Tight Approximation Ratio of a General Greedy Splitting Algorithm for the Minimum k-Way Cut Problem - Xiao, Mingyu; Cai, Leizhen; Yao, Andrew Chi-Chih (Algorithmica, 2011)
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.