A Fast Approximation Algorithm for TSP with Neighborhoods and Red-Blue Separation

In TSP with neighborhoods (TSPN) we are given a collection X of k polygonal regions, called neighborhoods, with totally n vertices, and we seek the shortest tour that visits each neighborhood. The Euclidean TSP is a special case of the TSPN problem, so TSPN is also NP-hard. In this paper we present a simple and fast algorithm that, given a start point, computes a TSPN tour of length O(log k) times the optimum in time O(n+k log k). When no start point is given we show how to compute a “good” start point in time O(n log n), hence we obtain a logarithmic approximation algorithm that runs in time O(n log n). We also present an algorithm which performs at least one of the following two tasks (which of these tasks is performed depends on the given input): (1) It outputs in time O(n log n) a TSPN tour of length O(log k) times the optimum. (2) It outputs a TSPN tour of length less than (1+ǫ) times the optimum in cubic time, where ǫ is an arbitrary real constant given as an optional parameter. The results above are significant improvements, since the best previously known logarithmic approximation algorithm runs in Ω(n) time in the worst case.