Waypoint Routing on Bounded Treewidth Graphs

TL;DR

This paper presents a new algorithm for the Waypoint Routing Problem on graphs with bounded treewidth, achieving optimal exponential time complexity and improving upon previous methods.

Contribution

The authors develop a $2^{O( ext{tw})} imes n$ time algorithm for waypoint routing on bounded treewidth graphs, proving its optimality under ETH.

Findings

New $2^{O( ext{tw})} imes n$ algorithm for the problem

Proved optimality of the algorithm under ETH

Significant improvement over previous algorithms

Abstract

In the \textsc{Waypoint Routing Problem} one is given an undirected capacitated and weighted graph $G$, a source-destination pair $s,t\in V(G)$ and a set $W\subseteq V(G)$, of \emph{waypoints}. The task is to find a walk which starts at the source vertex $s$, visits, in any order, all waypoints, ends at the destination vertex $t$, respects edge capacities, that is, traverses each edge at most as many times as is its capacity, and minimizes the cost computed as the sum of costs of traversed edges with multiplicities. We study the problem for graphs of bounded treewidth and present a new algorithm for the problem working in $2^{O(\mathrm{tw})}\cdot n$ time, significantly improving upon the previously known algorithms. We also show that this running time is optimal for the problem under Exponential Time Hypothesis.