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 $(2 - \varepsilon)$-approximation algorithm has been found for any $\varepsilon > 0$. Because of the many connections between the independence number $\alpha(G)$ and the Lovasz theta function $\vartheta(G)$, and because vc(G) = |V| - \alpha(G)$, it is natural to ask how well |V| - \vartheta(G)$ approximates vc(G). It is not difficult to show that these quantities are within a factor of 2 of each other ($|V| - \vartheta(G)$ 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 $(2 - \varepsilon)$ times $|V| - \vartheta(G)$ for any $\varepsilon > 0$. We also investigate a stronger lower bound than $|V|- \vartheta(G)$ for vc(G).