The maximum traveling salesman problem (Max TSP) consists of finding a Hamiltonian cycle with the maximum total weight of the edges in a given complete weighted graph. This problem is APX-hard in the general metric ca...
详细信息
The maximum traveling salesman problem (Max TSP) consists of finding a Hamiltonian cycle with the maximum total weight of the edges in a given complete weighted graph. This problem is APX-hard in the general metric case but admits polynomial-time approximation schemes in the geometric setting, when the edge weights are induced by a vector norm in fixed-dimensional real space. We propose the first approximation scheme for Max TSP in an arbitrary metric space of fixed doubling dimension. The proposed algorithm implements an efficient PTAS which, for any fixed epsilon is an element of (0,1), computes a (1 - epsilon)-approximate solution of the problem in cubic time. Additionally, we suggest a cubic-time algorithm which finds asymptotically optimal solutions of the metric Max TSP in fixed and sublogarithmic doubling dimensions.
We introduce a randomized approximation algorithm for NP-hard problem of finding a subset of m vectors chosen from n given vectors in multidimensional Euclidean space R-k such that the norm of the corresponding sum-ve...
详细信息
ISBN:
(纸本)9783319449142;9783319449135
We introduce a randomized approximation algorithm for NP-hard problem of finding a subset of m vectors chosen from n given vectors in multidimensional Euclidean space R-k such that the norm of the corresponding sum-vector is maximum. We derive the relation between algorithm's time complexity, relative error and failure probability parameters. We show that the algorithm implements Polynomial-time Randomized Approximation Scheme (PRAS) for the general case of the problem. Choosing particular parameters of the algorithm one can obtain asymptotically exact algorithm with significantly lower time complexity compared to known exactalgorithm. Another set of parameters provides polynomial-time 1/2-approximation algorithm for the problem. We also show that the algorithm is applicable for the related (minimization) clustering problem allowing to obtain better performance guarantees than existing algorithms.
暂无评论