On the Complexity of λ_∞ , Vertex Expansion, and Spread Constant of Trees
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Bobkov, Houdré, and the last author introduced a Poincaré-type functional parameter, λ_∞, of a graph G. They related λ_∞ to the vertex expansion of the graph via a Cheeger-type inequality, analogous to the inequality relating the spectral gap of the graph, λ_2, to its edge expansion. While λ_2 can be computed efficiently, the computational complexity of λ_∞ has remained an open question. Following the work of the second author with Raghavendra and Vempala, wherein the complexity of λ_∞ was related to the so-called small-set expansion (SSE) problem, it has been believed that computing λ_∞ is a hard problem. We confirm this conjecture by proving that computing λ_∞ is indeed NP-hard, even for weighted trees. Our gadget further proves NP-hardness of computing 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 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.
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