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Algorithmica

, Volume 59, Issue 4, pp 510–520 | Cite as

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

  • Mingyu Xiao
  • Leizhen Cai
  • Andrew Chi-Chih Yao
Article

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≤hk, 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 1h 2⋅⋅⋅h q and ∑ i=1 q (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.

Keywords

k-way cut Approximation algorithm 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Mingyu Xiao
    • 1
  • Leizhen Cai
    • 2
  • Andrew Chi-Chih Yao
    • 2
    • 3
  1. 1.School of Computer Science and EngineeringUniversity of Electronic Science and Technology of ChinaChengduChina
  2. 2.Department of Computer Science and EngineeringThe Chinese University of Hong KongShatinChina
  3. 3.Institute for Theoretical Computer ScienceTsinghua UniversityBeijingChina

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