Multicuts in unweighted graphs and digraphs with bounded degree and bounded tree-width

  • Authors:

  • Gruia Calinescu;Cristina G. Fernandes;Bruce Reed

  • Affiliations:

  • Department of Computer Science, Illinois Institute of Technology, Stuart Building, Room 236, 10 West 31st Street, Chicago, IL;Department of Computer Science, University of São Paulo, Brazil;Department of Computer Science, Illinois Institute of Technology, Stuart Building, Room 236, 10 West 31st Street, Chicago, IL and CNRS, Paris, France

  • Venue:
  • Journal of Algorithms
  • Year:
  • 2003

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Abstract

The Multicut problem can be defined as: given a graph G and a collection of pairs of distinct vertices {si, ti} of G, find a minimum set of edges of G whose removal disconnects each si from the corresponding ti. Multicut is known to be NP-hard and Max SNP-hard even when the input graph is restricted to being a tree. The main result of the paper is a polynomial-time approximation scheme (PTAS) for Multicut in unweighted graphs with bounded degree and bounded tree-width. That is, for any ε 0, we present a polynomial-time (1 + ε)-approximation algorithm. In the particular case when the input is a bounded-degree tree, we have a linear-time implementation of the algorithm. We also provide some hardness results: we prove that Multicut is still NP-hard for binary trees and that it is Max SNP-hard if we drop any of the three conditions (unweighted, bounded-degree, bounded tree-width). Finally we show that some of these results extend to the vertex version of Multicut and to a directed version of Multicut.