A Common Variable Minimax Theorem for Graphs

Ronald R. Coifman, Nicholas F. Marshall, Stefan Steinerberger

Abstract: Let \({\mathcal {G}} = \{G_1 = (V, E_1), \ldots , G_m = (V, E_m)\}\) be a collection of m graphs defined on a common set of vertices V but with different edge sets \(E_1, \ldots , E_m\). Informally, a function \(f :V \rightarrow {\mathbb {R}}\) is smooth with respect to \(G_k = (V,E_k)\) if \(f(u) \sim f(v)\) whenever \((u, v) \in E_k\). We study the problem of understanding whether there exists a nonconstant function that is smooth with respect to all graphs in \({\mathcal {G}}\), simultaneously, and how to find it if it exists.