Abstract The Steiner Team Orienteering Problem (STOP) is defined on a digraph in which arcs are associated with traverse times, and whose vertices are labeled as either mandatory or profitable, being the latter provided with rewards (profits). Given a homogeneous fleet of vehicles M, the goal is to find up to m = | M | disjoint routes (from an origin vertex to a destination one) that maximize the total sum of rewards collected while satisfying a given limit on the route’s duration. Naturally, all mandatory vertices must be visited. In this work, we show that solely finding a feasible solution for STOP is NP-hard and propose a Large Neighborhood Search (LNS) heuristic for the problem. The algorithm is provided with initial solutions obtained by means of the matheuristic framework known as Feasibility Pump (FP). In our implementation, FP uses as backbone a commodity-based formulation reinforced by three classes of valid inequalities. To our knowledge, two of them are also introduced in this work. The LNS heuristic itself combines classical local searches from the literature of routing problems with a long-term memory component based on Path Relinking. We use the primal bounds provided by a state-of-the-art cutting-plane algorithm from the literature to evaluate the quality of the solutions obtained by the heuristic. Computational experiments show the efficiency and effectiveness of the proposed heuristic in solving a benchmark of 387 instances. Overall, the heuristic solutions imply an average percentage gap of only 0.54% when compared to the bounds of the cutting-plane baseline. In particular, the heuristic reaches the best previously known bounds on 382 of the 387 instances. Additionally, in 21 of these cases, our heuristic is even able to improve over the best known bounds.

中文翻译：

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耦合可行性泵和大邻域搜索解决斯坦纳团队定向运动问题

摘要 Steiner Team Orienteering Problem (STOP) 被定义在一个有向图中，其中弧与遍历时间相关联，并且其顶点被标记为强制性或盈利，后者提供奖励（利润）。给定同质车队 M，目标是找到最多 m = | 男 | 不相交的路线（从起点到终点），在满足给定的路线持续时间限制的同时最大化收集到的奖励总和。自然地，必须访问所有强制顶点。在这项工作中，我们表明仅为 STOP 寻找可行的解决方案是 NP 难的，并针对该问题提出了大型邻域搜索 (LNS) 启发式方法。该算法提供了通过称为可行性泵 (FP) 的数学框架获得的初始解。在我们的实施中，FP 使用由三类有效不等式强化的基于商品的公式作为主干。据我们所知，在这项工作中也介绍了其中的两个。LNS 启发式本身将路由问题文献中的经典局部搜索与基于路径重新链接的长期记忆组件相结合。我们使用文献中最先进的切割平面算法提供的原始边界来评估启发式获得的解决方案的质量。计算实验显示了所提出的启发式在解决 387 个实例的基准方面的效率和有效性。总体而言，与切割平面基线的边界相比，启发式解决方案意味着平均百分比差距仅为 0.54%。特别是，启发式在 387 个实例中的 382 个上达到了先前已知的最佳边界。