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

TL;DR

GCS-TSP extends the Traveling Salesman Problem to graphs of convex sets where edge costs depend on the realized trajectory within convex regions, invalidating fixed-edge-cost approaches. GHOST solves this by a two-level hierarchical search: a high-level best-first RTSP over abstract tour triplets on the induced complete graph, guided by admissible lower bounds from a Lower-Bound Graph (LBG), and a low-level unfolding that computes and optimizes feasible convex trajectories for each unfolded path. Central to GHOST are the path-unfolding algorithm and the triplet-based lower bounds, which enable strong pruning while preserving optimality; a bounded-suboptimal variant, $\epsilon$-GHOST, offers scalable performance with guaranteed approximation bounds. Empirically, GHOST dramatically speeds up GCS-TSP solving compared with unified MICP baselines and uniquely handles complex trajectories and incomplete GCS instances, enabling effective planning for coverage, inspection, and multi-target visitation in robotics.

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.

Figures (6)

• Figure 1: GCS-TSP solution by GHOST. Gray polygons are convex sets. Trajectory color follows the bar (top-bottom).
• Figure 2: LBG and path unfolding. (a) The optimal trajectory (red dashed line) yielding lower-bound cost $lb_p$ for any trajectories (black dashed curves) conditioned on each possible $p$ along $\hat{\pi}$. (b)(c) Trajectories $\tau$ and $\tau'$ (blue solid curves) conditioned on $\pi$ and $\pi'$ unfolded from $\hat{\pi}$, respectively.
• Figure 3: Illustration of Alg. \ref{['alg:bfs']} for $\hat{\pi}=(v_1,\ldots)$ on the GCS from Fig. \ref{['fig:lbg']}: OPEN evolution (nodes in gray blocks sorted top-bottom in non-decreasing $f$), node expansions (dashed arrows), and appended triplets (blue).
• Figure 4: High level of GHOST. Each $T_i$ denotes a low-level search tree that explores unfolded paths for abstract tour $\hat{\pi}_i$. The search forest is pruned at $T_j$ where $\mathcal{L}(\hat{\pi}_j)$ exceeds the current best trajectory cost.
• Figure 5: A collection of GCS-TSP problem solutions produced by GHOST. (a) GCS-TSP with 23 convex sets (colored polygons) and 50 edges on Point-GCS, Linear-GCS, and Bézier-GCS. (b) Ground vehicle inspection planning for target location sets (squares) in a 2D maze. (c) Quadrotor coverage planning in a 3D obstacle-rich (gray cuboids) bounded region. (d) Task and motion planning for a 7-DoF manipulator, where black and red spheres indicate the trajectory and task poses, respectively.

Theorems & Definitions (5)

• theorem 1: Lower-Bound Path Cost
• Lemma 2
• theorem 3: Lower-Bound Tour Cost
• theorem 4
• Corollary 5