Simple and Improved Parameterized Algorithms for Multiterminal Cuts - Xiao, Mingyu (Theory of computing systems, 2010)
Given a graph G =( V , E ) with n vertices and m edges, and a subset T of k vertices called terminals , the Edge (respectively, Vertex ) Multiterminal Cut problem is to find a set of at most l edges (non-terminal vertices), whose removal from G separates each terminal from all the others. These two problems are NP-hard for k ≥3 but well-known to be polynomial-time solvable for k =2 by the flow technique. In this paper, based on a notion farthest minimum isolating cut , we design several simple and improved algorithms for Multiterminal Cut. We show that Edge Multiterminal Cut can be solved in O (2 l kT ( n , m )) time and Vertex Multiterminal Cut can be solved in O ( k l T ( n , m )) time, where T ( n , m )= O (min ( n 2/3 , m 1/2 ) m ) is the running time of finding a minimum ( s , t ) cut in an unweighted graph. Furthermore, the running time bounds of our algorithms can be further reduced for small values of k : Edge 3-Terminal Cut can be solved in O (1.415 l T ( n , m )) time, and Vertex {3,4,5,6}-Terminal Cuts can be solved in O (2.059 l T ( n , m )), O (2.772 l T ( n , m )), O (3.349 l T ( n , m )) and O (3.857 l T ( n , m )) time respectively. Our results on Multiterminal Cut can also be used to obtain faster algorithms for Multicut : -time algorithm for Edge Multicut and O ((2 k ) k + l /2 T ( n , m ))-time algorithm for Vertex Multicut.