In this paper, we unify several graph partitioning problems including multicut, multiway cut, and k-cut, into a single problem. The input to a requirement cut problem is an undirected edge-weighted graph G=(V,E), and g groups of vertices X 1, ⋯ ,X g  ⊆ V, each with a requirement r i between 0 and |X i |. The goal is to find a minimum cost set of edges whose removal separates each group X i into at least r i disconnected components.

We give an O(log n log (gR)) approximation algorithm for the requirement cut problem, where n is the total number of vertices, g is the number of groups, and R is the maximum requirement. We also show that the integrality gap of a natural LP relaxation for this problem is bounded by O(log n log (gR)). On trees, we obtain an improved guarantee of O(log (gR)). There is a natural Ω (log g) hardness of approximation for the requirement cut problem.


Approximation Algorithm Greedy Algorithm Steiner Tree Residual Graph Graph Partitioning Problem 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Viswanath Nagarajan
    • 1
  • Ramamoorthi Ravi
    • 1
  1. 1.Tepper School of BusinessCarnegie Mellon UniversityPittsburghUSA

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