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

Abstract

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 ∑ ^q_i=1 (h _i −1)=k−1. We prove that the approximation ratio of this general algorithm is $2-(\sum_{i=1}^{q}{h_{i}\choose2})/{k\choose2}$ , 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.