Approximating minimum subset feedback sets in undirected graphs with applications

Guy Even*, Joseph Naor, Baruch Schieber, Leonid Zosin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

26 Scopus citations


Let G = (V, E) be a weighted undirected graph where all weights are at least one. We consider the following generalization of feedback set problems. Let S ⊂ V be a subset of the vertices. A cycle is called interesting if it intersects the set S. A subset feedback edge (vertex) set is a subset of the edges (vertices) that intersects all interesting cycles. In minimum subset feedback problems the goal is to find such sets of minimum weight. This problem has a variety of applications, among them genetic linkage analysis and circuit testing. The case in which S consists of a single vertex is equivalent to the multiway cut problem, in which the goal is to separate a given set of terminals. Hence, the subset feedback problem is NP-complete and also generalizes the multiway cut problem. We provide a polynomial time algorithm for approximating the subset feedback edge set problem that achieves an approximation factor of two. This implies a Δ-approximation algorithm for the subset feedback vertex set problem, where Δ is the maximum degree in G. We also consider the multicut problem and show how to achieve an O(log τ*) approximation factor for this problem, where τ* is the value of the optimal fractional solution. To achieve the O(log τ*) factor we employ a bootstrapping technique.

Original languageEnglish
Pages (from-to)255-267
Number of pages13
JournalSIAM Journal on Discrete Mathematics
Issue number2
StatePublished - Apr 2000


  • Approximation algorithms
  • Combinatorial optimization
  • Feedback edge set
  • Feedback vertex set
  • Multicut
  • Subset feedback set


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