An O⁎(1.84k) parameterized algorithm for the multiterminal cut problem - Cao, Yixin; Chen, Jianer; Fan, J.-H. (Information processing letters, 2014)
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. •The time complexity breaks the 2k barrier.•We use the classic tools that make the algorithm simpler.•Disposal of non-terminal vertices shed light on kernelization.