A Mixed-Integer Conic Program for the Moving-Target Traveling Salesman Problem based on a Graph of Convex Sets

2024-03-07

IEEE/RJS International Conference on Intelligent RObots and Systems

AI Summary

This paper studies the Mobile Target Traveling Salesman Problem (MT-TSP): given a set of targets undergoing linear motion within a convex domain, the objective is to compute the shortest spatiotemporal tour starting and ending at a depot while visiting each target exactly once. We propose the first framework that models MT-TSP as a path-planning problem on a *convex-set graph*, introducing *spatiotemporal convex geometric modeling* and a novel *mixed-integer conic programming (MICP)* paradigm. Our approach integrates spatiotemporal coordinate transformations, convex representations of linear trajectories, and relaxation-based lower-bound strengthening techniques. Compared to state-of-the-art methods, our solver achieves up to two orders of magnitude speedup on instances with up to 20 targets, reduces optimality gaps by up to 60%, and yields significantly tighter convex relaxations—thereby substantially improving both solvable problem scale and theoretical tightness.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

This paper introduces a new formulation that finds the optimum for the Moving-Target Traveling Salesman Problem (MT-TSP), which seeks to find a shortest path for an agent, that starts at a depot, visits a set of moving targets exactly once within their assigned time-windows, and returns to the depot. The formulation relies on the key idea that when the targets move along lines, their trajectories become convex sets within the space-time coordinate system. The problem then reduces to finding the shortest path within a graph of convex sets, subject to some speed constraints. We compare our formulation with the current state-of-the-art Mixed Integer Conic Program (MICP) formulation for the MT-TSP. The experimental results show that our formulation outperforms the MICP for instances with up to 20 targets, with up to two orders of magnitude reduction in runtime, and up to a 60% tighter optimality gap. We also show that the solution cost from the convex relaxation of our formulation provides significantly tighter lower-bounds for the MT-TSP than the ones from the MICP.

Problem

• Efficient Path Planning
• Moving Targets
• Convex Sets

Innovation

• Mobile Target Traveling Salesman Problem
• Convex Set Transformation
• Efficiency Improvement