research-article

Free access

Improved approximating algorithms for Directed Steiner Forest

Authors:

Zeev NutovAuthors Info & Claims

SODA '09: Proceedings of the twentieth annual ACM-SIAM symposium on Discrete algorithms

Pages 922 - 931

Published: 04 January 2009 Publication History

Abstract

We consider the k-Directed Steiner Forest (k-DSF) problem: given a directed graph G = (V, E) with edge costs, a collection D ∈ V x V of ordered node pairs, and an integer k ≤ |D|, find a min-cost subgraph H of G that contains an st-path for (at least) k pairs (s, t) ∈ D. When k = |D|, we get the Directed Steiner Forest (DSF) problem. The best known approximation ratios for these problems are: Õ(k^2/3) for k-DSF by Charikar et al. [2], and O(k^1/2+ε) for DSF by Chekuri et al. [3].

For DSF we give an O(n^ε·min {n^4/5,m^2/3})-approximation scheme using a novel LP-relaxation seeking to connect pairs via "cheap" paths. This is the first sublinear (in terms of n = |V|) approximation ratio for the problem. For k-DSF we give a simple greedy O(k^1/2+ε)-approximation scheme, improving the best known ratio Õ(k^2/3) by Charikar et al. [2], and (almost) matching, in terms of k, the best ratio known for the undirected variant [11]. Even when used for the particular case DSF, our algorithm favorably compares to the one of [3], which repeatedly solves linear programs, and uses complex time and space consuming transformations.

References

[1]

A. Agrawal, P. Klein, and R. Ravi. When trees collide: an approximation algorithm for the generalized Steiner problem on networks. SIAM J. Computing, 24(3):440--456, 1995.

Digital Library

[2]

M. Charikar, C. Chekuri, T. Cheung, Z. Dai, A. Goel, S. Guha, and M. Li. Approximation algorithms for directed Steiner problems. Journal of Algorithms, 33:73--91, 1999.

Digital Library

[3]

C. Chekuri, G. Even, A. Gupta, and D. Segev. Set connectivity problems in undirected graphs and the directed Steiner network problem. In SODA, pages 532--541, 2008.

Digital Library

[4]

C. Chekuri, G. Even, and G. Kortsarz. A greedy approximation algorithms for the group Steiner problems. Discrete applied Math., 154(1):15--34, 2006.

Digital Library

[5]

C. Chekuri, M. T. Hajiaghayi, G. Kortsarz, and M. R. Salavatipour. Approximation algorithms for non-uniform buy-at-bulk network design. In FOCS, pages 677--686, 2006.

Digital Library

[6]

Y. Dodis and S. Khanna. Design networks with bounded pairwise distance. In STOC, pages 750--759, 1999.

Digital Library

[7]

G. Even. Recursive greedy methods, Ch. 5 in Approximation Algorithms and Metahueristics, T. F. Gonzales ed. Chapman and Hall/CRC, 2007.

[8]

U. Feige, G. Kortsarz, and D. Peleg. The dense k-Subgraph problem. Algorithmica, 29(3):410--421, 2001.

[9]

N. Garg. Saving an epsilon: a 2-approximation for the k-MST problem in graphs. In STOC, pages 396--402, 2005.

Digital Library

[10]

N. Garg, N. Konjevod, and R. Ravi. A polylogarithmic approximation algorithm for the group Steiner tree problem. J. Algorithms, 66(1):66--84, 2000.

Digital Library

[11]

A. Gupta, M. T. Hajiaghayi, V. Nagarajan, and R. Ravi. Dial a Ride from k-Forest. In ESA, pages 241--252, 2007.

Digital Library

[12]

M. T. Hajiaghayi and K. Jain. The prize-collecting generalized Steiner tree problem via a new approach of primal-dual schema. In SODA, pages 631--640, 2006.

Digital Library

[13]

E. Halperin and R. Krauthgamer. Polylogarithmic inapproximability. In STOC, pages 585--594, 2003.

Digital Library

[14]

R. Hassin. Approximation schemes for the restricted shortest path problem. Math. Oper. Res., 17(1):36--42, 1992.

Digital Library

[15]

C. H. Helvig, G. Robins, and A. Zelikovsky. Improved approximation scheme for the group Steiner problem. Networks, 37(1):8--20, 2001.

[16]

K. Jain. Factor 2 approximation algorithm for the generalized Steiner network problem. Combinatorica, 21(1):39--60, 2001.

[17]

S. Khot. On the unique games conjecture. In FOCS, page 3, 2005.

Digital Library

[18]

S. Khuller. Approximation algorithms for finding highly connected subgraphs, Chapter 6 in Approximation Algorithms for NP-hard problems, D. S. Hochbaum Ed., pages 236--265. PWS, 1995.

Digital Library

[19]

P. N. Klein and R. Ravi. A nearly best-possible approximation algorithm for node-weighted steiner trees. Journal of Algorithms, 19(1):104--115, 1995.

Digital Library

[20]

G. Kortsarz and Z. Nutov. Approximating minimum cost connectivity problems, Ch. 58 in Approximation Algorithms and Metahueristics, T. F. Gonzales ed.,. Chapman & Hall/CRC, 2007.

