An Optimal Separation Between Two Property Testing Models for Bounded Degree Directed Graphs
We revisit the relation between two fundamental property testing models for bounded-degree directed graphs: the bidirectional model in which the algorithms are allowed to query both the outgoing edges and incoming edges of a vertex, and the unidirectional model in which only queries to the outgoing edges are allowed. Czumaj, Peng and Sohler [STOC 2016] showed that for directed graphs with both maximum indegree and maximum outdegree upper bounded by d, any property that can be tested with query complexity O_{ε,d}(1) in the bidirectional model can be tested with n^{1-Ω_{ε,d}(1)} queries in the unidirectional model. In particular, {if the proximity parameter ε approaches 0, then the query complexity of the transformed tester in the unidirectional model approaches n}. It was left open if this transformation can be further improved or there exists any property that exhibits such an extreme separation.
We prove that testing subgraph-freeness in which the subgraph contains k source components, requires Ω(n^{1-1/k}) queries in the unidirectional model. This directly gives the first explicit properties that exhibit an O_{ε,d}(1) vs Ω(n^{1-f(ε,d)}) separation of the query complexities between the bidirectional model and unidirectional model, where f(ε,d) is a function that approaches 0 as ε approaches 0. Furthermore, our lower bound also resolves a conjecture by Hellweg and Sohler [ESA 2012] on the query complexity of testing k-star-freeness.
Graph property testing
Directed graphs
Lower bound
Subgraph-freeness
Theory of computation~Streaming, sublinear and near linear time algorithms
96:1-96:16
Track A: Algorithms, Complexity and Games
This work has been supported in part by NSFC grant 62272431 and "the Fundamental Research Funds for the Central Universities".
Pan
Peng
Pan Peng
School of Computer Science and Technology, University of Science and Technology of China, Hefei, China
https://orcid.org/0000-0003-2700-5699
Yuyang
Wang
Yuyang Wang
School of Computer Science and Technology, University of Science and Technology of China, Hefei, China
10.4230/LIPIcs.ICALP.2023.96
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Pan Peng and Yuyang Wang
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