On Ring Grooming in optical networks

An instance I of Ring Grooming consists of m sets A1, A2, . . . , Am 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 {P1, P2, . . . , Pk} of {1, 2, . . . , m} such that |Pi | ≤ g for each 1 ≤ i ≤ k and ∑k i=1 | ⋃ r∈Pi Ar | is minimized. The restricted variant of Ring Grooming, referred to as Restricted Ring Grooming, seeks a partition {P1, P2, . . . , P g } of {1, 2, . . . , m} such that |Pi | ≤ g for each 1 ≤ i ≤ g and ∑k i=1 | ⋃ r∈Pi Ar | 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 g2 in general while it is conjectured to be at most g4 + 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 g2 . 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 mg .