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Approximation Algorithms for k-hurdle Problems

Abstract

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.

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Correspondence to Brian C. Dean.

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Dean, B.C., Griffis, A., Parekh, O. et al. Approximation Algorithms for k-hurdle Problems. Algorithmica 59, 81–93 (2011). https://doi.org/10.1007/s00453-010-9408-y

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Keywords

  • Multiway cut
  • Multicut
  • Approximation algorithm
  • Randomized rounding