A Simpler Approach for Monotone Parametric Minimum Cut: Finding the Breakpoints in Order

TL;DR

This paper introduces PBFS, a simpler and more practical algorithm for finding breakpoints in the parametric minimum cut problem, outperforming existing methods especially on large instances with many breakpoints.

Contribution

The paper presents PBFS, a new algorithm that efficiently finds breakpoints in parametric minimum cut problems with a simpler approach and better practical performance.

Findings

PBFS outperforms state-of-the-art algorithms on benchmark instances.

PBFS scales efficiently to large instances with millions of vertices.

PBFS is particularly effective on instances with many breakpoints.

Abstract

We present parametric breadth-first search (PBFS), a new algorithm for solving the parametric minimum cut problem in a network with source-sink-monotone capacities. The objective is to find the set of breakpoints, i.e., the points at which the minimum cut changes. It is well known that this problem can be solved in the same asymptotic runtime as the static minimum cut problem. However, existing algorithms that achieve this runtime bound involve fairly complicated steps that are inefficient in practice. PBFS uses a simpler approach that discovers the breakpoints in ascending order, which allows it to achieve the desired runtime bound while still performing well in practice. We evaluate our algorithm on benchmark instances from polygon aggregation and computer vision. Polygon aggregation was recently proposed as an application for parametric minimum cut, but the monotonicity property has not been exploited fully. PBFS outperforms the state of the art on most benchmark instances, usually by a factor of 2-3. It is particularly strong on instances with many breakpoints, which is the case for polygon aggregation. Compared to the existing min-cut-based approach for polygon aggregation, PBFS scales much better with the instance size. On large instances with millions of vertices, it is able to compute all breakpoints in a matter of seconds.