Tight Approximation Ratio of a General Greedy Splitting Algorithm for the Minimum k-Way Cut Problem

• Published in: Algorithmica Vol. 59; no. 4; pp. 510 - 520
• Main Authors: Xiao, Mingyu, Cai, Leizhen, Yao, Andrew Chi-Chih
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.04.2011; Springer
• Subjects: Algorithm Analysis and Problem Complexity; Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Computer Science; Computer science; control theory; systems; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Exact sciences and technology; Mathematics of Computing; Theoretical computing; Theory of Computation; Approximation algorithm; way cut; Edge(graph); K cut problem; Graph connectivity; NP hard problem; Greedy algorithm; Connected graph; Iterative method; k-way cut; Weighted graph; Non directed graph
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-009-9316-1

For an edge-weighted connected undirected graph, the minimum k -way cut problem is to find a subset of edges of minimum total weight whose removal separates the graph into k connected components. The problem is NP-hard when k is part of the input and W[1]-hard when k is taken as a parameter. A simple algorithm for approximating a minimum k -way cut is to iteratively increase the number of components of the graph by h −1, where 2≤ h ≤ k , until the graph has k components. The approximation ratio of this algorithm is known for h ≤3 but is open for h ≥4. In this paper, we consider a general algorithm that successively increases the number of components of the graph by h i −1, where 2≤ h 1 ≤ h 2 ≤ ⋅⋅⋅ ≤ h q and ∑ i =1 q ( h i −1)= k −1. We prove that the approximation ratio of this general algorithm is , which is tight. Our result implies that the approximation ratio of the simple iterative algorithm is 2− h / k + O ( h 2 / k 2 ) in general and 2− h / k if k −1 is a multiple of h −1.