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An Algorithm for the Polyhedral Cycle Cover Problem with Constraints on the Number and Length of Cycles

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Abstract

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 where 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.

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Funding

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

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 630090, Russia

    V. V. Shenmaier

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  1. V. V. Shenmaier
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Correspondence to V. V. Shenmaier.

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Shenmaier, V.V. An Algorithm for the Polyhedral Cycle Cover Problem with Constraints on the Number and Length of Cycles. Proc. Steklov Inst. Math. 307 (Suppl 1), 142–150 (2019). https://doi.org/10.1134/S0081543819070113

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  • DOI: https://doi.org/10.1134/S0081543819070113

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