Polynomial-time approximation scheme for a Euclidean problem on a cycle covering of a graph

We study the Min-$k$-SCCP problem on a partition of a complete weighted digraph into $k$ vertex-disjoint cycles of minimum total weight. This problem is a natural generalization of the known Traveling salesman problem (TSP) and has a number of applications in operations research and data analysis. We show that the problem is strongly $NP$-hard and preserves intractability even in the geometric statement. For a metric special case of the problem, a new polynomial $2$-approximation algorithm is proposed. For the Euclidean Min-$2$-SCCP, a polynomial-time approximation scheme based on Arora's approach is built.