GHOST: Solving the Traveling Salesman Problem on Graphs of Convex Sets

2025-11-09

AI Summary

This paper introduces GCS-TSP, a novel variant of the Traveling Salesman Problem on Graphs of Convex Sets (GCS), where edge costs are trajectory-dependent—i.e., determined by the actual geometric path traversing each convex region—rendering classical TSP algorithms inapplicable. To address this, we propose GHOST, a hierarchical framework: its upper layer generates candidate visit sequences via abstract path expansion and combinatorial search; its lower layer jointly optimizes both continuous trajectories and switching points using mixed-integer convex programming, subject to continuity constraints. GHOST yields verifiable lower bounds, enabling efficient branch-and-bound pruning and formal optimality guarantees. Experiments demonstrate that GHOST achieves speedups of several orders of magnitude over baseline methods and is the first scalable approach capable of solving GCS-TSP instances with high-order continuity constraints, incomplete GCS graphs, and bounded-suboptimal trajectory planning.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning
• 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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We study GCS-TSP, a new variant of the Traveling Salesman Problem (TSP) defined over a Graph of Convex Sets (GCS) -- a powerful representation for trajectory planning that decomposes the configuration space into convex regions connected by a sparse graph. In this setting, edge costs are not fixed but depend on the specific trajectory selected through each convex region, making classical TSP methods inapplicable. We introduce GHOST, a hierarchical framework that optimally solves the GCS-TSP by combining combinatorial tour search with convex trajectory optimization. GHOST systematically explores tours on a complete graph induced by the GCS, using a novel abstract-path-unfolding algorithm to compute admissible lower bounds that guide best-first search at both the high level (over tours) and the low level (over feasible GCS paths realizing the tour). These bounds provide strong pruning power, enabling efficient search while avoiding unnecessary convex optimization calls. We prove that GHOST guarantees optimality and present a bounded-suboptimal variant for time-critical scenarios. Experiments show that GHOST is orders-of-magnitude faster than unified mixed-integer convex programming baselines for simple cases and uniquely handles complex trajectory planning problems involving high-order continuity constraints and an incomplete GCS.

Problem

• Solving TSP on Graphs of Convex Sets with variable edge costs
• Developing hierarchical framework combining tour search with trajectory optimization
• Handling trajectory planning with continuity constraints and incomplete graphs

Innovation

• Hierarchical framework combining tour search and trajectory optimization
• Abstract-path-unfolding algorithm for admissible lower bounds
• Best-first search with strong pruning for efficiency