Fixed Parameter Approximation Scheme for Min-Max k-Cut

Research output: Chapter in Book/Report/Conference proceedingConference contribution


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, …, Vk 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 ℓp -norm measures of k-partitioning for every p≥ 1.

Original languageEnglish (US)
Title of host publicationInteger Programming and Combinatorial Optimization - 22nd International Conference, IPCO 2021, Proceedings
EditorsMohit Singh, David P. Williamson
Number of pages14
ISBN (Print)9783030738785
StatePublished - 2021
Event22nd International Conference on Integer Programming and Combinatorial Optimization, IPCO 2021 - Virtual, Online
Duration: May 19 2021May 21 2021

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume12707 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference22nd International Conference on Integer Programming and Combinatorial Optimization, IPCO 2021
CityVirtual, Online


  • Min-max objective
  • Parameterized approximation scheme
  • k-cut

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)


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