The Number of Minimum $k$-Cuts: Improving the Karger-Stein Bound

Abstract

Given an edge-weighted graph, how many minimum $k$-cuts can it have? This is a fundamental question in the intersection of algorithms, extremal combinatorics, and graph theory. It is particularly interesting in that the best known bounds are algorithmic: they stem from algorithms that compute the minimum $k$-cut. In 1994, Karger and Stein obtained a randomized contraction algorithm that finds a minimum $k$-cut in $O(n^{(2-o(1))k})$ time. It can also enumerate all such $k$-cuts in the same running time, establishing a corresponding extremal bound of $O(n^{(2-o(1))k})$. Since then, the algorithmic side of the minimum $k$-cut problem has seen much progress, leading to a deterministic algorithm based on a tree packing result of Thorup, which enumerates all minimum $k$-cuts in the same asymptotic running time, and gives an alternate proof of the $O(n^{(2-o(1))k})$ bound. However, beating the Karger--Stein bound, even for computing a single minimum $k$-cut, has remained out of reach. In this paper, we give an algorithm to enumerate all minimum $k$-cuts in $O(n^{(1.981+o(1))k})$ time, breaking the algorithmic and extremal barriers for enumerating minimum $k$-cuts. To obtain our result, we combine ideas from both the Karger--Stein and Thorup results, and draw a novel connection between minimum $k$-cut and extremal set theory. In particular, we give and use tighter bounds on the size of set systems with bounded dual VC-dimension, which may be of independent interest.