1 research outputs found
Replacing spectral techniques for expander ratio, normalized cut and conductance by combinatorial flow algorithms
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 nodes and 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.Comment: The paper was submitted to ArXiv system by autho