Polynomial Time Approximation Scheme for the Minimum-weight k-Size Cycle Cover Problem in Euclidean space of an arbitrary fixed dimension

We study the Min-k-SCCP on the partition of a complete weighted digraph by k vertex-disjoint cycles of minimum total weight. This problem is the generalization of the well-known traveling salesman problem (TSP) and the special case of the classical vehicle routing problem (VRP). It is known that the problem Min-k-SCCP is strongly NP-hard and remains intractable even in the geometric statement. For the Euclidean Min-k-SCCP in Rd, we construct a polynomial-time approximation scheme, which generalizes the approach proposed earlier for the planar Min-2-SCCP. For any fixed c > 1, the scheme finds a (1 + 1/c)-approximate solution in time of O(nd+1(k log n)(O (√dc))d-1 2k).