Researchers have developed a new mathematical approach to solve a challenging routing problem that arises in robotics, logistics, and inspection tasks. This problem involves finding the shortest possible route that visits a set of required areas—represented as shapes called convex sets—while allowing flexible paths that can revisit points or pass through optional locations. The new method, based on an advanced search technique called branch-and-bound, efficiently navigates an otherwise infinite set of possible routes to find near-optimal solutions. This advancement could improve how autonomous robots plan inspection paths or how delivery systems optimize complex travel routes.
Key Takeaways
- The research addresses the Steiner Traveling Salesman Problem (Steiner-TSP) on Graphs of Convex Sets, a complex variant of the classic routing problem where targets are areas, not just points.
- A unified branch-and-bound search algorithm was developed that systematically explores possible routes using lower-bound cost estimates to prune less promising options.
- The method guarantees that the best route found is within a user-defined factor of the true optimal path, providing certified near-optimal solutions.
- Testing on benchmark scenarios showed that the approach reliably finds feasible routes much faster and more consistently than recent baseline methods, with typical optimality gaps around 28-30%.
At its core, the problem tackled by the researchers generalizes the classic Traveling Salesman Problem (TSP), a well-known challenge where the goal is to find the shortest possible loop visiting a list of cities. The Steiner-TSP on Graphs of Convex Sets (GCS) extends this by requiring visits to entire regions (convex sets) rather than fixed points, and allowing the route to pass through optional intermediate locations or revisit areas if needed. This flexibility models real-world tasks like robotic inspection, where a robot must cover certain zones but can plan its exact path with freedom.
To solve this, the team employed a branch-and-bound search strategy, a method that incrementally builds partial routes (“walk prefixes”) and uses mathematical lower bounds to estimate the minimum remaining cost needed to complete the route. These lower bounds help discard partial solutions that cannot lead to an optimal or near-optimal full route, dramatically reducing the search space. The researchers designed two key bounding techniques: one that evaluates the cost committed so far, and another based on a connected-flow relaxation that estimates the minimal cost to cover the remaining targets and return to the start.
The search can proceed in two ways: depth-first, which explores one route deeply before backtracking, or best-first, which expands the most promising partial route first. The paper proves that under reasonable assumptions about travel costs, both methods will terminate after a finite number of expansions, either once a sufficiently good solution is found or when the search exhausts the possibilities. Importantly, the approach can provide a global lower bound on the minimal route cost, certifying that the solution’s cost is within a user-specified factor (ε) of the true optimum.
To demonstrate practical value, the researchers applied their method to a mobile-manipulator robot inspection task. This complex scenario involved selecting sensing modes, ordering visits, and choosing continuous trajectories while respecting action precedences expressed in a formal logic framework called linear temporal logic over finite traces (LTLf). The algorithm successfully found feasible routes on all benchmark instances within 30 seconds, outperforming recent baseline methods that solved only about half the cases. Average certified optimality gaps were around 28–30%, meaning the solutions were provably close to the best possible.
This research opens the door to more reliable and efficient route planning for applications where targets are regions rather than points, and where flexibility in routing can lead to better overall performance. Potential real-world impacts include improved autonomous inspection systems, more efficient delivery route planning, and enhanced robotic navigation in complex environments. Future work may focus on further reducing optimality gaps, scaling to larger problem instances, or integrating dynamic changes in target sets and environments.
Based on research published on arXiv by Jingtao Tang, Hang Ma.