Approximation Algorithms for the Traveling Thief Problem
2026-07-06 • Data Structures and Algorithms
Data Structures and Algorithms
AI summaryⓘ
The authors study a problem called the Traveling Thief Problem (TTP), which combines deciding a route like the Traveling Salesperson Problem with choosing items to pick up like the Knapsack Problem. Picking more items slows down travel, so the challenge is to balance collecting valuable items and finishing quickly. They developed an algorithm that finds solutions within a factor of about 9 for both profit and travel time in reasonable time. They also looked at a simpler case where which items to collect is fixed, and provided a better approximation for that version.
Traveling Thief ProblemTraveling Salesperson ProblemKnapsack Problemapproximation algorithmPareto setmetric spaceheuristicspolynomial timeWeighted TSPtravel time optimization
Authors
Jan Eube, Kelin Luo, Heiko Röglin, Sarah Sturm
Abstract
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.