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**1 - 2**of**2**### Faster Exact Algorithms for Some Terminal Set Problems

"... Many problems on graphs can be expressed in the following language: given a graph G = (V, E) and a terminal set T ⊆ V, find a minimum size set S ⊆ V which intersects all “structures ” (such as cycles or paths) passing through the vertices in T. We call this class of problems as terminal set proble ..."

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Many problems on graphs can be expressed in the following language: given a graph G = (V, E) and a terminal set T ⊆ V, find a minimum size set S ⊆ V which intersects all “structures ” (such as cycles or paths) passing through the vertices in T. We call this class of problems as terminal set problems. In this paper we introduce a general method to obtain faster exact exponential time algorithms for many terminal set problems. More precisely, we show that – Node Multiway Cut can be solved in time O(1.4766 n). – Directed Unrestricted Node Multiway Cut can be solved in time O(1.6181 n). – There exists a deterministic algorithm for Subset Feedback Vertex Set running in time O(1.8980 n) and a randomized algorithm with expected running time O(1.8826 n). Furthermore, Subset Feedback Vertex Set on chordal graphs can be solved in time O(1.6181 n). – Directed Subset Feedback Vertex Set can be solved in time O(1.9993 n). A key feature of our method is that, it uses the existing best polynomial time, fixed parameter tractable and exact exponential time algorithms for the non-terminal version of the same problem (i.e. when T = V), as subroutines. Therefore faster algorithms for these special cases will imply further improvements in the running times of our algorithms. Our algorithms for Node Multiway Cut, and Subset Feedback Vertex Set on chordal graphs improve the current best algorithms for these problems and answers an open question posed in [15]. Furthermore, our algorithms for Directed Unrestricted Node Multiway Cut and Directed Subset Feedback Vertex Set are the first exact algorithms improving upon the brute force O ∗ (2 n)-algorithms.