Improved Lower and Upper Bounds for Universal TSP in Planar Metrics

A universal TSP tour of a. metric space is a total ordering of the points of the space such that for any finite subset, the tour which visits these points in the given order is not too much longer than the optimal tour. There is a vast literature on the TSP problem, and universal TSP tours have been studied since the 1980's when Bartholdi and Platzman [29] introduced the spacefilling curve heuristic for the Euclidean TSP problem and conjectured that there exists a constant-competitive universal TSP tour based on this heuristic. Here, we settle this conjecture negatively by proving an Omega ((6)root log n/log log n) lower bound for universal TSP tours of the n x n grid; this is the first known example of a. family of finite metrics with no constant-competitive universal tour. Generalizing from the n x n grid to arbitrary weighted planar graph metrics, and more generally H-minor-free metrics, we improve the best known upper bound for universal tours of such metrics from O(log(4) n/log log n) to O(log(2) n).