# V.V. Shenmaier. An algorithm for the polyhedral cycle cover problem with restrictions on the number and length of cycles

A cycle cover of a graph is a spanning subgraph whose connected components are simple cycles. Given a complete weighted directed graph, consider the intractable problem of finding a maximum-weight cycle cover which satisfies an upper bound on the number of cycles and a lower bound on the number of edges in each cycle. We suggest a polynomial-time algorithm for solving this problem in the geometric case when the vertices of the graph are points in a multidimensional real space and the distances between them are induced by a positively homogeneous function whose unit ball is an arbitrary convex polytope with a fixed number of facets. The obtained result extends the ideas underlying the well-known algorithm for the polyhedral Max TSP.

Keywords: cycle cover, Max TSP, polyhedral metric, optimal solution, polynomial-time algorithm

The paper was received by the Editorial Office on May 8, 2018.

Funding Agency: This work was supported by the Russian Science Foundation (project no. 16-11-10041).

Vladimir Vladimirovich Shenmaier, Cand. Sci. (Phys.-Math.), Sobolev Institute of Mathematics SB RAS, Novosibirsk, 630090 Russia, e-mail: shenmaier@mail.ru

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