Extensions of Karger's Algorithm: Why They Fail in Theory and How They Are Useful in Practice

TL;DR

This paper investigates the limitations of extending Karger's algorithm to other cut problems, proves some generalizations fail, but introduces a new practical algorithm for seeded segmentation that performs well in experiments.

Contribution

The paper demonstrates the failure of many Karger-based generalizations for certain cut problems and proposes a new, efficient algorithm for seeded segmentation with practical advantages.

Findings

Many natural extensions of Karger's algorithm cannot solve s-t mincut or normalized cut efficiently.

A new linear-time algorithm for seeded segmentation performs comparably to existing methods.

The proposed method relates to the random walker algorithm and performs well on image segmentation tasks.

Abstract

The minimum graph cut and minimum $s$-$t$-cut problems are important primitives in the modeling of combinatorial problems in computer science, including in computer vision and machine learning. Some of the most efficient algorithms for finding global minimum cuts are randomized algorithms based on Karger's groundbreaking contraction algorithm. Here, we study whether Karger's algorithm can be successfully generalized to other cut problems. We first prove that a wide class of natural generalizations of Karger's algorithm cannot efficiently solve the $s$-$t$-mincut or the normalized cut problem to optimality. However, we then present a simple new algorithm for seeded segmentation / graph-based semi-supervised learning that is closely based on Karger's original algorithm, showing that for these problems, extensions of Karger's algorithm can be useful. The new algorithm has linear asymptotic runtime and yields a potential that can be interpreted as the posterior probability of a sample belonging to a given seed / class. We clarify its relation to the random walker algorithm / harmonic energy minimization in terms of distributions over spanning forests. On classical problems from seeded image segmentation and graph-based semi-supervised learning on image data, the method performs at least as well as the random walker / harmonic energy minimization / Gaussian processes.