  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.
