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 edge weights w: E→ R₊ 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), where λ is the 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 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.