A Generalized Cheeger Inequality

The generalized conductance $\phi(G,H)$ between two graphs $G$ and $H$ on the same vertex set $V$ is defined as the ratio $$\phi(G,H) = \min_{S\subseteq V} \frac{cap_G(S,\bar{S})}{ cap_H(S,\bar{S})},$$ where $cap_G(S,\bar{S})$ is the total weight of the edges crossing from $S$ to $\bar{S}=V-S$. We show that the minimum generalized eigenvalue $\lambda(L_G,L_H)$ of the pair of Laplacians $L_G$ and $L_H$ satisfies $$\lambda(L_G,L_H) \geq \phi(G,H) \phi(G)/8,$$ where $\phi(G)$ is the usual conductance of $G$. A generalized cut that meets this bound can be obtained from the generalized eigenvector corresponding to $\lambda(L_G,L_H)$. The inequality complements a recent proof that $\phi(G)$ cannot be replaced by $\Theta(\phi(G,H))$ in the above inequality, unless the Unique Games Conjecture is false.