TY - GEN
T1 - Fixed Parameter Approximation Scheme for Min-Max k-Cut
AU - Chandrasekaran, Karthekeyan
AU - Wang, Weihang
N1 - Supported in part by NSF grants CCF-1814613 and CCF-1907937.
PY - 2021
Y1 - 2021
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 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.
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 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.
KW - Min-max objective
KW - Parameterized approximation scheme
KW - k-cut
UR - http://www.scopus.com/inward/record.url?scp=85106156097&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85106156097&partnerID=8YFLogxK
U2 - 10.1007/978-3-030-73879-2_25
DO - 10.1007/978-3-030-73879-2_25
M3 - Conference contribution
AN - SCOPUS:85106156097
SN - 9783030738785
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 354
EP - 367
BT - Integer Programming and Combinatorial Optimization - 22nd International Conference, IPCO 2021, Proceedings
A2 - Singh, Mohit
A2 - Williamson, David P.
PB - Springer
T2 - 22nd International Conference on Integer Programming and Combinatorial Optimization, IPCO 2021
Y2 - 19 May 2021 through 21 May 2021
ER -