Approximation Algorithms for the Traveling Thief Problem
This work provides the first theoretical performance guarantees for TTP, a problem previously only addressed by heuristics and exact algorithms, benefiting researchers in combinatorial optimization.
The paper introduces the first approximation algorithms for the Traveling Thief Problem (TTP), achieving a (9+ε, 9+ε)-approximate Pareto set in polynomial time, and a (2e+ε)-approximation for the Weighted TSP variant.
The Traveling Thief Problem (TTP) combines the Traveling Salesperson Problem with the Knapsack Problem. In this problem, a finite metric space is given, and at each location an item with some profit and weight is placed. An agent seeks to collect a subset of the items. To do so, the agent must decide which items to collect and to determine a cyclic tour visiting the corresponding locations. While collecting an item yields its profit as a reward, the agent's speed decreases as more weight is picked up. The problem involves two competing objectives: maximizing the total profit of the collected items and minimizing the travel time of the tour. While many heuristics and exact algorithms (with a non-polynomial running time) have been developed, no approximation algorithms are known for any variant of the TTP. We aim at computing an $(α_1,α_2)$-approximate Pareto set that, for every solution, contains another solution collecting at least a $\frac{1}{α_1}$ fraction of its profit while requiring at most $α_2$ times its travel time. Our main result is an algorithm that calculates a $(9 + ε,9 + ε)$-approximate Pareto set in polynomial time. We also consider the setting in which the set of items to be collected is given in advance, so that the agent only has to compute a tour through the corresponding locations that minimizes the total travel time. This is the so-called Weighted TSP. For this setting, we present a $(2e + ε)$-approximation algorithm.