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Monotone Bounded-Depth Complexity of Homomorphism Polynomials

Authors C.S. Bhargav , Shiteng Chen , Radu Curticapean , Prateek Dwivedi

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ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Circuit complexity
  • Mathematics of computing → Graph theory
Keywords
  • algebraic complexity
  • homomorphisms
  • monotone circuit complexity
  • bounded-depth circuits
  • treewidth
  • pathwidth

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Abstract

For every fixed graph H, it is known that homomorphism counts from H and colorful H-subgraph counts can be determined in O(n^{t+1}) time on n-vertex input graphs G, where t is the treewidth of H. On the other hand, a running time of n^{o(t / log t)} would refute the exponential-time hypothesis. Komarath, Pandey, and Rahul (Algorithmica, 2023) studied algebraic variants of these counting problems, i.e., homomorphism and subgraph polynomials for fixed graphs H. These polynomials are weighted sums over the objects counted above, where each object is weighted by the product of variables corresponding to edges contained in the object. As shown by Komarath et al., the monotone circuit complexity of the homomorphism polynomial for H is Θ(n^{tw(H)+1}).
In this paper, we characterize the power of monotone bounded-depth circuits for homomorphism and colorful subgraph polynomials. This leads us to discover a natural hierarchy of graph parameters tw_Δ(H), for fixed Δ ∈ ℕ, which capture the width of tree-decompositions for H when the underlying tree is required to have depth at most Δ. We prove that monotone circuits of product-depth Δ computing the homomorphism polynomial for H require size Θ(n^{tw_Δ(H^{†})+1}), where H^{†} is the graph obtained from H by removing all degree-1 vertices. This allows us to derive an optimal depth hierarchy theorem for monotone bounded-depth circuits through graph-theoretic arguments.

Cite As Get BibTex

C.S. Bhargav, Shiteng Chen, Radu Curticapean, and Prateek Dwivedi. Monotone Bounded-Depth Complexity of Homomorphism Polynomials. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.19

Author Details

C.S. Bhargav
  • Indian Institute of Technology, Kanpur, India
Shiteng Chen
  • Key Laboratory of System Software (Chinese Academy of Sciences) and State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China
  • University of Chinese Academy of Sciences, Beijing, China
Radu Curticapean
  • University of Regensburg, Germany
  • IT University of Copenhagen, Denmark
Prateek Dwivedi
  • IT University of Copenhagen, Denmark

Funding

Part of this work was carried out during the Copenhagen Summer of Counting & Algebraic Complexity, funded by research grants from VILLUM FONDEN (Young Investigator Grant 53093) and the European Union (ERC, CountHom, 101077083).
  • Chen, Shiteng: Supported by National Key R & D Program of China (2023YFA1009500), NSFC 61932002 and NSFC 62272448.
  • Curticapean, Radu: Funded by the European Union (ERC, CountHom, 101077083).
  • Dwivedi, Prateek: Funded by the Independent Research Fund Denmark (FLows 10.46540/3103-00116B) and supported by BARC, Villum Investigator Grant 54451.

Acknowledgements

We want to thank Vishwas Bhargava, Cornelius Brand, Deepanshu Kush, and Anurag Pandey for insightful discussions during this project. We also thank Marcin Pilipczuk for pointing out the graph parameter vertex integrity. We also thank an anonymous reviewer for pointing out a subtle issue with our definition of prunings. Part of this work was carried out during the Copenhagen Summer of Counting & Algebraic Complexity, funded by research grants from VILLUM FONDEN (Young Investigator Grant 53093) and the European Union (ERC, CountHom, 101077083). Views and opinions expressed are those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

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