The complexity of the Lin-Kernighan heuristic for the traveling salesman problem
SIAM Journal on Computing
Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems
Journal of the ACM (JACM)
Dynamic Programming Treatment of the Travelling Salesman Problem
Journal of the ACM (JACM)
P-Complete Approximation Problems
Journal of the ACM (JACM)
Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties
Some NP-complete geometric problems
STOC '76 Proceedings of the eighth annual ACM symposium on Theory of computing
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We present a quite simple, fast and practical algorithm to find a short cyclic tour that visits a set of points distributed on the plane. The algorithm runs in O(n log n) time with O(n) space, and is simple enough to easily implement on resource restricted machines. It constructs a tour essentially by axis-sorts of the points and takes a kind of the 'fixed dissection strategy,' though it neither tries to find best tours in subregions nor optimizes the order among subregions. As well as the worst-case approximation ratio of produced tours, we show that the algorithm is a 'probabilistic' constant-ratio approximation algorithm for uniform random distributions. We made computational comparisons of our algorithm, Karp's partitioning algorithm, Lin-Kernighan local search, Arora's randomized PTAS, etc. The results indicate that in running time our algorithm overwhelms existing ones, and the average approximation ratio is better or competitive.