Smoothed approximation ratio of the 2-opt heuristic for the TSP

The 2-Opt heuristic is a simple, easy-to-implement local search heuristic for the traveling salesman problem. While it usually provides good approximations to the optimal tour in experiments, its worst-case performance is poor. In an attempt to explain the approximation performance of 2-Opt, we prove an upper bound of exp(O(sqrt(log(1/sigma))) for the smoothed approximation ratio of 2-Opt. As a lower bound, we prove that the worst-case lower bound of Omega(log n/log log n) for the approximation ratio holds for sigma = O(1/ sqrt(n)).\ud Our main technical novelty is that, different from existing smoothed analyses, we do not separately analyze objective values of the global and the local optimum on all inputs, but simultaneously bound them on the same input

Channel allocation in elastic optical networks using traveling salesman problem algorithms

Elastic optical networks have been proposed to support high data rates in metro and core networks. However, frequency allocation of the channels (i.e., channel ordering) in such networks is a challenging problem. This requires arranging the optical channels within the frequency grid with the objective of ensuring a minimum signal-to-noise ratio (SNR). An optimal arrangement results in the highest SNR margin for the entire network. However, determining the optimal arrangement requires an exhaustive search through all possible arrangements (permutations) of the channels. The search space increases exponentially with the number of channels. This discourages an algorithm employing an exhaustive search for the optimal frequency allocation. We utilize the Gaussian noise (GN) model to formulate the frequency allocation (channel ordering) problem as a variant of the traveling salesman problem (TSP) using graph theory. Thereafter, we utilize graph-theoretic tools for the TSP from the existing literature to solve the channel ordering problem. Performance figures obtained for the proposed scheme show that it is marginally inferior to the optimal search (through all possible permutations) and outperforms any random allocation scheme. Moreover, the proposed scheme is implementable for a scenario with a large number of channels. In comparison, an exhaustive search with the GN model and split-step Fourier method simulations are shown to be feasible for a small number of channels only. It is also illustrated that the SNR decreases with an increase in bandwidth when the frequency separation is high