for the Survivable Network Design Problem with higher connectivities in nearly linear time. The current fastest implementation of this algorithm is due to Gabow, Goemans and Williamson [3] which takes O(c 2 n 2 cn p m log log n) time, where c is the maximum connectivity desired. ....

D. Williamson, M. Goemans, M. Mihail, V. Vazirani. A primal-dual approximation algorithm for generalized steiner network problems, Combinatorica, 15, 1995, pp. 435-454.

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D. P. Williamson, M. X. Goemans, M. Mihail, and V. V. Vazirani. A primal-dual approximation algorithm for generalized Steiner network problems. In Theory of Computing, pages 708 -- 717, San Diego, CA, May 1993.

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D. P. Williamson, M. X. Goemans, M. Mihail, and V. Vazirani, A primal-dual approximation algorithm for generalized Steiner network problems, Combinatorica 15 (1995), 435-454.

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David P. Williamson, Michel X. Goemans, Milena Mihail, and Vijay V. Vazirani. A primaldual approximation algorithm for generalized Steiner network problems. Combinatorica, 15(3):435--454, 1995. (Preliminary version in: 25th Annual ACM Symposium on Theory of Computing, pages 708--717, 1993).

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D. P. Williamson, M. X. Goemans, M. Mihail, and V. V. Vazirani. A primal-dual approximation algorithm for generalized Steiner network problems. Combinatorica, 15:435--454, 1995.

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D. P. Williamson, M. X. Goemans, M. Mihail, and V. V. Vazirani. A primal-dual approximation algorithm for generalized Steiner network problems. Combinatorica, 15:435-- 454, 1995. 10

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D. Williamson, M. Goemans, M. Mihail, and V. Vazirani, A Primal-dual approximation algorithm for generalized steiner network problems, Combinatorica, 15 (1995), pp. 435{ 454. 15

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