M.Yu. Khachai, E.D. Neznakhina, K.V. Ryzhenko. Polynomial approximability of the asymmetric problem of covering a graph by a bounded number of cycles ... P. 261-273

Recently, O. Svensson and V. Traub provided the first proof of the polynomial-time approximability with fixed ratios for the Asymmetric Traveling Salesman Problem (ATSP). Just as the famous Christofides—Serdyukov algorithm for the symmetric routing problems, this breakthrough result, applied as a "black box", has opened an opportunity for developing the first polynomial-time approximation algorithms with constant ratios for several related combinatorial problems. At the same time, problems have been revealed in which this simple approach, based on reducing a given instance to one or more auxiliary ATSP instances, does not succeed. In the present paper, we extend the Svensson—Traub approach to the wider class of problems about finding a minimum-weight cycle cover of an edge-weighted directed graph with an additional constraint on the number of cycles. In particular, it is shown for the first time that the Minimum Weight Cycle Cover Problem with at most $k$ cycles admits a polynomial-time approximation with constant ratio $\max\{22+\varepsilon,4+k\}$ for arbitrary $\varepsilon>0$.

Keywords: cycle cover of a graph, asymmetric traveling salesman problem, fixed ratio approximation algorithm

Received August 4, 2023

Revised August 18, 2023

Accepted August 21, 2023

Funding Agency: This work was supported by the Russian Science Foundation (project no. 22-21-00672).

Mikhail Yur’evich Khachai, Dr. Phys.-Math. Sci, RAS Corresponding Member, Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: mkhachay@imm.uran.ru

Ekaterina Dmitrievna Neznakhina, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: eneznakhina@yandex.ru

Kseniya Valer’evna Ryzhenko, doctoral student, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: kseniarizhenko@gmail.com

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Cite this article as: M.Yu. Khachai, E.D. Neznakhina, K.V. Ryzhenko. Polynomial approximability of the asymmetric problem of covering a graph by a bounded number of cycles. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 3, pp. 261–273; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2023, Vol. 323, Suppl. 1, pp. S121–S132.