Monotone Arithmetic Complexity of Graph Homomorphism Polynomials

Authors Balagopal Komarath, Anurag Pandey, Chengot Sankaramenon Rahul

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Author Details

Balagopal Komarath
  • Indian Institute of Technology Gandhinagar, India
Anurag Pandey
  • Department of Computer Science, Universität des Saarlances, Saarland Informatics Campus, Saarbrücken, Germany
Chengot Sankaramenon Rahul
  • School of Mathematics and Computer Science, Indian Institute of Technology Goa, India


We thank Christian Engels and Marc Roth for bringing missing and incorrect references to our attention. We thank Markus Bläser for valuable discussions. We also thank anonymous reviewers for their valuable comments on the draft that helped improve the presentation of our results.

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Balagopal Komarath, Anurag Pandey, and Chengot Sankaramenon Rahul. Monotone Arithmetic Complexity of Graph Homomorphism Polynomials. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 83:1-83:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Computational complexity and cryptography
  • Theory of computation → Circuit complexity
  • Theory of computation → Parameterized complexity and exact algorithms
  • Mathematics of computing → Graph algorithms
  • 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


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