Multicuts in Unweighted Graphs with Bounded Degree and Bounded Tree-Width

The Multicut problem can be defined as: given a graph G and a collection of pairs of distinct vertices (s i ; t i ) of G, find a minimum set of edges of G whose removal disconnects each s i from the corresponding t i . The fractional Multicut problem is the dual of the well-known Multicommodity Flow problem. 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 polynomialtime approximation scheme (PTAS) for Multicut in unweighted graphs with bounded degree and bounded tree-width. That is, for any ffl ? 0, we present a polynomial-time (1 + ffl)-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 relax any of the three condition (unweighted, bounded-degree, bounded tree-width). We als..