The multi allocation p-hub median problem (MApHM), the multi allocation uncapacitated hub location problem (MAuHLP) and the multi allocation p-hub location problem (MApHLP) are common hub location problems with several practical applications. HLPs aim to construct a network for routing tasks between different locations. Specifically, a set of hubs must be chosen and each routing must be performed using one or two hubs as stopovers. The costs between two hubs are discounted. The objective is to minimize the total transportation cost in the MApHM and additionally to minimize the set-up costs for the hubs in the MAuHLP and MApHLP. In this paper, an approximation algorithm to solve these problems is developed, which improves the approximation bound for MApHM to 3.451, for MAuHLP to 2.173 and for MApHLP to 4.552 when combined with the algorithm of Benedito & Pedrosa. The proposed algorithm is capable of solving much bigger instances than any exact algorithm in the literature. New benchmark instances have been created and published for evaluation, such that HLP algorithms can be tested and compared on huge instances. The proposed algorithm performs on most instances better than the algorithm of Benedito & Pedrosa, which was the only known approximation algorithm for these problems by now.

翻译：多分配p-中位枢纽问题（MApHM）、多分配无容量枢纽选址问题（MAuHLP）及多分配p-枢纽选址问题（MApHLP）是几类具有多种实际应用的常见枢纽选址问题。枢纽选址问题旨在构建用于不同位置间任务路由的网络。具体而言，必须选择一组枢纽，且每个路由任务需经由一个或两个枢纽作为中转站完成。枢纽之间的运输成本享受折扣。MApHM的目标是最小化总运输成本，而MAuHLP与MApHLP还需额外最小化枢纽的建设成本。本文提出一种求解这些问题的近似算法，结合Benedito & Pedrosa的算法后，将MApHM的近似比改进至3.451，MAuHLP改进至2.173，MApHLP改进至4.552。该算法能够求解远大于现有文献中任何精确算法所能处理的实例规模。研究创建并发布了新的基准测试实例用于评估，使得大规模实例上的枢纽选址算法测试与比较成为可能。在大多数测试实例上，本文算法性能优于此前已知求解这些问题的唯一近似算法——Benedito & Pedrosa算法。