An O ^*(1.84^k) Parameterized Algorithm for the Multiterminal Cut Problem

Abstract

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. One is max volume min (s,t)-cuts by [Ford and Fulkerson, Flows in Networks. Princeton University Press, 1962], and the other is isolating cuts by [Dahlhaus et al., The complexity of multiterminal cuts. SIAM J. Comp. 23(4), 1994]. 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 $1.84^k\cdot n^{{\cal O}(1)}$, thereby breaking the $2^k\cdot n^{{\cal O}(1)}$ barrier. As a by-product, it gives a $1.36^k\cdot n^{{\cal O}(1)}$ algorithm for 3-terminal cut. The preprocessing applied on non-terminal vertices might be of use for study of this problem from other aspects.