The Steiner Multicycle problem consists of, given a complete graph, a weight function on its vertices, and a collection of pairwise disjoint non-unitary sets called terminal sets, finding a minimum weight collection of vertex-disjoint cycles in the graph such that, for every terminal set, all of its vertices are in a same cycle of the collection. This problem generalizes the Traveling Salesman problem and therefore is hard to approximate in general. On the practical side, it models a collaborative less-than-truckload problem with pickup and delivery locations. Using an algorithm for the Survivable Network Design problem and T -joins, we obtain a 3-approximation for the metric case, improving on the previous best 4-approximation. Furthermore, we present an (11/9)-approximation for the particular case of the Steiner Multicycle in which each edge weight is 1 or 2. This algorithm can be adapted to obtain a (7/6)-approximation when every terminal set contains at least 4 vertices. Finally, we devise an O(lg n)-approximation algorithm for the asymmetric version of the problem.

翻译：斯坦纳多循环问题：给定一个完全图、其顶点上的权重函数，以及一组两两不相交的非单元终端集合（称为终端集），寻找图中一组最小权重的顶点不相交循环，使得每个终端集合中的所有顶点都位于同一循环中。该问题推广了旅行商问题，因此在一般情况下难以近似。在实际应用中，它模拟了涉及取货和送货地点的协作式零担运输问题。利用可生存网络设计问题的算法和T-连接，我们获得了度规情况下的3-近似，改进了此前最优的4-近似。此外，针对每条边权重为1或2的斯坦纳多循环特例，我们提出了(11/9)-近似算法。当下每个终端集合至少包含4个顶点时，该算法可调整为(7/6)-近似。最后，我们针对该问题的不对称版本设计了O(lg n)-近似算法。