Sanjaya Gajurel & Roger Bielefeld International Journal of Experimental Algorithms (IJEA), Volume (3) : Issue (1) : 2012 9 A Fast Near Optimal Vertex Cover Algorithm (NOVCA) Sanjaya Gajurel sxg125@case.edu Advanced Research Computing Case Western Reserve University Cleveland, OH, US Roger Bielefeld rab5@case.edu Advanced Research Computing Case Western Reserve University Cleveland, OH, US Abstract This paper describes an extremely fast polynomial time algorithm, the Near Optimal Vertex Cover Algorithm (NOVCA) that produces an optimal or near optimal vertex cover for any known undirected graph G (V, E). NOVCA is based on the idea of (i) including the vertex having maximum degree in the vertex cover and (ii) rendering the degree of a vertex to zero by including all its adjacent vertices. The two versions of algorithm, NOVCA-I and NOVCA-II, have been developed. The results identifying bounds on the size of the minimum vertex cover as well as polynomial complexity of algorithm are given with experimental verification. Future research efforts will be directed at tuning the algorithm and providing proof for better approximation ratio with NOVCA compared to any other available vertex cover algorithms. Keywords: Vertex Cover Problem, Combinatorial Problem, NP-Complete Problem, Approximation Algorithm. 1. INTRODUCTION The Vertex Cover (VC) of a graph G(V,E) with vertex set V and edge set E is a subset of vertices C of V (C ⊆ V) such that every edge of G has at least one endpoint in C. In 1972 Richard Karp [1] showed that identification of minimal VC in a graph is an NP-complete problem. Various algorithmic approaches have been used to tackle NP complete problems. The Vertex Cover problem, one of the NP complete problems, has been actively studied because of its important research and application implications. Polynomial-time approximation and heuristic algorithms for VC have been developed but none of them guarantee optimality. By using the definition of approximation ratio, VC has an approximation ratio of ρ(n) for any input of size n. The solution C produced by approximation algorithm is within the factor of ρ(n) of the solution C* of an optimal algorithm i.e. C*/C ≤ ρ(n). Also, the approximation algorithm has approximation ratio of 2 – ε, where 0 < ε < 1. A 2-approximation [2] algorithm has been trivially obtained and similar approximation algorithms have been developed [3], [4] with an approximation of (2 – (ln (ln n)/2ln n)), where n is the number of vertices. Halperin [5] achieved an approximation factor of (2 – (1 – o(1))(2ln (ln Δ)/ ln Δ)) with maximum degree at most Δ. Karakostas [6] attained an approximation factor of (2 – θ(1/(log n) 1/2 ))), the best approximation yet, by using the semidefinite programming relaxation of VC. Evolutionary algorithms (EA) that are randomized search heuristics have also been used for solving combinatorial optimization problems including VC [7], [8]. Vertex Cover problems have been solved in O (1.2738k + kn) time [9] by using a bounded search technique where a function of a parameter restricts the search space. Abu-Khazm et al. have identified crown structure to reduce the size of both n and k [10]. It has been known that when relevant parameters are fixed, NP-complete problems can be solved in polynomial time. In both [10] and [11], n is the input size and k is the positive integer parameter. Though not guaranteed to