Testing $C_k$-freeness in bounded-arboricity graphs

TL;DR

This work develops sublinear-time, one-sided property testers for C_k-freeness in graphs with bounded arboricity, identifying tight or near-tight bounds across several regimes. The authors introduce a unified framework: if a graph is ε-far from C_k-freeness, there exists a large collection of edge-disjoint C_k’s whose structure relative to vertex degrees can be exploited by sampling and short random walks. They prove Ω(n^{1/4}) lower bounds for C_4 and C_5 and Ω(n^{1/3}) lower bounds for k≥6, while providing constructive upper bounds, including a tilde-O(n^{1/2}) tester for C_6 and a general upper bound of O(n^{1-1/loor{k/2}}) for fixed k, with extensions to general fixed F via ell(F). The results answer open questions about moving from bounded-degree models to bounded-arboricity graphs and reveal a nuanced dependency on n, α, and ε, including a negative resolution to Goldreich’s open problem. Overall, the paper advances understanding of sublinear testing for cycle-freeness in sparse, unbounded-degree graphs and offers techniques likely applicable to broader F-freeness properties.

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 $C_3$-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 $C_4$-freeness, and that $\widetilde{O}(n^{1/4})$ are sufficient. The same bounds hold for $C_5$. (2) For every fixed $k \geq 6$, 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 $\widetilde{O}(n^{1/2})$. (4) For any fixed $k$, the query complexity of testing $C_k$-freeness is upper bounded by ${O}(n^{1-1/\lfloor k/2\rfloor})$. Our $Ω(n^{1/4})$ lower bound for testing $C_4$-freeness in constant arboricity graphs provides a negative answer to an open problem posed by (Goldreich, 2021).

Figures (4)

• Figure 1: An illustration for some of the cases considered in the analysis of the algorithm for $C_4$-freeness. On the left side are two examples in which there is a single vertex $v'$ with degree greater than $\theta_1$, so that there is a vertex $v$ with degree at most $\theta_1$ with two neighbors whose degree is at most $\theta_0$. On the right is an illustration when there are two such vertices with degree greater than $\theta_1$.
• Figure 2: An illustration for the lower bound construction. The graph on the left is $C_4$-free while the graph on the right contains $\Omega(m)$ edge-disjoint $C_4$s and is hence $\Omega(1)$-far from being $C_4$-free.
• Figure 3: An illustration of the auxiliary (multi-)graph $G'$ in the $C_6$-freeness testing algorithm. The dashed lines represent edges in $G'$, each one corresponding to a length-2 path in $G$ that passes through a vertex with degree at most $\theta_0$.
• Figure 4: An illustration for the lower bound construction for $C_k$-freeness in constant arboricity graphs when $k=9$. The three circles in the middle and the dashed lines represent a graph $G'\in {\mathcal{G}}_{n'}$. The outer circles represent the additional vertices in $G$. Since $k=9$ in this example, each edge in $G'$ is replaced by a path of length 3 in $G$.

Theorems & Definitions (33)

• Theorem 1
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• Theorem 7
• Claim 1
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• Claim 3