We consider the problem of collectively delivering a package from a specified source to a designated target location in a graph, using multiple mobile agents. Each agent starts from some vertex of the graph; it can move along the edges of the graph and can pick up the package from a vertex and drop it in another vertex during the course of its movement. However, each agent has limited energy budget allowing it to traverse a path of bounded length B; thus, multiple agents need to collaborate to move the package to its destination. Given the positions of the agents in the graph and their energy budgets, the problem of finding a feasible movement schedule is called the Collaborative Delivery problem and has been studied before.

One of the open questions from previous results is what happens when the delivery must follow a fixed path given in advance. Although this special constraint reduces the search space for feasible solutions, the problem remains NP hard, as the general version of the problem. We consider the optimization version of the problem that asks for the optimal energy budget B per agent which allows for a feasible delivery schedule along a fixed path, given the initial positions of the agents. We provide polynomial time approximation algorithms for both directed and undirected graphs, and establish hardness of approximation for the directed case. Note that the fixed path version of collaborative delivery requires completely different techniques since a single agent may be used multiple times, unlike the general version of collaborative delivery studied before. We show that restricting each agent to a single pickup allows better approximations for fixed path collaborative delivery compared to the original problem. Finally, we provide a polynomial time algorithm for determining a feasible delivery strategy, if any exists, for a given budget B when the number of available agents is bounded by a constant.

我们考虑使用多个移动代理将包裹从指定来源集中交付到图形中的指定目标位置的问题。每个代理都从图的某个顶点开始；它可以沿着图的边缘移动，并且可以在移动过程中从一个顶点拾取包装并将其放置在另一个顶点中。但是，每个主体的能量预算都有限，因此它只能穿越边界长度为B的路径。因此，多个代理需要协作才能将包裹移至其目的地。给定代理在图中的位置及其能量预算，找到可行的移动计划的问题称为协作交付问题，并且之前已进行过研究。

先前结果中未解决的问题之一是，当交付必须遵循预先给定的固定路径时，会发生什么情况。尽管此特殊约束减少了可行解的搜索空间，但作为该问题的常规版本，该问题仍然是NP难题。我们考虑要求最佳能源预算B的问题的优化版本给定每个代理商的数量，这可以在给定代理商的初始位置的情况下，沿着固定路径制定可行的交货计划。我们为有向图和无向图提供多项式时间逼近算法，并为有向情况建立近似硬度。请注意，协作交付的固定路径版本需要完全不同的技术，因为单个代理可以多次使用，这与之前研究的协作交付的一般版本不同。我们证明，与原始问题相比，将每个座席限制在一个接机范围内，可以更好地近似固定路径协作交付。最后，我们提供了多项式时间算法，用于针对给定的预算B确定可行的投放策略（如果有） 当可用代理的数量受常数限制时。