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Fixed parameter approximation scheme for min-max k-cut.
- Source :
-
Mathematical Programming . Feb2023, Vol. 197 Issue 2, p1093-1144. 52p. - Publication Year :
- 2023
-
Abstract
- 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 integral edge weights w : E → Z + 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) + 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 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 . [ABSTRACT FROM AUTHOR]
- Subjects :
- *APPROXIMATION algorithms
*INTEGERS
*ALGORITHMS
*INTEGRALS
Subjects
Details
- Language :
- English
- ISSN :
- 00255610
- Volume :
- 197
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- Mathematical Programming
- Publication Type :
- Academic Journal
- Accession number :
- 161716903
- Full Text :
- https://doi.org/10.1007/s10107-022-01842-3