Approximability of the minimum-weight k-size cycle cover problem

Abstract

The cycle cover problem is a combinatorial optimization problem, which is to find a minimum cost cover of a given weighted digraph by a family of vertex-disjoint cycles. We consider a special case of this problem, where, for a fixed number k, all feasible cycle covers are restricted to be of the size k. We call this case the minimum weight k-size cycle cover problem (Min-k-SCCP). Since each cycle in a cover can be treated as a tour of some vehicle visiting an appropriate set of clients, the problem in question is closely related to the vehicle routing problem. Moreover, the studied problem is a natural generalization of the well-known traveling salesman problem (TSP), since the Min-1-SCCP is equivalent to the TSP. We show that, for any fixed $k>1$, the Min-k-SCCP is strongly NP-hard in the general setting. The Metric and Euclidean special cases of the problem are intractable as well. Also, we prove that the Metric Min-k-SCCP belongs to APX class and has a 2-approximation polynomial-time algorithm, even if k is not fixed. For the Euclidean Min-2-SCCP in the plane, we present a polynomial-time approximation scheme extending the famous result obtained by S. Arora for the Euclidean TSP. Actually, for any fixed $c>1$, the scheme finds a $(1+1/c)$-approximate solution of the Euclidean Min-2-SCCP in $O(n^3(\log n)^{O(c)})$ time.