A Polynomial-Time Algorithm for (2-2/k)-stable Instances of the k-terminal cut Problem

06/15/2018
by   Mark Velednitsky, et al.
0

The k-terminal cut problem is defined on an edge-weighted graph with k distinct vertices called "terminals." The goal is to remove a minimum weight collection of edges from the graph such that there is no path between any pair of terminals. The k-terminal cut problem is known to be NP-hard. There has been interest in determining special classes of graphs for which k-terminal cut can be solved in polynomial time. One special class of graphs is the class of γ-stable graphs, a notion introduced by Bilu and Linial. An instance of k-terminal cut is said to be γ-stable if edges in the cut can be multiplied by up to γ without changing the optimal solution. For several years, the best-known result for γ-stable instances of k-terminal cut stated that the problem can be solved in polynomial time for γ≥ 4 by solving a certain linear program. This result was recently improved to γ≥ 2 - 2/k using the same linear program. In this paper, we match the result, showing that γ-stable instances of k-terminal cut can be solved in polynomial time for γ≥ 2 - 2/k, with a faster algorithm. The result is surprising: we show that a known (2 - 2/k)-approximation algorithm for the problem actually delivers the unique optimal solution for (2 - 2/k)-stable graphs. The algorithm utilizes only minimum cut procedures, obviating the use of linear programming.

READ FULL TEXT

Please sign up or login with your details

Forgot password? Click here to reset