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 $O_{\varepsilon,d}(1)$ in the bidirectional model can be tested with $n^{1-\Omega_{\varepsilon,d}(1)}$ queries in the unidirectional model. In particular, if the proximity parameter $\varepsilon$ 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 $\Omega(n^{1-\frac{1}{k}})$ queries in the unidirectional model. This directly gives the first explicit properties that exhibit an $O_{\varepsilon,d}(1)$ vs $\Omega(n^{1-f(\varepsilon,d)})$ separation of the query complexities between the bidirectional model and unidirectional model, where $f(\varepsilon,d)$ is a function that approaches $0$ as $\varepsilon$ 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.

翻译：我们重新审视有界度有向图的两种基本性质测试模型间的关系：双向模型（允许算法查询顶点的出边和入边）与单向模型（仅允许查询出边）。Czumaj、Peng与Sohler [STOC 2016] 证明：对于最大入度和最大出度均以$d$为上界的有向图，任何在双向模型中可用$O_{\varepsilon,d}(1)$查询复杂度测试的性质，在单向模型中可用$n^{1-\Omega_{\varepsilon,d}(1)}$次查询测试。特别地，当接近参数$\varepsilon$趋近于0时，单向模型转换测试器的查询复杂度趋近于$n$。该转换能否进一步改进，或是否存在性质展现出如此极端的分离？本文证明：测试包含$k$个源组件的子图自由性，在单向模型中需要$\Omega(n^{1-\frac{1}{k}})$次查询。这直接给出了首批显式性质，其在双向模型与单向模型之间展现出$O_{\varepsilon,d}(1)$对$\Omega(n^{1-f(\varepsilon,d)})$的查询复杂度分离，其中$f(\varepsilon,d)$是随$\varepsilon$趋近0而趋近0的函数。此外，下界结果同时解决了Hellweg与Sohler [ESA 2012]关于测试$k$-星自由性查询复杂度的猜想。