Abstract
Many divide-and-conquer algorithms employ the fact that the vertex set of a graph of bounded treewidth can be separated in two roughly balanced subsets by removing a small subset of vertices, referred to as a separator. In this paper we prove a trade-off between the size of the separator and the sharpness with which we can fix the size of the two sides of the partition. Our result appears to be a handy and powerful tool for the design of exact and parameterized algorithms for NP-hard problems. We illustrate that by presenting two applications.
Our first application is a parameterized algorithm with running time O(16k + o(k) + n O(1)) for the Maximum Internal Subtree problem in directed graphs. This is a significant improvement over the best previously known parameterized algorithm for the problem by [Cohen et al.’09], running in time O(49.4k + n O(1)).
The second application is a O(2n + o(n)) time and space algorithm for the Degree Constrained Spanning Tree problem: find a spanning tree of a graph with the maximum number of nodes satisfying given degree constraints. This problem generalizes some well-studied problems, among them Hamiltonian Path, Full Degree Spanning Tree, Bounded Degree Spanning Tree, Maximum Internal Spanning Tree and their edge weighted variants.
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Fomin, F.V., Lokshtanov, D., Grandoni, F., Saurabh, S. (2010). Sharp Separation and Applications to Exact and Parameterized Algorithms. In: López-Ortiz, A. (eds) LATIN 2010: Theoretical Informatics. LATIN 2010. Lecture Notes in Computer Science, vol 6034. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12200-2_8
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DOI: https://doi.org/10.1007/978-3-642-12200-2_8
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