An FPT Algorithm Beating 2-Approximation for $k$-Cut

TL;DR

This paper presents the first fixed-parameter tractable algorithm parameterized by $k$ that achieves a better than 2-approximation for the $k$-Cut problem, improving upon prior approximation ratios within FPT time.

Contribution

It introduces a novel FPT algorithm parameterized solely by $k$ that guarantees a $(2 - ext{constant})$-approximation for $k$-Cut, surpassing the longstanding 2-approximation barrier.

Findings

Achieves a $(2 - ext{constant})$-approximation in $2^{O(k^6)} ilde{O}(n^4)$ time.

First FPT algorithm parameterized by $k$ to improve upon the 2-approximation.

Demonstrates that better approximations are possible within fixed-parameter tractability.

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. Prior work on this problem gives, for all $h \in [2,k]$, a $(2-h/k)$-approximation algorithm for $k$-cut that runs in time $n^{O(h)}$. Hence to get a $(2 - \varepsilon)$-approximation algorithm for some absolute constant $\varepsilon$, the best runtime using prior techniques is $n^{O(k\varepsilon)}$. Moreover, it was recently shown that getting a $(2 - \varepsilon)$-approximation for general $k$ is NP-hard, assuming the Small Set Expansion Hypothesis. If we use the size of the cut as the parameter, an FPT algorithm to find the exact $k$-Cut is known, but solving the $k$-Cut problem exactly is $W[1]$-hard if we parameterize only by the natural parameter of $k$. An immediate question is: \emph{can we approximate $k$-Cut better in FPT-time, using $k$ as the parameter?} We answer this question positively. We show that for some absolute constant $\varepsilon > 0$, there exists a $(2 - \varepsilon)$-approximation algorithm that runs in time $2^{O(k^6)} \cdot \widetilde{O} (n^4)$. This is the first FPT algorithm that is parameterized only by $k$ and strictly improves the $2$-approximation.