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Graph Search Trees and the Intermezzo Problem

Authors Jesse Beisegel , Ekkehard Köhler, Fabienne Ratajczak , Robert Scheffler , Martin Strehler

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Author Details

Jesse Beisegel
  • Institute of Mathematics, Brandenburg University of Technology, Cottbus, Germany
Ekkehard Köhler
  • Institute of Mathematics, Brandenburg University of Technology, Cottbus, Germany
Fabienne Ratajczak
  • Institute of Mathematics, Brandenburg University of Technology, Cottbus, Germany
Robert Scheffler
  • Institute of Mathematics, Brandenburg University of Technology, Cottbus, Germany
Martin Strehler
  • Department of Mathematics, Westsächsische Hochschule Zwickau, Germany

Acknowledgements

The authors would like to thank Matjaž Krnc, Martin Milanič, and Nevena Pivač for fruitful discussions about first-in and last-in trees as well as about Paul and Mary.

Cite AsGet BibTex

Jesse Beisegel, Ekkehard Köhler, Fabienne Ratajczak, Robert Scheffler, and Martin Strehler. Graph Search Trees and the Intermezzo Problem. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.MFCS.2024.22

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • graph search trees
  • intermezzo problem
  • algorithm
  • parameterized complexity

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