凸集图上的斯坦纳旅行商问题的统一分支定界搜索

中文摘要 AI 辅助

我们将凸集图（Graphs of Convex Sets, GCS）上的斯坦纳旅行商问题（Steiner-TSP）形式化，该问题旨在寻找一条经过所需凸集的最小代价闭合轨迹，允许可选的中转顶点和重复访问。为探索由此产生的无限解空间，我们提出一种基于带根行走前缀的统一分支定界搜索方法：加法下界图代价约束已确定的前缀，而割分离的连通流松弛则对访问所有剩余目标并返回根节点的残余代价进行下界估计。在统一正代价假设下，最佳优先遍历在每个可行实例上无需初始可行解即可在有限次扩展后终止；深度优先遍历则在获得有限可行解后终止。对于用户指定的因子ε≥1，全局下界可保证两种策略的可行解代价至多为全局最优解的ε倍。我们进一步在移动操作臂检查任务中演示了感知模式、访问顺序和连续轨迹的联合选择，包括用有限迹线性时序逻辑（LTL_f）表示的动作优先级。两种遍历策略均在30秒内为所有基准实例找到可行解，平均已认证最优性间隙分别为28.1%和29.7%，而两个近期基线方法仅在约一半实例上成功。

英文摘要

We formalize the Steiner Traveling Salesman Problem (Steiner-TSP) on Graphs of Convex Sets (GCS), which seeks a minimum-cost closed trajectory through required convex sets while allowing optional transit vertices and revisits. To explore the resulting infinite solution space, we propose a unified branch-and-bound search over rooted walk prefixes. Additive lower-bound-graph costs bound committed prefixes, while a cut-separated connected-flow relaxation lower-bounds the residual cost of visiting every remaining target and returning to the root. Under a uniform positive-cost assumption, best-first traversal terminates after finitely many expansions on every feasible instance without an initial incumbent, whereas depth-first traversal does so once a finite incumbent is available. For a user-specified factor $ε\geq1$, a global lower bound certifies that either strategy's incumbent cost is at most $ε$ times the global optimum. We further demonstrate joint sensing-mode, visitation-order, and continuous-trajectory selection for a mobile-manipulator inspection task, including action precedences expressed in linear temporal logic over finite traces (LTL$_f$). Both traversal strategies find feasible solutions on all benchmark instances within 30s with mean certified optimality gaps of 28.1% and 29.7%, respectively, whereas two recent baselines succeed on only about half of the instances

发表机构

• School of Computing Science, Simon Fraser University（西蒙菲莎大学计算科学学院）

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