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 (α₁,α₂)-approximate Pareto set that, for every solution, contains another solution collecting at least a 1/(α₁) fraction of its profit while requiring at most α₂ 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.

As a major unsupervised learning method, clustering has received a lot of attention over multiple decades. The various clustering problems that have been studied intensively include, e.g., the $k$-means problem and the $k$-center problem. However, in applications, it is common that good clusterings should optimize multiple objectives (e.g., visualizing data on a map by clustering districts into areas that are both geographically compact but also homogeneous with respect to the data). We study combinations of different objectives, for example optimizing $k$-center and $k$-means simultaneously or optimizing $k$-center with respect to two different metrics. Usually these objectives are conflicting and cannot be optimized simultaneously, making it necessary to find trade-offs. We develop novel algorithms for computing the set of Pareto-optimal solutions (approximately) for various combinations of two objectives. Our algorithms achieve provable approximation guarantees and we demonstrate in several experiments that the (approximate) Pareto set contains good clusterings that cannot be found by considering one of the objectives separately.