Approximability of the minimum-weight k-size cycle cover problem / Khachay M., Neznakhina K. // Journal of Global Optimization. - 2016. - V. 66, l. 1. - P. 65-82.

ISSN:
09255001
Type:
Article
Abstract:
The cycle cover problem is a combinatorial optimization problem, which is to find a minimum cost cover of a given weighted digraph by a family of vertex-disjoint cycles. We consider a special case of this problem, where, for a fixed number k, all feasible cycle covers are restricted to be of the size k. We call this case the minimum weight k-size cycle cover problem (Min-k-SCCP). Since each cycle in a cover can be treated as a tour of some vehicle visiting an appropriate set of clients, the problem in question is closely related to the vehicle routing problem. Moreover, the studied problem is a natural generalization of the well-known traveling salesman problem (TSP), since the Min-1-SCCP is equivalent to the TSP. We show that, for any fixed k> 1 , the Min-k-SCCP is strongly NP-hard in the general setting. The Metric and Euclidean special cases of the problem are intractable as well. Also, we prove that the Metric Min-k-SCCP belongs to APX class and has a 2-approximation polynomial-time algorithm, even if k is not fixed. For the Euclidean Min-2-SCCP in the plane, we present a polynomial-time approximation scheme extending the famous result obtained by S. Arora for the Euclidean TSP. Actually, for any fixed c> 1 , the scheme finds a (1 + 1 / c) -approximate solution of the Euclidean Min-2-SCCP in O(n3(log n) O ( c )) time. © 2015, Springer Science+Business Media New York.
Author keywords:
Cycle cover problem (CCP); NP-hard problem; Polynomial-time approximation scheme (PTAS); Traveling salesman problem (TSP)
Index keywords:
Algorithms; Approximation theory; Combinatorial optimization; Computational complexity; Optimization; Polynomial approximation; Polynomials; Vehicle routing; Approximate solution; Combinatorial optimi
DOI:
10.1007/s10898-015-0391-3
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Affiliations Krasovsky Institute of Mathematics and Mechanics, Ural Federal University, Ekaterinburg, Russian Federation
Author Keywords Cycle cover problem (CCP); NP-hard problem; Polynomial-time approximation scheme (PTAS); Traveling salesman problem (TSP)
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Correspondence Address Khachay, M.; Krasovsky Institute of Mathematics and Mechanics, Ural Federal UniversityRussian Federation; email: mkhachay@imm.uran.ru
Publisher Springer New York LLC
CODEN JGOPE
Language of Original Document English
Abbreviated Source Title J of Global Optim
Source Scopus