Replacing spectral techniques for expander ratio, normalized cut and conductance by combinatorial flow algorithms

TL;DR

This paper introduces a combinatorial flow algorithm that efficiently replaces spectral techniques for solving complex graph partitioning problems, often yielding better results and handling large-scale data within minutes.

Contribution

The authors propose a novel combinatorial algorithm that relaxes the orthogonality constraint, solves it optimally in polynomial time, and outperforms spectral methods in quality and scalability.

Findings

Algorithm solves problems on graphs with millions of nodes and hundreds of millions of edges in under 10 minutes.

The combinatorial method often produces significantly better partitioning results than spectral techniques.

The approach is strongly polynomial and leverages Hochbaum's Pseudo-Flow as a subroutine.

Abstract

Several challenging problem in clustering, partitioning and imaging have traditionally been solved using the "spectral technique". These problems include the normalized cut problem, the graph expander ratio problem, the Cheeger constant problem and the conductance problem. These problems share several common features: all seek a bipartition of a set of elements; the problems are formulated as a form of ratio cut; the formulation as discrete optimization is shown here to be equivalent to a quadratic ratio, sometimes referred to as the Raleigh ratio, on discrete variables and a single sum constraint which we call the balance or orthogonality constraint; when the discrete nature of the variables is disregarded, the continuous relaxation is solved by the spectral method. Indeed the spectral relaxation technique is a dominant method providing an approximate solution to these problems. We propose an algorithm for these problems which involves a relaxation of the orthogonality constraint only. This relaxation is shown here to be solved optimally, and in strongly polynomial time, in O(mn log((n^2) / m) for a graph on $n$ nodes and $m$ edges. The algorithm, using HPF (Hochbaum's Pseudo-Flow) as subroutine, is efficient enough to be used to solve these bi-partitioning problems on millions of elements and more than 300 million edges within less than 10 minutes. It is also demonstrated, via a preliminary experimental study, that the results of the combinatorial algorithm proposed often improve dramatically on the quality of the results of the spectral method.