There are numerous examples of the so-called ``square root phenomenon'' in the field of parameterized algorithms: many of the most fundamental graph problems, parameterized by some natural parameter k, become significantly simpler when restricted to planar graphs and in particular the best possible running time is exponential in O(k−−√) instead of O(k) (modulo standard complexity assumptions). We consider a classic optimization problem Subset Traveling Salesman, where we are asked to visit all the terminals T by a minimum-weight closed walk. We investigate the parameterized complexity of this problem in planar graphs, where the number k=|T| of terminals is regarded as the parameter. We show that Subset TSP can be solved in time 2O(k√logk)⋅nO(1) even on edge-weighted directed planar graphs. This improves upon the algorithm of Klein and Marx [SODA 2014] with the same running time that worked only on undirected planar graphs with polynomially large integer weights.
2022-04
2024-09-10