, Volume 59, Issue 1, pp 81–93 | Cite as

Approximation Algorithms for k-hurdle Problems

  • Brian C. DeanEmail author
  • Adam Griffis
  • Ojas Parekh
  • Adam Whitley


The polynomial-time solvable k-hurdle problem is a natural generalization of the classical s-t minimum cut problem where we must select a minimum-cost subset S of the edges of a graph such that |pS|≥k for every s-t path p. In this paper, we describe a set of approximation algorithms for “k-hurdle” variants of the NP-hard multiway cut and multicut problems. For the k-hurdle multiway cut problem with r terminals, we give two results, the first being a pseudo-approximation algorithm that outputs a (k−1)-hurdle solution whose cost is at most that of an optimal solution for k hurdles. Secondly, we provide a \(2(1-\frac{1}{r})\)-approximation algorithm based on rounding the solution of a linear program, for which we give a simple randomized half-integrality proof that works for both edge and vertex k-hurdle multiway cuts that generalizes the half-integrality results of Garg et al. for the vertex multiway cut problem. We also describe an approximation-preserving reduction from vertex cover as evidence that it may be difficult to achieve a better approximation ratio than \(2(1-\frac{1}{r})\). For the k-hurdle multicut problem in an n-vertex graph, we provide an algorithm that, for any constant ε>0, outputs a ⌈(1−ε)k⌉-hurdle solution of cost at most O(log n) times that of an optimal k-hurdle solution, and we obtain a 2-approximation algorithm for trees.


Multiway cut Multicut Approximation algorithm Randomized rounding 


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Brian C. Dean
    • 1
    Email author
  • Adam Griffis
    • 1
  • Ojas Parekh
    • 2
  • Adam Whitley
    • 1
  1. 1.School of ComputingClemson UniversityClemsonUSA
  2. 2.Mathematics and Computer Science DepartmentEmory UniversityAtlantaUSA

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