Effective Traveling for Metric Instances of the Traveling Thief Problem

2026-04-21

AI Summary

This study addresses the route optimization problem in the Traveling Thief Problem (TTP) under a fixed packing plan, where the challenge lies in the dependence of travel cost on the cumulative weight of collected items. The problem is modeled as a weighted variant of the Traveling Salesman Problem. We propose, for the first time, a dynamic programming algorithm with $O(n^2)$ time complexity and prove that the problem remains NP-hard even under star metrics. Furthermore, we design constant-factor approximation algorithms for both star and general metric spaces. Experimental results demonstrate that our approach significantly improves solution quality over existing benchmarks on standard TTP route instances, thereby validating its effectiveness in optimizing the routing component of the TTP.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

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.

Problem

• Traveling Thief Problem
• TSP
• Knapsack Problem
• metric TSP
• route optimization

Innovation

• Traveling Thief Problem
• weighted TSP
• dynamic programming
• approximation algorithm
• path metric