Fixed parameter approximation scheme for min-max k-cut

We consider the graph k-partitioning problem under the min-max objective, termed as Minmaxk-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 V₁, … , V_k so as to minimize maxi=1kw(δ(Vi)). Although minimizing the sum objective ∑i=1kw(δ(Vi)), termed as Minsumk-cut, has been studied extensively in the literature, very little is known about minimizing the max objective. We initiate the study of Minmaxk-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 Minmaxk-cut that runs in time (λk)O(k2)nO(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 Minsumk-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.