On the Complexity of $lambda_infty,,$ Vertex Expansion, and Spread Constant of Trees

Bobkov, Houdre, and the last author introduced a Poincare-type functional parameter, $lambda_infty$, of a graph $G$. They related $lambda_infty$ to the {em vertex expansion} of the graph via a Cheeger-type inequality, analogous to the inequality relating the spectral gap of the graph, $lambda_2$, 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 confirm this conjecture by proving that computing $lambda_infty$ is indeed NP-hard, even for weighted trees. Our gadget further proves NP-hardness of computing emph{spread constant} of a weighted tree; i.e., a geometric measure of the graph, introduced by Alon, Boppana, and Spencer, in the context of deriving an asymptotic isoperimetric inequality of Cartesian products of graphs. We conclude this case by providing a fully polynomial time approximation scheme. We further study a generalization of spread constant in machine learning literature, namely the {em maximum variance embedding} problem. For trees, we provide fast combinatorial algorithms that avoid solving a semidefinite relaxation of the problem. On the other hand, for general graphs, we propose a randomized projection method that can outperform the optimal orthogonal projection, i.e., PCA, classically used for rounding of the optimum lifted solution (to SDP relaxation) of the problem.