Fixed Parameter Approximation Scheme for Min-max k-cut
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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-negative edge weights w:E→ℝ_+ 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), where λ 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 ℓ_p-norm measures of k-partitioning for every p≥ 1.
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