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

We study GCS-TSP, a 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 GCS-TSP, edge costs are not fixed but depend on the specific trajectory passing through each convex region, making classical TSP methods inapplicable. We introduce GHOST, a hierarchical framework that optimally solves 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, reducing unnecessary optimization calls. We prove that GHOST guarantees optimality and present a bounded-suboptimal variant for time-critical settings. Experiments show that GHOST is orders-of-magnitude faster than unified mixed-integer convex programming baseline while uniquely handling complex problems involving high-order continuity constraints and incomplete GCSs.