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

TL;DR

This paper presents a new subexponential parameterized algorithm for the Directed Subset Traveling Salesman Problem on planar graphs, improving previous results and extending applicability to directed graphs with weighted edges.

Contribution

It introduces a subexponential time algorithm for Directed Subset TSP on planar graphs, extending prior undirected results to directed and weighted cases.

Findings

Algorithm runs in 2^{O(√k log k)}·n^{O(1)} time for directed planar graphs.

Improves upon previous algorithms limited to undirected graphs.

Extends the square root phenomenon to directed planar graphs with weights.

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.