# Improved approximations for buy-at-bulk and shallow-light \(k\)-Steiner trees and \((k,2)\)-subgraph

- 103 Downloads

## Abstract

In this paper we give improved approximation algorithms for some network design problems. In the bounded-diameter or shallow-light \(k\)-Steiner tree problem (SL\(k\)ST), we are given an undirected graph \(G=(V,E)\) with terminals \(T\subseteq V\) containing a root \(r\in T\), a cost function \(c:E\rightarrow \mathbb {R}^+\), a length function \(\ell :E\rightarrow \mathbb {R}^+\), a bound \(L>0\) and an integer \(k\ge 1\). The goal is to find a minimum \(c\)-cost \(r\)-rooted Steiner tree containing at least \(k\) terminals whose diameter under \(\ell \) metric is at most \(L\). The input to the buy-at-bulk \(k\)-Steiner tree problem (BB\(k\)ST) is similar: graph \(G=(V,E)\), terminals \(T\subseteq V\) containing a root \(r\in T\), cost and length functions \(c,\ell :E\rightarrow \mathbb {R}^+\), and an integer \(k\ge 1\). The goal is to find a minimum total cost \(r\)-rooted Steiner tree \(H\) containing at least \(k\) terminals, where the cost of each edge \(e\) is \(c(e)+\ell (e)\cdot f(e)\) where \(f(e)\) denotes the number of terminals whose path to root in \(H\) contains edge \(e\). We present a bicriteria \((O(\log ^2 n),O(\log n))\)-approximation for SL\(k\)ST: the algorithm finds a \(k\)-Steiner tree with cost at most \(O(\log ^2 n\cdot \text{ opt }^*)\) where \(\text{ opt }^*\) is the cost of an LP relaxation of the problem and diameter at most \(O(L\cdot \log n)\). This improves on the algorithm of Hajiaghayi et al. (2009) (APPROX’06/Algorithmica’09) which had ratio \((O(\log ^4 n), O(\log ^2 n))\). Using this, we obtain an \(O(\log ^3 n)\)-approximation for BB\(k\)ST, which improves upon the \(O(\log ^4 n)\)-approximation of Hajiaghayi et al. (2009). We also consider the problem of finding a minimum cost \(2\)-edge-connected subgraph with at least \(k\) vertices, which is introduced as the \((k,2)\)-subgraph problem in Lau et al. (2009) (STOC’07/SICOMP09). This generalizes some well-studied classical problems such as the \(k\)-MST and the minimum cost \(2\)-edge-connected subgraph problems. We give an \(O(\log n)\)-approximation algorithm for this problem which improves upon the \(O(\log ^2 n)\)-approximation algorithm of Lau et al. (2009).

## Keywords

Combinatorial optimization Approximation algorithms Network design Steiner tree \(k\)-edge connected## Notes

### Acknowledgments

We would like to thank an anonymous referee for her/his careful reading of this paper and the comments and suggestions.

