We develop fast algorithms for solving regression problems on graphs where one is given the value of a function at some vertices, and must find its smoothest possible extension to all vertices. The extension we compute is the absolutely minimal Lipschitz extension, and is the limit for large $p$ of $p$-Laplacian regularization. We present an algorithm that computes a minimal Lipschitz extension in expected linear time, and an algorithm that computes an absolutely minimal Lipschitz extension in expected time $O (m n)$. The latter algorithm has variants that seem to run much faster in practice. These extensions are particulary amenable to regularization: we can perform $l_{0}$ regularization on the given values in polynomial time and $l_{1}$ regularization on the graph edge weights in time $\widetilde{O} (m^{3/2})$. Our algorithms naturally extend to directed graphs.