Date of this Version
SODA '93 Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms
Given a set S of n points in the unit square [0, 1)2, an optimal traveling salesman tour of S is a tour of S that is of minimum length. A worst-case point set for the Traveling Salesman Problem in the unit square is a point set S(n) whose optimal traveling salesman tour achieves the maximum possible length among all point sets S C [0, 1)2, where JSI = n. An open problem is to determine the structure of S(n). We show that for any rectangle R contained in [0, 1 F, the number of points in S(n) n R is asymptotic to n times the area of R. One corollary of this result is an 0( n log n) approximation algorithm for the worst-case Euclidean TSP. Analogous results are proved for the minimum spanning tree, minimum-weight matching, and rectilinear Steiner minimum tree. These equidistribution theorems are the first results concerning the structure of worst-case point sets like S(n).
Snyder, T. L., & Steele, J. M. (1993). Equidistribution of Point Sets for the Traveling Salesman and Related Problems. SODA '93 Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms, 462-466. Retrieved from https://repository.upenn.edu/oid_papers/260
Date Posted: 27 November 2017