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Testing C_k-Freeness in Bounded-Arboricity Graphs

Authors Talya Eden , Reut Levi , Dana Ron

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Talya Eden
  • Department of Computer Science, Bar-Ilan University, Ramat-Gan, Israel
Reut Levi
  • Efi Arazi School of Computer Science, Reichman University, Herzliya, Israel
Dana Ron
  • School of Electrical Engineering, Tel Aviv University, Israel

Funding

  • Levi, Reut: ISF grant number 1867/20
  • Ron, Dana: ISF grant number 1146/18

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Talya Eden, Reut Levi, and Dana Ron. Testing C_k-Freeness in Bounded-Arboricity Graphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 60:1-60:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.60

Abstract

We study the problem of testing C_k-freeness (k-cycle-freeness) for fixed constant k > 3 in graphs with bounded arboricity (but unbounded degrees). In particular, we are interested in one-sided error algorithms, so that they must detect a copy of C_k with high constant probability when the graph is ε-far from C_k-free.
We next state our results for constant arboricity and constant ε with a focus on the dependence on the number of graph vertices, n. The query complexity of all our algorithms grows polynomially with 1/ε.  
1) As opposed to the case of k = 3, where the complexity of testing C₃-freeness grows with the arboricity of the graph but not with the size of the graph (Levi, ICALP 2021) this is no longer the case already for k = 4. We show that Ω(n^{1/4}) queries are necessary for testing C₄-freeness, and that Õ(n^{1/4}) are sufficient. The same bounds hold for C₅. 
2) For every fixed k ≥ 6, any one-sided error algorithm for testing C_k-freeness must perform Ω(n^{1/3}) queries. 
3) For k = 6 we give a testing algorithm whose query complexity is Õ(n^{1/2}). 
4) For any fixed k, the query complexity of testing C_k-freeness is upper bounded by {O}(n^{1-1/⌊k/2⌋}). 
The last upper bound builds on another result in which we show that for any fixed subgraph F, the query complexity of testing F-freeness is upper bounded by O(n^{1-1/𝓁(F)}), where 𝓁(F) is a parameter of F that is always upper bounded by the number of vertices in F (and in particular is k/2 in C_k for even k).
We extend some of our results to bounded (non-constant) arboricity, where in particular, we obtain sublinear upper bounds for all k.
Our Ω(n^{1/4}) lower bound for testing C₄-freeness in constant arboricity graphs provides a negative answer to an open problem posed by (Goldreich, 2021).

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Property Testing
  • Cycle-Freeness
  • Bounded Arboricity

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