8 research outputs found
Optimal Bounds for the -cut Problem
In the -cut problem, we want to find the smallest set of edges whose
deletion breaks a given (multi)graph into connected components. Algorithms
of Karger & Stein and Thorup showed how to find such a minimum -cut in time
approximately . The best lower bounds come from conjectures about
the solvability of the -clique problem, and show that solving -cut is
likely to require time . Recent results of Gupta, Lee & Li have
given special-purpose algorithms that solve the problem in time , and ones that have better performance for special classes of graphs
(e.g., for small integer weights).
In this work, we resolve the problem for general graphs, by showing that the
Contraction Algorithm of Karger outputs any fixed -cut of weight with probability , where
denotes the minimum -cut size. This also gives an extremal bound of
on the number of minimum -cuts and an algorithm to compute a
minimum -cut in similar runtime. Both are tight up to lower-order factors,
with the algorithmic lower bound assuming hardness of max-weight -clique.
The first main ingredient in our result is a fine-grained analysis of how the
graph shrinks -- and how the average degree evolves -- in the Karger process.
The second ingredient is an extremal bound on the number of cuts of size less
than , using the Sunflower lemma.Comment: Final version of arXiv:1911.09165 with new and more general proof