Abstract
We study the minimum weight k-size cycle cover problem (Min-k-SCCP), which consists in partitioning a complete weighted digraph into k vertex-disjoint cycles of minimum total weight. This problem is a generalization of the known traveling salesman problem and a special case of the classical vehicle routing problem. It is known that Min-k-SCCP is strongly NP-hard in the general case and preserves its intractability even in the geometric statement. For Euclidean Min-k-SCCP in ℝd with k = O(log n), we construct a polynomialtime approximation scheme (PTAS), which generalizes the approach proposed earlier for planar Min-2-SCCP. For each fixed c > 1 the scheme finds a (1 + 1/c)-approximate solution in time \(O\left( {{n^{O\left( d \right)}}{{\left( {\log n} \right)}^{{{\left( {O\left( {\sqrt {dc} } \right)} \right)}^{^{d - 1}}}}}} \right)\).
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Original Russian Text © E.D. Neznakhina, 2015, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2015, Vol. 21, No. 3.
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Neznakhina, E.D. A PTAS for Min-k-SCCP in Euclidean space of arbitrary fixed dimension. Proc. Steklov Inst. Math. 295 (Suppl 1), 120–130 (2016). https://doi.org/10.1134/S0081543816090133
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DOI: https://doi.org/10.1134/S0081543816090133