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 integral edge weights w : E → Z + and an integer k ≥ 2 and the goal is to partition the vertices into k non-empty parts V 1 , … , V k so as to minimize max i = 1 k w ( δ ( V i ) ) . Although minimizing the sum objective ∑ i = 1 k w ( δ ( V i ) ) , 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 ( λ k ) O ( k 2 ) n O ( 1 ) + O ( m ) , where λ is value of the optimum, n is the number of vertices, and m is the number of edges. 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 ℓ p -norm measures of k -partitioning for every p ≥ 1 .