License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/LIPIcs.ICALP.2022.83
URN: urn:nbn:de:0030-drops-164245
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16424/
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Komarath, Balagopal ; Pandey, Anurag ; Rahul, Chengot Sankaramenon

Monotone Arithmetic Complexity of Graph Homomorphism Polynomials

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Abstract

We study homomorphism polynomials, which are polynomials that enumerate all homomorphisms from a pattern graph H to n-vertex graphs. These polynomials have received a lot of attention recently for their crucial role in several new algorithms for counting and detecting graph patterns, and also for obtaining natural polynomial families which are complete for algebraic complexity classes VBP, VP, and VNP. We discover that, in the monotone setting, the formula complexity, the ABP complexity, and the circuit complexity of such polynomial families are exactly characterized by the treedepth, the pathwidth, and the treewidth of the pattern graph respectively.
Furthermore, we establish a single, unified framework, using our characterization, to collect several known results that were obtained independently via different methods. For instance, we attain superpolynomial separations between circuits, ABPs, and formulas in the monotone setting, where the polynomial families separating the classes all correspond to well-studied combinatorial problems. Moreover, our proofs rediscover fine-grained separations between these models for constant-degree polynomials.

BibTeX - Entry

@InProceedings{komarath_et_al:LIPIcs.ICALP.2022.83,
  author =	{Komarath, Balagopal and Pandey, Anurag and Rahul, Chengot Sankaramenon},
  title =	{{Monotone Arithmetic Complexity of Graph Homomorphism Polynomials}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{83:1--83:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16424},
  URN =		{urn:nbn:de:0030-drops-164245},
  doi =		{10.4230/LIPIcs.ICALP.2022.83},
  annote =	{Keywords: Homomorphism polynomials, Monotone complexity, Algebraic complexity, Graph algorithms, Fine-grained complexity, Fixed-parameter algorithms and complexity, Treewidth, Pathwidth, Treedepth, Graph homomorphisms, Algebraic circuits, Algebraic branching programs, Algebraic formulas}
}

Keywords: Homomorphism polynomials, Monotone complexity, Algebraic complexity, Graph algorithms, Fine-grained complexity, Fixed-parameter algorithms and complexity, Treewidth, Pathwidth, Treedepth, Graph homomorphisms, Algebraic circuits, Algebraic branching programs, Algebraic formulas
Collection: 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)
Issue Date: 2022
Date of publication: 28.06.2022

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