Walking Through Waypoints

TL;DR

This paper studies the problem of finding shortest walks through specified waypoints in capacitated graphs, providing polynomial-time algorithms for certain graph classes and bounds on computational complexity.

Contribution

It introduces an exact polynomial-time algorithm for graphs of bounded treewidth and shows that with logarithmically bounded waypoints, similar algorithms exist for general graphs.

Findings

Polynomial-time algorithm for bounded treewidth graphs.

Existence of algorithms for general graphs with logarithmically bounded waypoints.

Intractability results for more general cases.

Abstract

We initiate the study of a fundamental combinatorial problem: Given a capacitated graph $G=(V,E)$, find a shortest walk ("route") from a source $s\in V$ to a destination $t\in V$ that includes all vertices specified by a set $\mathscr{W}\subseteq V$: the \emph{waypoints}. This waypoint routing problem finds immediate applications in the context of modern networked distributed systems. Our main contribution is an exact polynomial-time algorithm for graphs of bounded treewidth. We also show that if the number of waypoints is logarithmically bounded, exact polynomial-time algorithms exist even for general graphs. Our two algorithms provide an almost complete characterization of what can be solved exactly in polynomial-time: we show that more general problems (e.g., on grid graphs of maximum degree 3, with slightly more waypoints) are computationally intractable.