Fast and Deterministic Approximations for $k$-Cut

TL;DR

This paper introduces a fast, deterministic algorithm that approximates the minimum $k$-cut problem within a factor of $(2 + ext{small epsilon})$ in near-linear time, improving the efficiency and determinism of previous methods.

Contribution

The authors develop a deterministic algorithm achieving a $(2 + ext{epsilon})$ approximation for $k$-cut in near-linear time, utilizing a novel LP relaxation approach.

Findings

Achieves $(2 + ext{epsilon})$-approximation in $O(m rac{ ext{log}^3 n}{ ext{epsilon}^2})$ time.

Provides a deterministic alternative matching the speed of randomized algorithms.

Uses LP relaxation to improve approximation and runtime.

Abstract

In an undirected graph, a $k$-cut is a set of edges whose removal breaks the graph into at least $k$ connected components. The minimum weight $k$-cut can be computed in $O(n^{O(k)})$ time, but when $k$ is treated as part of the input, computing the minimum weight $k$-cut is NP-Hard [Holdschmidt and Hochbaum 1994]. For $\operatorname{poly}(m,n,k)$-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi 2017]. Saran and Vazirani [1995] showed that a $(2 - 2/k)$-approximately minimum weight $k$-cut can be computed by $O(k)$ minimum cuts, which implies an $\tilde{O}(mk)$ randomized running time via the nearly linear time randomized min-cut algorithm of Karger [2000]. Nagamochi and Kamidoi [2007] showed that the minimum weight $k$-cut can be computed deterministically in $O(mn + n^2 \log n)$ time. These results prompt two basic questions. The first concerns the role of randomization. Is there a deterministic algorithm for 2-approximate $k$-cuts matching the randomized running time of $\tilde{O}(mk)$? The second question qualitatively compares minimum cut to 2-approximate minimum $k$-cut. Can 2-approximate $k$-cuts be computed as fast as the (exact) minimum cut - in $\tilde{O}(m)$ randomized time? We make progress on these questions with a deterministic approximation algorithm that computes $(2 + \epsilon)$-minimum $k$-cuts in $O(m \log^3(n) / \epsilon^2)$ time, via a $(1 + \epsilon)$-approximate for an LP relaxation of $k$-cut.