Faster Exact and Approximate Algorithms for $k$-Cut

TL;DR

This paper introduces faster algorithms for the $k$-cut problem, improving exact and approximate solutions with better runtime complexities and approximation ratios, advancing the state-of-the-art in graph partitioning algorithms.

Contribution

It presents a new $O(k^{O(k)} n^{(2 ext{w}/3 + o(1))k})$-time exact algorithm and improved approximation algorithms, including a $(1+ ext{epsilon})$-approximation and a fixed-parameter $1.81$-approximation.

Findings

Faster exact $k$-cut algorithm with improved runtime.

New $(1+ ext{epsilon})$-approximation algorithm with better efficiency.

A $1.81$-approximation algorithm in fixed-parameter time.

Abstract

In the $k$-cut problem, we are given an edge-weighted graph $G$ and an integer $k$, and have to remove a set of edges with minimum total weight so that $G$ has at least $k$ connected components. The current best algorithms are an $O(n^{(2-o(1))k})$ randomized algorithm due to Karger and Stein, and an $\smash{\tilde{O}}(n^{2k})$ deterministic algorithm due to Thorup. Moreover, several $2$-approximation algorithms are known for the problem (due to Saran and Vazirani, Naor and Rabani, and Ravi and Sinha). It has remained an open problem to (a) improve the runtime of exact algorithms, and (b) to get better approximation algorithms. In this paper we show an $O(k^{O(k)} \, n^{(2\omega/3 + o(1))k})$-time algorithm for $k$-cut. Moreover, we show an $(1+\epsilon)$-approximation algorithm that runs in time $O((k/\epsilon)^{O(k)} \,n^{k + O(1)})$, and a $1.81$-approximation in fixed-parameter time $O(2^{O(k^2)}\,\text{poly}(n))$.