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 |p ∩ S| ≥ 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 two different \(2(1-\frac{1}{r})\)-approximation algorithms. The first is based on rounding the solution of a linear program that embeds our graph into a simplex, and although this same linear program yields stronger approximation guarantees for the traditional multiway cut problem, we show that its integrality gap increases to \(2(1 -\frac{1}{r})\) in the k-hurdle case. Our second approximation result is based on half-integrality, for which we provide 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. 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(logn) times that of an optimal k-hurdle solution, and we obtain a 2-approximation algorithm for trees.
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Dean, B.C., Griffis, A., Whitley, A. (2008). Approximation Algorithms for k-Hurdle Problems. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds) LATIN 2008: Theoretical Informatics. LATIN 2008. Lecture Notes in Computer Science, vol 4957. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78773-0_39
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DOI: https://doi.org/10.1007/978-3-540-78773-0_39
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