The Thief Orienteering Problem on Series-Parallel Graphs

In the thief orienteering problem an agent called a thief carries a knapsack of capacity W and has a time limit T to collect a set of items of total weight at most W and maximum profit along a simple path in a weighted graph \(G = (V, E)\) from a start vertex s to an end vertex t. There is a set I of items each with weight \(w_{i}\) and profit \(p_{i}\) that are distributed among \(V \setminus \{s,t\}\) . The time needed by the thief to travel an edge depends on the length of the edge and the weight of the items in the knapsack at the moment when the edge is traversed. There is a polynomial-time approximation scheme for the thief orienteering problem on directed acyclic graphs that produces solutions that use time at most \(T(1 + \epsilon )\) for any constant \(\epsilon > 0\) . We give a polynomial-time algorithm for transforming instances of the problem on series-parallel graphs into equivalent instances of the thief orienteering problem on directed acyclic graphs; therefore, yielding a polynomial-time approximation scheme for the thief orienteering problem on this graph class that produces solutions that use time at most \(T(1 + \epsilon )\) for any constant \(\epsilon > 0\) .