[21]

G. Kortsarz and Z. Nutov. Tight approximation for connectivity augmentation problems. J. Comput. Syst. Sci., 74(5):662--670, 2008.

Digital Library

[22]

G. Kortsarz and D. Peleg. Approximating the weight of shallow Steiner trees. Discrete Applied Math., 93(2--3):265--285, 1999.

Digital Library

[23]

Z. Nutov. Approximating Steiner networks with node weights. In LATIN, pages 411--422, 2008.

Digital Library

[24]

D. Peleg. Approximation algorithms for the Label-Cover _MAX and red-blue set cover problems. J. of Discrete Algorithms, 5(1):55--64, 2007.

Digital Library

[25]

R. Raz. A parallel repetition theorem. SIAM Journal on Computing, 27(3):763--803, 1998.

Digital Library

[26]

G. Robins and A. Zelikovsky. Tighter bounds for graph Steiner tree approximation. SIAM J. Discrete Math., 19(1):122--134, 2005.

Digital Library

[27]

A. Zelikovsky. A series of approximation algorithms for the acyclic directed Steiner tree problem. Algorithmica, 18(1):99--110, 1997.

Cited By

Das AGansner EKaufmann MKobourov SSpoerhase JWolff A(2011)Approximating minimum manhattan networks in higher dimensionsProceedings of the 19th European conference on Algorithms10.5555/2040572.2040579(49-60)Online publication date: 5-Sep-2011
https://dl.acm.org/doi/10.5555/2040572.2040579
Berman PBhattacharyya AMakarychev KRaskhodnikova SYaroslavtsev G(2011)Improved approximation for the directed spanner problemProceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I10.5555/2027127.2027129(1-12)Online publication date: 4-Jul-2011
https://dl.acm.org/doi/10.5555/2027127.2027129
Ganian R(2011)New results on the complexity of the max- and min-rep problemsProceedings of the 37th international conference on Current trends in theory and practice of computer science10.5555/1946370.1946390(238-247)Online publication date: 22-Jan-2011
https://dl.acm.org/doi/10.5555/1946370.1946390
Show More Cited By

Index Terms

Improved approximating algorithms for Directed Steiner Forest

Recommendations

Improved approximation algorithms for Directed Steiner Forest

We consider the k-Directed Steiner Forest (k-DSF) problem: Given a directed graph G=(V,E) with edge costs, a collection D@__ __VxV of ordered node pairs, and an integer k=<|D|, find a minimum cost subgraph H of G that contains an st-path for (at least) ...
Approximation Algorithms for Directed Steiner Problems

We give the first nontrivial approximation algorithms for the Steiner tree problem and the generalized Steiner network problem on general directed graphs. These problems have several applications in network design and multicast routing. For both ...
Approximating Steiner Networks with Node-Weights

The (undirected) Steiner Network problem is as follows: given a graph $G=(V,E)$ with edge/node-weights and edge-connectivity requirements $\{r(u,v):u,v\in U\subseteq V\}$, find a minimum-weight subgraph $H$ of $G$ containing $U$ so that the $uv$-edge-...

Comments

Information & Contributors

Information

Published In

cover image Guide Proceedings

SODA '09: Proceedings of the twentieth annual ACM-SIAM symposium on Discrete algorithms

January 2009

1297 pages

Program Chair:
Claire Mathieu

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 04 January 2009

Qualifiers

Research-article

Acceptance Rates

Overall Acceptance Rate 411 of 1,322 submissions, 31%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

4
Total Citations
View Citations
320
Total Downloads

Downloads (Last 12 months)13
Downloads (Last 6 weeks)6

Reflects downloads up to 16 Oct 2024

Other Metrics

View Author Metrics

Citations

Cited By

Das AGansner EKaufmann MKobourov SSpoerhase JWolff A(2011)Approximating minimum manhattan networks in higher dimensionsProceedings of the 19th European conference on Algorithms10.5555/2040572.2040579(49-60)Online publication date: 5-Sep-2011
https://dl.acm.org/doi/10.5555/2040572.2040579
Berman PBhattacharyya AMakarychev KRaskhodnikova SYaroslavtsev G(2011)Improved approximation for the directed spanner problemProceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I10.5555/2027127.2027129(1-12)Online publication date: 4-Jul-2011
https://dl.acm.org/doi/10.5555/2027127.2027129
Ganian R(2011)New results on the complexity of the max- and min-rep problemsProceedings of the 37th international conference on Current trends in theory and practice of computer science10.5555/1946370.1946390(238-247)Online publication date: 22-Jan-2011
https://dl.acm.org/doi/10.5555/1946370.1946390
Nutov Z(2009)Approximating Node-Connectivity Augmentation ProblemsProceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques10.1007/978-3-642-03685-9_22(286-297)Online publication date: 21-Aug-2009
https://dl.acm.org/doi/10.1007/978-3-642-03685-9_22

View Options

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Media

Figures

Other

Tables

View Table of Contents