  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.
