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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-negativeedge weights $w:E\rightarrow \mathbb{R}_+$ and an integer $k\geq 2$ and thegoal is to partition the vertices into $k$ non-empty parts $V_1, \ldots, V_k$so as to minimize $\max_{i=1}^k w(\delta(V_i))$. Although minimizing the sumobjective $\sum_{i=1}^k w(\delta(V_i))$, termed as Minsum $k$-cut, has beenstudied extensively in the literature, very little is known about minimizingthe max objective. We initiate the study of Minmax $k$-cut by showing that itis NP-hard and W[1]-hard when parameterized by $k$, and design a parameterizedapproximation scheme when parameterized by $k$. The main ingredient of ourparameterized approximation scheme is an exact algorithm for Minmax $k$-cutthat runs in time $(\lambda k)^{O(k^2)}n^{O(1)}$, where $\lambda$ is value ofthe optimum and $n$ is the number of vertices. Our algorithmic technique buildson the technique of Lokshtanov, Saurabh, and Surianarayanan (FOCS, 2020) whoshowed a similar result for Minsum $k$-cut. Our algorithmic techniques are moregeneral and can be used to obtain parameterized approximation schemes forminimizing $\ell_p$-norm measures of $k$-partitioning for every $p\geq 1$.

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