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Simple and improved parameterized algorithms for multiterminal cuts. (English) Zbl 1213.68472
Summary: 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 $$k\geq 3$$ 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 $$T(n,m)=O(\text{min} (n ^{2/3},m ^{1/2})m)$$ 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: $$O((\min(\sqrt{2k},l)+1)^{2k}2^{l}T(n,m))$$-time algorithm for Edge Multicut and $$O((2k)^{k+l/2} T(n,m))$$-time algorithm for Vertex Multicut.

##### MSC:
 68R10 Graph theory (including graph drawing) in computer science 05C85 Graph algorithms (graph-theoretic aspects) 68Q25 Analysis of algorithms and problem complexity
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