Approximation Algorithms for k-hurdle Problems
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 vertical bar p boolean AND S vertical bar >= 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 - 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 - 1/r). For the k-hurdle multicut problem in an n-vertex graph, we provide an algorithm that, for any constant epsilon > 0, outputs a inverted right perpendicular(1 - epsilon)kinverted left perpendicular-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.
approximation algorithm multiway cut multicut randomized rounding