Solving the Rural Postman problem using the Adleman-Lipton model

TL;DR

This paper explores applying the Adleman-Lipton quantum model to efficiently solve the NP-complete Rural Postman problem, providing a polynomial-time algorithm that improves upon traditional methods.

Contribution

It introduces a polynomial-time algorithm based on the Adleman-Lipton model for solving the NP-complete Rural Postman problem, demonstrating potential for quantum approaches in network optimization.

Findings

Provides a polynomial $ ext{O}(n^2)$ time algorithm for RPP

Shows the applicability of the Adleman-Lipton model to NP-complete problems

Highlights potential for quantum algorithms in solving complex network problems

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.