Solving the Rural Postman problem using the Adleman-Lipton model

In this survey we investigate the application of the Adleman-Lipton model on Rural Postman problem, which given an undirected graph G = (V,E) with positive integer lengths on each of its edges and a subset E ′ ⊆ E, asks whether there exists a hamiltonian circuit that includes each edge of E ′ and has total cost (sum of edge lengths) less or equal to a given integer B (we are allowed to use any edges of the set E − E ′ , but we must use all edges of the set E). The Rural Postman problem (RPP) is a very interesting NP-complete problem used, especially, in network optimization. RPP is actually a special case of the Route Inspection problem, where we need to traverse all edges of an undirected graph at a minimum total cost. As all NP-complete problems, it currently admits no efficient solution and if actually P 6= NP as it is widely accepted to be, it cannot admit a polynomial time algorithm to solve it. The application of the Adleman-Lipton model on this problem, provides an efficient way to solve RPP, as it is the fact for many other hard problems on which the Adleman-Lipton model has been applied. In this survey, we provide a polynomial algorithm based on the Lipton-Adleman model, which solves the RPP in O(n) time, where n refers to the input of the problem.