Approximate Maximum Flow on Separable Undirected Graphs

TL;DR

This paper introduces faster algorithms for approximate maximum flow in undirected graphs with good separator structures, leveraging grouped L2 flow and spectral sparsification techniques to improve efficiency.

Contribution

It presents novel algorithms that significantly speed up approximate maximum flow computations in separable undirected graphs using spectral methods.

Findings

Achieves $ ilde{O}(m^{6/5} ext{poly}(rac{1}{\e}))$ time for approximate max flow

Extends to image problems in 2D and 3D with similar speedups

Provides faster spectral vertex sparsification algorithms for well-separated graphs

Abstract

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.