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arXiv · 1707.02190

A subexponential parameterized algorithm for Directed Subset Traveling Salesman Problem on planar graphs

Dániel Marx·Marcin Pilipczuk·Michał Pilipczuk
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Abstract

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 $O(\sqrt{k})$ 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 $2^{O(\sqrt{k}\log k)}\cdot n^{O(1)}$ 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.

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Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk. 2017-07-07. A subexponential parameterized algorithm for Directed Subset Traveling Salesman Problem on planar graphs. https://doi.org/10.1137/19m1304088

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