License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.IPEC.2022.6
URN: urn:nbn:de:0030-drops-173626
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Bodlaender, Hans L. ; Groenland, Carla ; Jacob, Hugo ; Pilipczuk, Marcin ; Pilipczuk, MichaƂ

On the Complexity of Problems on Tree-Structured Graphs

LIPIcs-IPEC-2022-6.pdf (1 MB)


In this paper, we introduce a new class of parameterized problems, which we call XALP: the class of all parameterized problems that can be solved in f(k)n^O(1) time and f(k)log n space on a non-deterministic Turing Machine with access to an auxiliary stack (with only top element lookup allowed). Various natural problems on "tree-structured graphs" are complete for this class: we show that List Coloring and All-or-Nothing Flow parameterized by treewidth are XALP-complete. Moreover, Independent Set and Dominating Set parameterized by treewidth divided by log n, and Max Cut parameterized by cliquewidth are also XALP-complete.
Besides finding a "natural home" for these problems, we also pave the road for future reductions. We give a number of equivalent characterisations of the class XALP, e.g., XALP is the class of problems solvable by an Alternating Turing Machine whose runs have tree size at most f(k)n^O(1) and use f(k)log n space. Moreover, we introduce "tree-shaped" variants of Weighted CNF-Satisfiability and Multicolor Clique that are XALP-complete.

BibTeX - Entry

  author =	{Bodlaender, Hans L. and Groenland, Carla and Jacob, Hugo and Pilipczuk, Marcin and Pilipczuk, Micha{\l}},
  title =	{{On the Complexity of Problems on Tree-Structured Graphs}},
  booktitle =	{17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
  pages =	{6:1--6:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-260-0},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{249},
  editor =	{Dell, Holger and Nederlof, Jesper},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-173626},
  doi =		{10.4230/LIPIcs.IPEC.2022.6},
  annote =	{Keywords: Parameterized Complexity, Treewidth, XALP, XNLP}

Keywords: Parameterized Complexity, Treewidth, XALP, XNLP
Collection: 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)
Issue Date: 2022
Date of publication: 14.12.2022

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