The Lovász Theta Function and a Semidefinite Programming Relaxation of Vertex Cover

Let vc(G) denote the minimum size of a vertex cover of a graph G = (V,E). It is well known that one can approximate vc(G) to within a factor of 2 in polynomial time; and despite considerable investigation, no (2−ε)-approximation algorithm has been found for any ε > 0. Because of the many connections between the independence number α(G) and the Lovász theta function θ(G), and because vc(G) = |V | − α(G), it is natural to ask how well |V | − θ(G) approximates vc(G). It is not difficult to show that these quantities are within a factor of 2 of each other (|V | − θ(G) is never less than the value of the canonical linear programming relaxation of vc(G)); our main result is that vc(G) can be more than (2− ε) times |V | − θ(G) for any ε > 0. We also investigate a stronger lower bound than |V | − θ(G) for vc(G).