An O⁎(1.84k) parameterized algorithm for the multiterminal cut problem

We study the multiterminal cut problem, which, given an n-vertex graph whose edges are integer-weighted and a set of terminals, asks for a partition of the vertex set such that each terminal is in a distinct part, and the total weight of crossing edges is at most k. Our weapons shall be two classical results known for decades: maximum volume minimum(s,t)-cuts by Ford and Fulkerson [11] and isolating cuts by Dahlhaus et al. [9]. We sharpen these old weapons with the help of submodular functions, and apply them to this problem, which enable us to design a more elaborated branching scheme on deciding whether a non-terminal vertex is with a terminal or not. This bounded search tree algorithm can be shown to run in 1.84k⋅nO(1) time, thereby breaking the 2k⋅nO(1) barrier. As a by-product, it gives a 1.36k⋅nO(1) time algorithm for 3-terminal cut. The preprocessing applied on non-terminal vertices might be of use for study of this problem from other aspects.