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URN: urn:nbn:de:0030-drops-119000
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Relational Width of First-Order Expansions of Homogeneous Graphs with Bounded Strict Width

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Abstract

Solving the algebraic dichotomy conjecture for constraint satisfaction problems over structures first-order definable in countably infinite finitely bounded homogeneous structures requires understanding the applicability of local-consistency methods in this setting. We study the amount of consistency (measured by relational width) needed to solve CSP(๐”ธ) for first-order expansions ๐”ธ of countably infinite homogeneous graphs โ„‹ := (A; E), which happen all to be finitely bounded. We study our problem for structures ๐”ธ that additionally have bounded strict width, i.e., for which establishing local consistency of an instance of CSP(๐”ธ) not only decides if there is a solution but also ensures that every solution may be obtained from a locally consistent instance by greedily assigning values to variables, without backtracking. Our main result is that the structures ๐”ธ under consideration have relational width exactly (2, ๐•ƒ_โ„‹) where ๐•ƒ_โ„‹ is the maximal size of a forbidden subgraph of โ„‹, but not smaller than 3. It beats the upper bound: (2 m, 3 m) where m = max(arity(๐”ธ)+1, ๐•ƒ, 3) and arity(๐”ธ) is the largest arity of a relation in ๐”ธ, which follows from a sufficient condition implying bounded relational width given in [Manuel Bodirsky and Antoine Mottet, 2018]. Since ๐•ƒ_โ„‹ may be arbitrarily large, our result contrasts the collapse of the relational bounded width hierarchy for finite structures ๐”ธ, whose relational width, if finite, is always at most (2,3).

BibTeX - Entry

@InProceedings{wrona:LIPIcs:2020:11900,
  author =	{Michaล‚ Wrona},
  title =	{{Relational Width of First-Order Expansions of Homogeneous Graphs with Bounded Strict Width}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{39:1--39:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Christophe Paul and Markus Bl{\"a}ser},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/11900},
  URN =		{urn:nbn:de:0030-drops-119000},
  doi =		{10.4230/LIPIcs.STACS.2020.39},
  annote =	{Keywords: Constraint Satisfaction, Homogeneous Graphs, Bounded Width, Strict Width, Relational Width, Computational Complexity}
}

Keywords: Constraint Satisfaction, Homogeneous Graphs, Bounded Width, Strict Width, Relational Width, Computational Complexity
Seminar: 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)
Issue date: 2020
Date of publication: 04.03.2020


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