A Polynomial-Time Approximation Scheme for Thief Orienteering on Directed Acyclic Graphs

Abstract

We consider the scenario of routing an agent called a thief through a weighted graph $G = (V, E)$ from a start vertex s to an end vertex t. A set I of items each with weight $w_{i}$ and profit $p_{i}$ is distributed among $V \setminus \{s,t\}$. In the thief orienteering problem, the thief, who has a knapsack of capacity W, must follow a simple path from s to t within a given time T while packing in the knapsack a set of items, taken from the vertices along the path, of total weight at most W and maximum profit. The travel time across an edge depends on the edge length and current knapsack load.

The thief orienteering problem is a generalization of the orienteering problem and the 0–1 knapsack problem. We present a polynomial-time approximation scheme (PTAS) for the thief orienteering problem when G is directed and acyclic, and adapt the PTAS for other classes of graphs and special cases of the problem. In addition, we prove there exists no approximation algorithm for the thief orienteering problem with constant approximation ratio, unless P = $\textsf {NP}$.

Andrew Bloch-Hansen and Roberto Solis-Oba were partially supported by the Natural Sciences and Engineering Research Council of Canada, grants 6636-548083-2020 and RGPIN-2020-06423, respectively. Daniel Page was partially supported publicly via Patreon.