Abstract
Traffic grooming is a central problem in optical networks. It refers to pack low rate signals into higher speed streams, in order to improve bandwidth utilization and reduce network cost. In WDM networks, the most accepted criterion is to minimize the number of electronic terminations, namely the number of SONET Add-Drop Multiplexers (ADMs). In this article we focus on ring and path topologies. On the one hand, we provide the first inapproximability result for Traffic Grooming for fixed values of the grooming factor g, answering affirmatively the conjecture of Chow and Lin (Networks, 44:194-202, 2004). More precisely, we prove that Ring Traffic Grooming for fixed g ≥ 1 and Path Traffic Grooming for fixed g ≥ 2 are APX-complete. That is, they do not accept a PTAS unless P= NP. Both results rely on the fact that finding the maximum number of edge-disjoint triangles in a graph (and more generally cycles of length 2g + 1 in a graph of girth 2g + 1) is APX-complete.
On the other hand, we provide a polynomial-time approximation algorithm for Ring and Path Traffic Grooming, based on a greedy cover algorithm, with an approximation ratio independent of g. Namely, the approximation guarantee is \(\mathcal{O}(n^{1/3} \log^2 n)\) for any g ≥ 1, n being the size of the network. This is useful in practical applications, since in backbone networks the grooming factor is usually greater than the network size. As far as we know, this is the first approximation algorithm with this property. Finally, we improve this approximation ratio under some extra assumptions about the request graph.
This work has been partially supported by European project IST FET AEOLUS, Ministerio de Educación y Ciencia of Spain, European Regional Development Fund under project TEC2005-03575, Catalan Research Council under project 2005SGR00256, and COST action 293 GRAAL, and has been done in the context of the crc Corso with France Telecom.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Amini, O., Pérennes, S., Sau, I.: Hardness and Approximation of Traffic Grooming. Research Report 6236, INRIA, Accessible from 3rd author’s homepage (2007)
Andersen, R.: Finding large and small dense subgraphs. Submitted for publication (arXiv:cs/0702032v1) (February 2007)
Bermond, J.-C., Coudert, D.
Bermond, J.-C., Coudert, D., Muñoz, X., Sau, I.: Traffic Grooming in Bidirectional WDM Ring Networks. In: IEEE-LEOS ICTON / COST 293 GRAAL, vol. 3, pp. 19–22 (2006)
Chiu, A., Modiano, E.: Traffic grooming algorithms for reducing electronic multiplexing costs in WDM ring networks. IEEE/OSA Journal of Lightwave Technology 18(1), 2–12 (2000)
Chow, T., Lin, P.: The ring grooming problem. Networks 44, 194–202 (2004)
Demaine, E., Hajiaghayi, M.T., Kawarabayashi, K.C.: Algorithmic graph minor theory: Decomposition, approximation and coloring. In: 46th Annual IEEE Symposium on Fondations of Computer Science (FOCS), pp. 637–646 (2005)
Dutta, R., Rouskas, G.: On Optimal Traffic Grooming in WDM Rings. In: Proceedings of ACM Sigmetrics/Performance, pp. 164–174, Cambridge (2001)
Dutta, R., Rouskas, N.: Traffic grooming in WDM networks: Past and future. IEEE Network 16(6), 46–56 (2002)
Epstein, L., Levin, A.: Better Bounds for Minimizing SONET ADMs. In: Persiano, G., Solis-Oba, R. (eds.) WAOA 2004. LNCS, vol. 3351, pp. 281–294. Springer, Heidelberg (2005)
Feige, U., Peleg, D., Kortsarz, G.: The Dense k-Subgraph Problem. Algorithmica 29(3), 410–421 (2001)
Flammini, M., Moscardelli, L., Shalom, M., Zaks, S.: Approximating the traffic grooming problem. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 915–924. Springer, Heidelberg (2005)
Gerstel, O., Ramaswami, R., Sasaki, G.: Cost Effective Traffic Grooming in WDM Rings. In: Proceedings of INFOCOM, San Francisco, pp. 69–77 (1998)
Holyer, I.: The NP-Completeness of Some Edge-Partition Problems. SIAM Journal on Computing 10(4), 713–717 (1981)
Huang, S., Dutta, R., Rouskas, G.N.: Traffic Grooming in Path, Star, and Tree Networks: Complexity, Bounds, and Algorithms. IEEE Journal on Selected Areas in Communications 24(4), 66–82 (2006)
Kann, V.: Maximum bounded 3-dimensional matching is MAX SNP-complete. Information Processing Letters 37, 27–35 (1991)
Khot, S.: Ruling out PTAS for graph min-bisection, densest subgraph and bipartite clique. In: FOCS, pp. 136–145 (2004)
Modiano, E., Lin, P.: Traffic grooming in WDM networks. IEEE Communications Magazine 39(7), 124–129 (2001)
Shalom, M., Unger, W., Zaks, S.: On the Complexity of the Traffic Grooming Problem in Optical Networks
Suzuki, A., Tokuyama, T.: Dense subgraph problem revisited. In: Joint Workshop ”New Horizons in Computing” and ”Statistical Mechanical Approach to Probabilistic Information Processing”, Japan (July 2005)
Zhu, K., Mukherjee, B.: A review of traffic grooming in WDM optical networks: Architectures and challenges. Optical Networks 4(2), 55–64 (2003)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Amini, O., Pérennes, S., Sau, I. (2007). Hardness and Approximation of Traffic Grooming. In: Tokuyama, T. (eds) Algorithms and Computation. ISAAC 2007. Lecture Notes in Computer Science, vol 4835. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77120-3_49
Download citation
DOI: https://doi.org/10.1007/978-3-540-77120-3_49
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-77118-0
Online ISBN: 978-3-540-77120-3
eBook Packages: Computer ScienceComputer Science (R0)