A Parameterized Approximation Scheme for Min $k$-Cut

In the Min $k$-Cut problem, input consists of an edge weighted graph $G$ and integer $k$, task is to partition vertex set into $k$ nonempty sets, such that total weight edges with endpoints in different parts minimized. When part input, problem NP-complete hard approximate within any factor less than 2. Recently, has received significant attention from perspective parameterized approximation. Gupta, Lee, Li [Proceedings 29th Annual ACM-SIAM Symposium on Discrete Algorithms, A. Czumaj, ed., SIAM, Philadelphia, 2018, pp. 2821--2837] initiated study FPT-approximation for gave a 1.9997-approximation algorithm running time $2^{\mathcal{O}(k^6)}n^{\mathcal{O}(1)}$. Later, same authors 59th IEEE Foundations Computer Science, M. Thorup, 113--123] designed $(1 +\epsilon)$-approximation runs $(k/\epsilon)^{\mathcal{O}(k)}n^{k+\mathcal{O}(1)}$ 1.81-approximation $2^{\mathcal{O}(k^2)}n^{\mathcal{O}(1)}$. More, recently, Kawarabayashi Lin 31st S. Chawla, 2020, 990--999] $(5/3 + \epsilon)$-approximation $2^{\mathcal{O}(k^2 \log k)}n^{\mathcal{O}(1)}$. this paper, we give approximation best possible guarantee dependence said (up exponential hypothesis constants exponent). particular, every $\epsilon > 0$, obtains +\epsilon)$-approximate solution $(k/\epsilon)^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$. The main ingredients our are simple sparsification procedure, new polynomial decomposing highly connected parts, exact $s^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$ unweighted (multi-) graphs. Here, $s$ denotes number minimum $k$-cut. latter two independent interest.