## References

- Andrews M (2004) Hardness of buy-at-bulk network design. Proceedings of IEEE FOCS, pp 115–124.Google Scholar
- Andrews M, Zhang L (2002) Approximation algorithms for access network design. Algorithmica 32(2):197–215CrossRefGoogle Scholar
- Antonakopoulos S, Chekuri C, Shepherd B, Zhang L (2011) Buy-at-bulk network design with protection. Mathe Oper Res 36(1):71–87MathSciNetCrossRefMATHGoogle Scholar
- Awerbuch B, Azar Y. (1997) Buy-at-bulk network design, In Proceedings of the 38th Annual Symposium on Foundations of Computer Science (FOCS ’97), pp 542–547.Google Scholar
- Awerbuch B, Azar Y, Blum A, Vempala S (1999) New approximation guarantees for minimum-weight \(k\)-trees and prize-collecting salesmen. SIAM J Comput 28(1):254–262MathSciNetCrossRefGoogle Scholar
- Bartal Y (1997) On approximating arbitrary metrics by tree metrics. In: Proceedings of ACM STOC, pp 161–168.Google Scholar
- Bar-Ilan J, Kortsarz G, Peleg D (2001) Generalized submodular cover problems and applications. Theor Comput Sci 250(1):179–200MathSciNetCrossRefMATHGoogle Scholar
- Bateni M, Chuzhoy J (2013) Approximation algorithms for the directed \(k\)-Tour and \(k\)-Stroll problems. Algorithmica 65(3):545–561MathSciNetCrossRefMATHGoogle Scholar
- Bhaskara A, Charikar M, Chlamtac E, Feige U, Vijayaraghavan A (2010) Detecting high log-densities: an \(O(n^{1/4})\)-approximation for densest \(k\)-subgraph. In: Proceedings of ACM Symposium on the Theory of Computing (STOC), pp 201–210.Google Scholar
- Blum A, Ravi R, Vempala S (1999) A constant-factor approximation algorithm for the \(k\)-MST problem. J Comput Syst Sci 58(1):101–108MathSciNetCrossRefMATHGoogle Scholar
- Chekuri C, Kumar A (2004) Maximum coverage problem with group budget constraints and applications. In: Approximation algorithms for combinatorial optimization, pp 72–83Google Scholar
- Chekuri C, Khanna S, Naor J (2001) A deterministic approximation algorithm for the cost-distance problem. Short paper in Proceedings of ACM-SIAM SODA, pp 232–233.Google Scholar
- Chekuri C, Hajiaghayi M, Kortsarz G, Salavatipour M (2006) Approximation algorithms for non-uniform buy-at-bulk network design problems. In: Proceedings of IEEE FOCS, pp 677–686.Google Scholar
- Chekuri C, Hajiaghayi M, Kortsarz G, Salavatipour M (2009) Approximation algorithms for non-uniform buy-at-bulk network design. SIAM J Comput 39(5):1772–1798MathSciNetCrossRefGoogle Scholar
- Chuzhoy J, Gupta A, Naor S, Sinha A (2005) On the approximability of some network design problems. Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, pp 943–951. Society for Industrial and Applied Mathematics.Google Scholar
- Even G, Garg N, Konemann J, Ravi R, Sinha A (2004) Covering graphs using trees and stars. Oper Res Lett 32(4):309–315MathSciNetCrossRefMATHGoogle Scholar
- Fakcharoenphol J, Rao S, Talwar K (2004) A tight bound on approximating arbitrary metrics by tree metrics. J Comput Syst Sci 69(3):485–497MathSciNetCrossRefMATHGoogle Scholar
- Garg N (2005) Saving an epsilon: a 2-approximation for the k-MST problem in graphs. In: Proceedings of the thirty-seventh annual ACM symposium on Theory of computing (STOC), pp 396–402.Google Scholar
- Guha S, Meyerson A, Munagala K (2009) A constant factor approximation for the single sink edge installation problem. SIAM J Comput 38(6):2426–2442Google Scholar
- Gupta A, Krishnaswamy R, Ravi R (2010) Tree embeddings for two-edge-connected network design. In: Proceedings of of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pp 1521–1538, SODA.Google Scholar
- Gupta A, Kumar A, Pal M, Roughgarden T (2003) Approximation via cost-sharing: a simple approximation algorithm for the multicommodity rent-or-buy problem. In: Proceedings of the 44rd Symposium on Foundations of Computer Science (FOCS ’03), pp 606–615. IEEE.Google Scholar
- Gupta A, Kumar A, Roughgarden T. Simpler and better approximation algorithms for network design, In Proceedings of the thirty-fifth ACM symposium on Theory of computing (STOC ’03), pp 365–372. ACM.Google Scholar
- Hajiaghayi MT, Kortsarz G, Salavatipour MR (2009) Approximating buy-at-bulk and shallow-light \(k\)-steiner tree. Algorithmica 53(1):89–103MathSciNetCrossRefMATHGoogle Scholar
- Hassin R (1992) Approximation schemes for the restricted shortest path problem. Math Oper Res 17(1):36–42MathSciNetCrossRefMATHGoogle Scholar
- Hassin R, Levin A (2002) Minimum restricted diameter spanning trees. Springer, Berlin, pp 175–184Google Scholar
- Jain K (2001) A factor 2 approximation algorithm for the generalized steiner network problem. Combinatorica 21(1):39–60MathSciNetCrossRefMATHGoogle Scholar
- Khani MR, Salavatipour MR (2011) Approximation algorithms for min-max tree cover and bounded tree cover problems. In: Proceedings of APPROX.Google Scholar
- Kortsarz G, Nutov Z (2009) Approximating some network design problems with vertex costs. In: Proceedings APPROX-RANDOM, pp 231–243.Google Scholar
- Kumar A, Gupta A, Roughgarden T (2002) A constant-factor approximation algorithm for the multicommodity rent-or-buy problem. In: Proceedings of the 43rd Symposium on Foundations of Computer Science (FOCS ’02), pp 333–342. IEEE.Google Scholar
- Lau L, Naor S, Salavatipour MR, Singh M (2009) Survivable network design with degree or order constraints. SIAM J Comput 39(3):1062–1087 Special issue for selected papers of STOC.Google Scholar
- Marathe MV, Ravi R, Sundaram R, Ravi SS, Rosenkrantz D, Hunt HB (1998) Bicriteria network design. J Algorithms 28(1):142–171Google Scholar
- Meyerson A, Munagala K, Plotkin S (2008) Cost-Distance: Two Metric Network Design. SIAM J. on Computing, 2648–1659.Google Scholar
- Ravi R, Sundaram R, Marathe MV, Rosenkrants DJ, Ravi SS (1996) Spanning trees short or small. SIAM J Discret Math 9(2):178–200CrossRefMATHGoogle Scholar
- Safari MA, Salavatipour MR (2011) A constant factor approximation for minimum \(\lambda \)-edge-connected \(k\)-subgraph with metric costs. SIAM J Discrete Math 25(3):1089–1102MathSciNetCrossRefMATHGoogle Scholar
- Salman FS, Cheriyan J, Ravi R, Subramanian S (2000) Approximating the single-sink link-installation problem in network design. SIAM J Optim 11(3):595–610MathSciNetCrossRefMATHGoogle Scholar
- Schrijver A (2003) Combinatorial optimization: Polyhedra and Efficiency. Springer-Verlag, BerlinGoogle Scholar
- Seymour PD (1981) Nowhere-zero 6-flows. J Comb Theory B 30(2):130–135Google Scholar