Multicut in trees viewed through the eyes of vertex cover

We take a new look at the multicut problem in trees, denoted multicut on trees henceforth, through the eyes of the vertex cover problem. This connection, together with other techniques that we develop, allows us to give an upper bound of O(k3)O(k3) on the kernel size for multicut on trees, significantly improving the O(k6)O(k6) upper bound given by Bousquet et al. We exploit this connection further to present a parameterized algorithm for multicut on trees that runs in time O⁎(ρk)O⁎(ρk), where ρ=(5+1)/2≈1.618. This improves the previous (time) upper bound of O⁎(2k)O⁎(2k), given by Guo and Niedermeier, for the problem.

► We observe and establish close connections between the multicut problem in trees and the vertex cover problem.
► We derive an upper bound of O(k3)O(k3) on the kernel size for multicut, significantly improving the previous O(k6)O(k6) upper bound.
► We present a parameterized algorithm for multicut that runs in time O⁎(ρk)O⁎(ρk), where ρ=(5+1)/2≈1.618, improving the previous (time) upper bound of O⁎(2k)O⁎(2k).