An O^*(1.84^k) Parameterized Algorithm for the Multiterminal Cut Problem
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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, Flows in Networks, 1962] and isolating cuts by [Dahlhaus et al., SIAM J. Comp. 23(4):864-894, 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· n^O(1) time, thereby breaking the 2^k· n^O(1) barrier. As a by-product, it gives a 1.36^k· n^O(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.
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