  We develop a polynomial time 3/2-approximation algorithm to solve the vertex
cover problem on a class of graphs satisfying a property called ``active edge
hypothesis''. The algorithm also guarantees an optimal solution on specially
structured graphs. Further, we give an extended algorithm which guarantees a
vertex cover $S_1$ on an arbitrary graph such that $|S_1|\leq {3/2} |S^*|+\xi$
where $S^*$ is an optimal vertex cover and $\xi$ is an error bound identified
by the algorithm. We obtained $\xi = 0$ for all the test problems we have
considered which include specially constructed instances that were expected to
be hard. So far we could not construct a graph that gives $\xi \not= 0$.
