Kelin Luo

Jan Eube, Kelin Luo, Aneta Neumann et al.

The Traveling Thief Problem (TTP) is a multi-component optimization problem that captures the interplay between routing and packing decisions by combining the classical Traveling Salesperson Problem (TSP) and the Knapsack Problem (KP). The TTP has gained significant attention in the evolutionary computation literature and a wide range of approaches have been developed over the last 10 years. Judging the performance of these algorithms in particular in terms of how close the get to optimal solutions is a very challenging task as effective exact methods are not available due to the highly challenging traveling component. In this paper, we study the tour-optimization component of TTP under a fixed packing plan. We formulate this task as a weighted variant of the TSP, where travel costs depend on the cumulative weight of collected items, and investigate how different distance metrics and cost functions affect computational complexity. We present an $(O(n^2))$-time dynamic programming algorithm for the path metric with general cost functions, prove that the problem is NP-hard even on a star metric, and develop constant-factor approximation algorithms for star metrics. Finally, we also develop an approximation algorithm for the problem under a general metric with a linear cost function. We complement our theoretical results with experimental evaluations on standard TTP instances adjusted to a path metric. Our experimental results demonstrate the practical effectiveness of our approaches by comparing it to solutions produced by popular iterative search algorithms. The results show that our methods are able to significantly improve the quality of solutions for some benchmark instances by optimizing the traveling part while pointing out the optimality of the travel component for other solutions obtained by iterative search methods.

Jan Eube, Kelin Luo, Heiko Röglin et al.

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.