Molecular solutions for the Maximum K-colourable Sub graph Problem in Adleman-Lipton model

Abstract

Adleman (1) showed that deoxyribonucleic acid (DNA) strands could be employed towards calculating solutions to an instance of the Hamiltonian path problem (HPP). Lipton (5) also demonstrated that Adleman's techniques could be used to solve the Satisfiability problem. In this paper, we use Adleman-Lipton model for developing a DNA algorithm to solve Maximum k-colourable Sub graph problem. In spite of the NP-hardness of Maximum k-colourable Sub graph problem our DNA procedures is done in a polynomial time. Recently, DNA computing has considerable attention as one of non-silicon based computing. Watson-Crick complementarity and massive parallelism are two important features of DNA. By using these features, one can solve an NP-complete problem, which usually needs exponential time on a silicon-based computer, in a polynomial number of steps with DNA molecules (3). Adleman (1) solved Hamiltonian path problem of size n in spite of NP-hardness of the problem in O(n) steps using DNA molecules. That is the first work for DNA computing. The second NP-hard problem that has solved by DNA computing is Satisfiability (SAT), Lipton (5) showed that the Adelman's manner could be used to determine SAT. Moreover, procedures for primitive operations, such as logic or arithmetic operations, have also been proposed so as to apply DNA computing in a wide range of problems (3-4, 6-14). In this paper, the DNA operations proposed by Adleman (1) and Lipton (5) are used for figuring out solutions of Maximum k- colourable Sub graph problem. Given an undirected graph G = (V, E) with an assignment of weights to the edges w: E → N and an integer k ∈ {2, 3, …, |V|}, we try to find maximum