Efficient PTAS for the maximum traveling salesman problem in a metric space of fixed doubling dimension

The maximum traveling salesman problem (Max TSP) consists of finding a Hamiltonian cycle with the total weight edges in given complete weighted graph. This is APX-hard general metric case but admits polynomial-time approximation schemes geometric setting, when edge weights are induced by vector norm fixed-dimensional real space. We propose first scheme for Max TSP an arbitrary space fixed doubling dimension. proposed algorithm implements efficient PTAS which, any \(\varepsilon \in (0,1)\), computes \((1-\varepsilon )\)-approximate solution cubic time. Additionally, we suggest cubic-time which finds asymptotically optimal solutions and sublogarithmic dimensions.