Approximation algorithms for geometric tour and network design problems (extended abstract)

Red-Blue Separation Problem (RBSP): Consider the problem of finding a minimum-perimeter Jordan curve (necessarily, a simple polygon) that separates a set of “red” points, R, from a set of “blue” points, B. This problem is seen to be NP-hard, using a reduction from the Euclidean traveling salesman problem [3, 12]. (Replace each city in the TSP instance by a pair of points, one red and one blue, very close together.) While Euclidean TSP can be approximated to within a factor of 1.5 times optimal (using Christofides’ heuristic [9]), the seemingly related RBSP has defied previous attempts to devise a provably good approximation algorithm. We provide an O(log m) approximation bound algorithm for RBSP, where m < n is the minimum number of sides

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