eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-08-23
22:1
22:18
10.4230/LIPIcs.MFCS.2024.22
article
Graph Search Trees and the Intermezzo Problem
Beisegel, Jesse
1
https://orcid.org/0000-0002-8760-0169
Köhler, Ekkehard
1
Ratajczak, Fabienne
1
https://orcid.org/0000-0002-5823-1771
Scheffler, Robert
1
https://orcid.org/0000-0001-6007-4202
Strehler, Martin
2
https://orcid.org/0000-0003-4241-6584
Institute of Mathematics, Brandenburg University of Technology, Cottbus, Germany
Department of Mathematics, Westsächsische Hochschule Zwickau, Germany
The last in-tree recognition problem asks whether a given spanning tree can be derived by connecting each vertex with its rightmost left neighbor of some search ordering. In this study, we demonstrate that the last-in-tree recognition problem for Generic Search is NP-complete. We utilize this finding to strengthen a complexity result from order theory. Given a partial order π and a set of triples, the NP-complete intermezzo problem asks for a linear extension of π where each first element of a triple is not between the other two. We show that this problem remains NP-complete even when the Hasse diagram of the partial order forms a tree of bounded height. In contrast, we give an XP-algorithm for the problem when parameterized by the width of the partial order. Furthermore, we show that - under the assumption of the Exponential Time Hypothesis - the running time of this algorithm is asymptotically optimal.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol306-mfcs2024/LIPIcs.MFCS.2024.22/LIPIcs.MFCS.2024.22.pdf
graph search trees
intermezzo problem
algorithm
parameterized complexity