In the Minimum Clique Routing Problem on Cycles \textsc{MCRPC} we are given a cycle together with a set of demands (weighted origin-destination pairs) and the goal is to route all the pairs minimizing the maximum weighted clique of the intersection graph induced by the routing. The vertices of this graph are the demands with their corresponding weights and two demands are adjacent when their routes share at least one arc. In this work we are not only interested in the \textsc{MCRPC} but also in two natural subproblems. First, we consider the situation where the demands are disjoint, in the sense that every two demands do not share any of their corresponding ends. Second, we analyze the subproblem where the weights of the routes are all equal. We first show that the problem is NP-complete even in the subproblem of disjoint demands. For the case of arbitrary weights, we exhibit a simple combinatorial 2-approximation algorithm and a $\frac{3}{2}$-approximation algorithm based on rounding a solution of a relaxation of an integer linear programming formulation of our problem. Finally, we give a Fixed Parameter Tractable algorithm for the case of uniform weights, whose parameter is related to the maximum degree of the intersection graph induced by any routing.

翻译：在《环上的最小团路由问题》（\textsc{MCRPC}）中，给定一个环及一组需求（带权重的起止点对），目标是路由所有需求，最小化由路由生成的交叉图的最大加权团。该图的顶点是各需求及其对应权重，两个需求相邻当且仅当它们的路由共享至少一条弧。本文不仅关注\textsc{MCRPC}，还研究两个自然子问题：首先，考虑需求不相交的情形，即任意两个需求对应的起止点均不重叠；其次，分析所有路由权重相等的子问题。我们首先证明，即使在需求不相交的子问题中，该问题也是NP完全的。对于一般权重情形，我们给出一个简单的组合2-近似算法，以及一个基于整数线性规划松弛解舍入的$\frac{3}{2}$-近似算法。最后，针对均匀权重情形，给出一个参数固定可解算法，其参数与任意路由生成的交叉图的最大度相关。