Complexity analysis and optimization of the shortest path tour problem

The shortest path tour problem (SPTP) consists in finding a shortest path from a given origination node s to a given destination node d in a directed graph with nonnegative arc lengths with the constraint that the optimal path P should successively and sequentially pass through at least one node from given node subsets T1, T2, . . . , TN, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T_i \cap T_j = \emptyset, \forall\ i, j=1,\ldots,N,\ i \neq j}$$\end{document}. In this paper, it will proved that the SPTP belongs to the complexity class P and several alternative techniques will be presented to solve it.