TY - JOUR

T1 - Fixed parameter approximation scheme for min-max k-cut

AU - Chandrasekaran, Karthekeyan

AU - Wang, Weihang

N1 - Funding Information:
Supported in part by NSF grants CCF-1814613 and CCF-1907937.
Publisher Copyright:
© 2022, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.

PY - 2022

Y1 - 2022

N2 - 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, … , 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)+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 ℓp-norm measures of k-partitioning for every p≥ 1.

AB - 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, … , 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)+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 ℓp-norm measures of k-partitioning for every p≥ 1.

KW - k-cut

KW - Min-max objective

KW - Parameterized approximation scheme

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U2 - 10.1007/s10107-022-01842-3

DO - 10.1007/s10107-022-01842-3

M3 - Article

AN - SCOPUS:85132109143

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

ER -