Solving the Rural Postman problem using the Adleman-Lipton model

Abstract

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 $E^{'}\subseteq E$, 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 $P\neq NP$ 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 $\mathcal{O}(n^{2})$ time, where n refers to the input of the problem.