On the Complexity and Approximation of $\lambda_\infty\,,$ Spread Constant and Maximum Variance Embedding

Twenty years ago, Bobkov, Houdr\'e, and the last author introduced a Poincar\'e-type functional graph parameter, $\lambda_\infty(G)$, of a graph $G$, and related it to the {\em vertex expansion} of $G$ via a Cheeger-type inequality. This is analogous to the Cheeger-type inequality relating the spectral gap, $\lambda_2(G)$, of the graph to its {\em edge expansion}. While $\lambda_2$ can be computed efficiently, the computational complexity of $\lambda_\infty$ has remained an open question. Following the work of the second author with Raghavendra and Vempala, wherein the complexity of $\lambda_\infty$ was related to the so-called Small-Set Expansion (SSE) problem, it has been believed that computing $\lambda_\infty$ is a hard problem. We settle this question by proving that computing $\lambda_\infty$ is indeed NP-hard. Additionally, we use our techniques to prove NP-hardness of computing the spread constant (of a graph), a geometric measure introduced by Alon, Boppana, and Spencer, in the context of deriving an asymptotic isoperimetric inequality on cartesian products of graphs. We complement our hardness results by providing approximation schemes for computing $\lambda_\infty$ and the spread constant of star graphs, and investigate constant approximability for weighted trees. Finally, we provide improved approximation results for the maximum variance embedding problem for general graphs, by replacing the optimal orthogonal projection (PCA) with a randomized projection approach.