Complexity of independency and cliquy trees
| Autor: | Casel, Katrin, Dreier, Jan, Fernau, Henning, Gobbert, Moritz, Kuinke, Philipp, Sánchez Villaamil, Fernando, Schmid, Markus L., van Leeuwen, E.J., Sub Algorithms and Complexity, Algorithms and Complexity |
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| Přispěvatelé: | Sub Algorithms and Complexity, Algorithms and Complexity |
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Polynomial hierarchy
Vertex (graph theory) Spanning tree Applied Mathematics 0211 other engineering and technologies Parameterized complexity 021107 urban & regional planning 0102 computer and information sciences 02 engineering and technology 01 natural sciences Cliquy tree Combinatorics 010201 computation theory & mathematics Polynomial kernel Independent set Discrete Mathematics and Combinatorics Maximum size Kernelization algorithms Independency tree Exact algorithms Mathematics |
| Zdroj: | Discrete Applied Mathematics, 272, 2. Elsevier |
| ISSN: | 0166-218X |
| Popis: | An independency (cliquy) tree of an n -vertex graph G is a spanning tree of G in which the set of leaves induces an independent set (clique). We study the problems of minimizing or maximizing the number of leaves of such trees, and fully characterize their parameterized complexity. We show that all four variants of deciding if an independency/cliquy tree with at least/most l leaves exists parameterized by l are either Para - NP - or W[1] -hard. We prove that minimizing the number of leaves of a cliquy tree parameterized by the number of internal vertices is Para - NP -hard too. However, we show that minimizing the number of leaves of an independency tree parameterized by the number k of internal vertices has an O ∗ ( 4 k ) -time algorithm and a 2 k vertex kernel. Moreover, we prove that maximizing the number of leaves of an independency/cliquy tree parameterized by the number k of internal vertices both have an O ∗ ( 1 8 k ) -time algorithm and an O ( k 2 k ) vertex kernel, but no polynomial kernel unless the polynomial hierarchy collapses to the third level. Finally, we present an O ( 3 n ⋅ f ( n ) ) -time algorithm to find a spanning tree where the leaf set has a property that can be decided in f ( n ) time and has minimum or maximum size. |
| Databáze: | OpenAIRE |
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