Abstract
We generalize and unify techniques from several papers to obtain relatively simple and general technique for designing approximation algorithms for finding min-cost k-node connected spanning subgraphs. For the general instance of the problem, the previously best known algorithm has approximation ratio 2k. For k ≤ 5, algorithms with approximation ratio [(k+1)/2] are known. For metric costs Khuller and Raghavachari gave a \( \left( {2 + \frac{{2(k - 1)}} {n}} \right) \)-approximation algorithm. We obtain the following results.
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(i)
An I(k-k 0)-approximation algorithm for the problem of making a k0-connected graph k-connected by adding a minimum cost edge set, where \( I\left( k \right) = 2 + \sum\nolimits_{j = 1}^{\left\lfloor {\tfrac{k} {2}} \right\rfloor - 1} {\tfrac{1} {j}\left\lfloor {\tfrac{k} {{j + 1}}} \right\rfloor } \)
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(ii)
A \( (2 + {\mathbf{ }}\frac{{k - 1}} {n}) \)-approximation algorithm for metric costs.
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(iv)
A [(k + 1)/2]-approximation algorithm fork= 6, 7.
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(v)
A fast [(k + 1)/2]-approximation algorithm fork= 4. The multiroot problem generalizes the min-cost k-connected subgraph problem. In the multiroot problem, requirements ku for every node u are given, and the aim is to find a minimum-cost subgraph that contains max{ku, kv} internally disjoint paths between every pair of nodes u, v. For the general instance of the problem, the best known algorithm has approximation ratio 2k, wherek= maxku. For metric costs there is a 3-approximation algorithm. We consider the case of metric costs, and, using our techniques, improve fork= 7 the approximation guarantee from 3 to \( 2 + \frac{{\left\lfloor {\left( {k - 1} \right)/2} \right\rfloor }} {k} < 2.5 \)
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References
R. P. Anstee: A polynomial time algorithm for b-matchings: an alternative approach. Information Processing Letters 24 (1987), 153–157.
V. Auletta, Y. Dinitz, Z. Nutov, and D. Parente, “A 2-approximation algorithm for finding an optimum 3-vertex connected spanning subgraph”, Journal of Algorithms 32, 1999, 21–30.
J. Cheriyan, T. Jordán, and Z. Nutov, “On rooted node-connectivity problems”, to appear in Algorithmica special issue on APPROX’98.
D. Coppersmith and S. Winograd: Matrix multiplication via arithmetic progressions, J. Symbolic Comp., 9 (1990), 251–280.
Y. Dinitz and Z. Nutov, “A 3-approximation algorithm for finding optimum 4,5-vertex-connected spanning subgraphs”, Journal of Algorithms 32, 1999, 31–40.
A. Frank, “Connectivity augmentation problems in network design”, Mathematical Programming, State of the Art, J. R. Birge and K. G. Murty eds., 1994, 34–63.
A. Frank and É. Tardos, An application of submodular flows, Linear Algebra and its Applications, 114/115 (1989), 329–348.
H. N. Gabow, A representation for crossing set families with application to submodular flow problems, Proc. 4th Annual ACM-SIAM Symp. on Discrete Algorithms 1993, 202–211.
T. Jordán, “On the optimal vertex-connectivity augmentation”, J. Comb. Theory B 63, 1995, 8–20.
S. Khuller, Approximation algorithms for finding highly connected subgraphs, In Approximation algorithms for NP-hard problems, Ed.D. S. Hochbaum, PWS publishing co., Boston, 1996.
S. Khuller and B. Raghavachari, Improved approximation algorithms for uniform connectivity problems, J. of Algorithms 21, 1996, 434–450.
J. B. Orlin, “A faster strongly polynomial minimum cost flow algorithm”, Operations Research 41, 1993, 338–350.
R. Ravi and D. P. Williamson, An approximation algorithm for minimum-cost vertex-connectivity problems, Algorithmica 18, (1997), 21–43.
R. Ravi and D. P. Williamson, private communication.
W. Mader, “Ecken vom Grad n in minimalen n-fach zusammenhängenden Graphen”, Archive der Mathematik 23, 1972, 219–224.
W. Mader, “Degree and local connectivity in finite graphs”, Recent advances in graph theory (Proc. Second Czechoslovak Sympos., Prague, 1974), Academia, Prague (1975), 341–344.
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Kortsarz, G., Nutov, Z. (2000). Approximating Node Connectivity Problems via Set Covers. In: Jansen, K., Khuller, S. (eds) Approximation Algorithms for Combinatorial Optimization. APPROX 2000. Lecture Notes in Computer Science, vol 1913. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44436-X_20
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DOI: https://doi.org/10.1007/3-540-44436-X_20
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