Improved parameterized and exact algorithms for cut problems on trees

We study the Multicut on Trees and the Generalized Multiway Cut on Trees problems. For the Multicut on Trees problem, we present a parameterized algorithm that runs in time O⁎(ρk)O⁎(ρk), where ρ=2+1<1.554 is the positive root of the polynomial x4−2x2−1x4−2x2−1. This improves the current-best algorithm of Chen et al. that runs in time O⁎(1.619k)O⁎(1.619k). For the Generalized Multiway Cut on Trees problem, we show that this problem is solvable in polynomial time if the number of terminal sets is fixed; this answers an open question posed in a recent paper by Liu and Zhang. By reducing the Generalized Multiway Cut on Trees problem to the Multicut on Trees problem, our results give a parameterized algorithm that solves the Generalized Multiway Cut on Trees problem in time O⁎(ρk)O⁎(ρk).