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Asymptotically optimal approach to the approximate solution of several problems of covering a graph by nonadjacent cycles

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Abstract

We consider the m-Cycle Cover Problem of covering a complete undirected graph by m vertex-nonadjacent cycles of extremal total edge weight. The so-called TSP approach to the construction of an approximation algorithm for this problem with the use of a solution of the traveling salesman problem (TSP) is presented. Modifications of the algorithm for the Euclidean Max m-Cycle Cover Problem with deterministic instances (edge weights) in a multidimensional Euclidean space and the Random Min m-Cycle Cover Problem with random instances UNI(0,1) are analyzed. It is shown that both algorithms have time complexity O(n 3) and are asymptotically optimal for the number of covering cycles m = o(n) and \(m \leqslant \frac{{n^{1/3} }} {{\ln n}}\), respectively.

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Authors and Affiliations

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

    E. Kh. Gimadi & I. A. Rykov

Authors
  1. E. Kh. Gimadi
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  2. I. A. Rykov
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Correspondence to E. Kh. Gimadi.

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Original Russian Text © E.Kh.Gimadi, I.A. Rykov, 2015, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2015, Vol. 21, No. 3.

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Gimadi, E.K., Rykov, I.A. Asymptotically optimal approach to the approximate solution of several problems of covering a graph by nonadjacent cycles. Proc. Steklov Inst. Math. 295 (Suppl 1), 57–67 (2016). https://doi.org/10.1134/S0081543816090078

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  • DOI: https://doi.org/10.1134/S0081543816090078

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