Fixed parameter approximation scheme for min-max k-cut

• 成果类型: Article
• 署名作者: Chandrasekaran, Karthekeyan; Wang, Weihang
• 署名单位: University of Illinois System; University of Illinois Urbana-Champaign
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-022-01842-3
• 发表日期: 2023
• 页码: 1093-1144
• 关键词: algorithms

摘要

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(delta(V-i)). Although minimizing the sum objective Sigma(k)(i=1) w(delta(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 Minmaxk-cut that runs in time (lambda k)(O(k2)) n(O(1)) + O(m), where lambda 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.