Fixed Parameter Approximation Scheme for Min-max $k$-cut

We consider the graph $k$-partitioning problem under the min-max objective, termed as Minmax $k$-cut. The input here is a graph $G=(V,E)$ with non-negative edge weights $w:E\rightarrow \mathbb{R}+$ and an integer $k\geq 2$ and the goal is to partition the vertices into $k$ non-empty parts $V1, \ldots, Vk$ so as to minimize $\max{i=1}^k w(\delta(Vi))$. Although minimizing the sum objective $\sum{i=1}^k w(\delta(Vi))$, termed as Minsum $k$-cut, has been studied extensively in the literature, very little is known about minimizing the max objective. We initiate the study of Minmax $k$-cut by showing that it is NP-hard and W[1]-hard when parameterized by $k$, and design a parameterized approximation scheme when parameterized by $k$. The main ingredient of our parameterized approximation scheme is an exact algorithm for Minmax $k$-cut that runs in time $(\lambda k)^{{O(k2)}n{O(1)}$,} where $\lambda$ is value of the optimum and $n$ is the number of vertices. Our algorithmic technique builds on the technique of Lokshtanov, Saurabh, and Surianarayanan (FOCS, 2020) who showed a similar result for Minsum $k$-cut. Our algorithmic techniques are more general and can be used to obtain parameterized approximation schemes for minimizing $\ellp$-norm measures of $k$-partitioning for every $p\geq 1$.