constant-ratio polynomial time approximation of the asymmetric minimum weight cycle cover problem with limited number of cycles

We consider polynomial time approximation of the minimum cost cycle cover problem for an edge-weighted digraph, where feasible covers are restricted to have at most $k$ disjoint cycles. in the literature this problem is referred to as minimum-weight $k$-size cycle cover problem.  the problem is closely related to the classic traveling salesman problem and vehicle routing problem and has many important applications in algorithms design and operations research. unlike its unconstrained variant, the studied problem is strongly np-hard even on undirected graphs and remains intractable in very specific settings. for any metric, the problem can be approximated in polynomial time within ratio 2, while   in fixed-dimensional euclidean spaces it admits polynomial time approximation schemes. in the same time, approximation of the more general asymmetric variant of the problem still remained weakly studied. in this paper, we propose the first constant-ratio approximation algorithm for this problem, which extends the recent breakthrough results of svensson-tarnawski-végh and traub-vygen for the asymmetric traveling salesman problem.