Sub-Exponential Time Lower Bounds for #VC and #Matching on 3-Regular Graphs
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Sub-Exponential Time Lower Bounds for #VC and #Matching on 3-Regular Graphs

Authors Ying Liu , Shiteng Chen

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Author Details

Ying Liu
  • State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China
  • University of Chinese Academy of Sciences, Beijing, China
Shiteng Chen
  • State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China
  • University of Chinese Academy of Sciences, Beijing, China

Funding

  • Liu, Ying: Supported by NSFC 61932002 and NSFC 62272448.
  • Chen, Shiteng: Supported by National Key R&D Program of China (2023YFA1009500), NSFC 61932002 and NSFC 62272448

Acknowledgements

The authors are very grateful to Prof. Mingji Xia for his beneficial guidance and advice.

Cite As Get BibTex

Ying Liu and Shiteng Chen. Sub-Exponential Time Lower Bounds for #VC and #Matching on 3-Regular Graphs. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 49:1-49:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.STACS.2024.49

Abstract

This article focuses on the sub-exponential time lower bounds for two canonical #P-hard problems: counting the vertex covers of a given graph (#VC) and counting the matchings of a given graph (#Matching), under the well-known counting exponential time hypothesis (#ETH). 
Interpolation is an essential method to build reductions in this article and in the literature. We use the idea of block interpolation to prove that both #VC and #Matching have no 2^{o(N)} time deterministic algorithm, even if the given graph with N vertices is a 3-regular graph. However, when it comes to proving the lower bounds for #VC and #Matching on planar graphs, both block interpolation and polynomial interpolation do not work. We prove that, for any integer N > 0, we can simulate N pairwise linearly independent unary functions by gadgets with only O(log N) size in the context of #VC and #Matching. Then we use log-size gadgets in the polynomial interpolation to prove that planar #VC and planar #Matching have no 2^{o(√{N/(log N)})} time deterministic algorithm. The lower bounds hold even if the given graph with N vertices is a 3-regular graph. 
Based on a stronger hypothesis, randomized exponential time hypothesis (rETH), we can avoid using interpolation. We prove that if rETH holds, both planar #VC and planar #Matching have no 2^{o(√N)} time randomized algorithm, even that the given graph with N vertices is a planar 3-regular graph. The 2^{Ω(√N)} time lower bounds are tight, since there exist 2^{O(√N)} time algorithms for planar #VC and planar #Matching.
We also develop a fine-grained dichotomy for a class of counting problems, symmetric Holant*.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • computational complexity
  • planar Holant
  • polynomial interpolation
  • rETH
  • sub-exponential
  • #ETH
  • #Matching
  • #VC

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