Abstract
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(2l 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(1.415l T(n,m)) time, and Vertex {3,4,5,6}-Terminal Cuts can be solved in O(2.059l T(n,m)), O(2.772l T(n,m)), O(3.349l T(n,m)) and O(3.857l T(n,m)) time respectively. Our results on Multiterminal Cut can also be used to obtain faster algorithms for Multicut: \(O((\min(\sqrt{2k},l)+1)^{2k}2^{l}T(n,m))\) -time algorithm for Edge Multicut and O((2k)k+l/2 T(n,m))-time algorithm for Vertex Multicut.
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A preliminary version of this paper was presented at the 3rd International Computer Science Symposium in Russia (CSR 2008), and appeared in [1].
The work was supported in part by UESTC Youth Science Funds and Earmarked Research Grant 410206 of the Research Grants Council of Hong Kong SAR. Most of the work was done when the author was a Ph.D. student in Department of Computer Science and Engineering, the Chinese University of Hong Kong.
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Xiao, M. Simple and Improved Parameterized Algorithms for Multiterminal Cuts. Theory Comput Syst 46, 723–736 (2010). https://doi.org/10.1007/s00224-009-9215-5
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DOI: https://doi.org/10.1007/s00224-009-9215-5