Geodesic
Asymptotically optimal approach to the approximate solution of several problems of covering a graph by nonadjacent cycles
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 3, pp. 89-99 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the $m$-Cycle Cover Problem, which consists in covering a complete undirected graph by $m$ vertex-nonadjacent cycles with extremal total edge weight. The so-called TSP approach to the construction of an approximate algorithm for this problem with the use of a solution of the traveling salesman problem (TSP) is presented. Modifications of the algorithm for the problems Euclidean Max $m$-Cycle Cover with deterministic instances (edge weights) in a multidimensional Euclidean space and Random Min $m$-Cycle Cover with random instances $UNI(0,1)$ are analyzed. It is shown that both algorithms have time complexity $\mathcal{O}(n^3)$ and are asymptotically optimal for the number of covering cycles $m=o(n)$ and $ m\le \frac{n^{1/3}}{\ln n}$, respectively. 
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E. Kh. Gimadi; I. A. Rykov. Asymptotically optimal approach to the approximate solution of several problems of covering a graph by nonadjacent cycles. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 3, pp. 89-99. http://geodesic.mathdoc.fr/item/TIMM_2015_21_3_a9/

[1] Gimadi E.Kh., Glebov N.I., Perepelitsa V.A., “Algoritmy s otsenkami dlya zadach diskretnoi optimizatsii ”, Problemy kibernetiki, no. 31, Nauka, M., 1975, 35–42 | MR

[2] Gimadi E.Kh., “Novaya versiya asimptoticheski tochnogo algoritma resheniya evklidovoi zadachi kommivoyazhera ”, Metody optimizatsii i ikh prilozheniya, tr. XII Baikal. mezhdunar. konf., 1, Irkutsk, 2001, 117–124

[3] Serdyukov A.I., “Asimptoticheski tochnyi algoritm dlya zadachi kommivoyazhera na maksimum v evklidovom prostranstve”, Upravlyaemye cistemy, 1987, no. 27, 79–87, Novosibirsk | MR | Zbl

[4] Khachai M.Yu., Neznakhina E.D., “Polinomialnaya priblizhennaya skhema dlya evklidovoi zadachi o tsiklovom pokrytii grafa”, Tr. In-ta matematiki i mekhaniki UrO RAN, 20:4 (2014), 297–311 | MR

[5] Khachai M.Yu., Neznakhina E.D., “Approksimiruemost zadachi o minimalnom po vesu tsiklovom pokrytii grafa”, Dokl. RAN, 461:6 (2015), 644–649 | DOI | Zbl

[6] Frieze A.M., “On random symmetric travelling salesman problems”, Math. Oper. Res., 29:4 (2004), 878–890 | DOI | MR | Zbl

[7] Frieze A.M., “On the value of a random minimum spanning tree problem”, Discrete Appl. Math., 10:1 (1985), 47–56 | DOI | MR | Zbl

[8] Gabow H.N., “An efficient reduction technique for degree-restricted subgraph and bidirected network flow problems”, Proc. 15th Annual ACM Simposium on Theory of Computing, N.Y., 1983, 448–456