Approximate Maximum Flow on Separable Undirected Graphs

We present faster algorithms for approximate maximum flow in undirected graphs with good separator structures, such as bounded genus, minor free, and geometric graphs. Given such a graph with $n$ vertices, $m$ edges along with a recursive $\sqrt{n}$-vertex separator structure, our algorithm finds an $1-\epsilon$ approximate maximum flow in time $\tilde{O}(m^{6/5} \poly{\epsilon^{-1}})$, ignoring poly-logarithmic terms. Similar speedups are also achieved for separable graphs with larger size separators albeit with larger run times. These bounds also apply to image problems in two and three dimensions. Key to our algorithm is an intermediate problem that we term grouped $L_2$ flow, which exists between maximum flows and electrical flows. Our algorithm also makes use of spectral vertex sparsifiers in order to remove vertices while preserving the energy dissipation of electrical flows. We also give faster spectral vertex sparsification algorithms on well separated graphs, which may be of independent interest.