Abstract
An instance I of Ring Grooming consists of m sets A 1,A 2,…, A m from the universe {0, 1,…, n − 1} and an integer g ≥ 2. The unrestricted variant of Ring Grooming, referred to as Unrestricted Ring Grooming, seeks a partition {P 1 , P 2, …,P k } of {1, 2, …, m} such that \( \vert P_{i} \vert \leq g\) for each 1 ≤ i ≤ k and \( \sum _{i=1}^{k}\vert \bigcup_{r\in P_{i}}A_{r}\vert \) is minimized. The restricted variant of Ring Grooming, referred to as Restricted Ring Grooming, seeks a partition \(\{ P_{1},P_{2},\ldots,P_{\lceil \frac{m}{g}\rceil}\}\) of {1,2,…,m} such that | P i | ≤ g for each \(1\leq i\leq\lceil \frac {m}{g}\rceil \) and \( \sum_{i=1}^{k}\vert \bigcup_{r\in P_{i}} A_{r}\vert \) is minimized. If g = 2, we provide an optimal polynomial-time algorithm for both variants. If g > 2, we prove that both both variants are NP-hard even with fixed g. When g is a power of two, we propose an approximation algorithm called iterative matching. Its approximation ratio is exactly 1.5 when g = 4, at most 2.5 when g = 8, and at most \(\frac{g}{2}\) in general while it is conjectured to be at most \(\frac{g}{4}+\frac{1}{2}\). The iterative matching algorithm is also extended for Unrestricted Ring Grooming with arbitrary g, and a loose upper bound on its approximation ratio is \(\lceil \frac{g}{2}\rceil \) . In addition, set-cover based approximation algorithms have been proposed for both Unrestricted Ring Grooming and Restricted Ring Grooming. They have approximation ratios of at most 1 + log g, but running time in polynomial of m g.
Similar content being viewed by others
References
Chvátal V (1979) A greedy heuristic for the set-covering problem. Math Oper Res 4(3):233–235
Coffman EG Jr, Garey MR, Johnson DS (1997) Approximation algorithms for bin-packing—a survey. In: Approximation algorithms for NP-hard problems. PWS Publishing Company, Boston, pp 46–93
Dinits EA, Karzanov AV, Lomonosov ML (1976) On the structure of a family of minimal weighted cuts in a graph. In: Studies in discrete mathematics, pp 290–306
Garey MR, Johnson DS (1979) Computers and intractability. W.H. Freeman and Co
Graham RL, Grötschel M, Lóvasz L (1995) Handbook of combinatorics. MIT Press
Kann V (1994) Maximum bounded H-matching is MAX SNP-complete. Inf Process Lett (49):309–318
Lovasz L (1979) Combinatorial problems and exercises North Holland
Wan P-J, Liu L-W, (1999) Frieder O Grooming of arbitrary traffic in SONET/WDM rings. IEEE GLOBECOM’99 1B:1012–1016
Zhang X, Qiao C (1998) Effective and comprehensive solution to traffic grooming and wavelength assignment in SONET/WDM rings. SPIE Proc. of All-Optical Networking: Architecture, Control, and Management Issues (Boston, MA) 3531:221–232
Author information
Authors and Affiliations
Corresponding author
Additional information
Work supported by a DIMACS postdoctoral fellowship.
Rights and permissions
About this article
Cite this article
Călinescu, G., Wan, PJ. On Ring Grooming in optical networks. J Comb Optim 13, 103–122 (2007). https://doi.org/10.1007/s10878-006-9012-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-006-9012-x