发表机构
The Academic College of Tel Aviv-Yaffo（特拉维夫雅福学术学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种自适应双面误差测试算法，以多项式查询复杂度$O(d^3\log(d+1)/\epsilon^2)$测试$0,1$矩阵的二元秩，并支持近似二元分解，解决了开放问题。
AI 中文摘要
我们针对一个$0,1$矩阵$M$的二元秩提出了一种自适应双面误差测试算法，其查询复杂度为$O(d^3\log(d+1)/\epsilon^2)$，其中$d$是被测试的二元秩上界，$\epsilon$是距离参数。这回答了Parnas、Ron和Shraibman提出的一个开放问题，他们询问二元秩是否可以用关于$d$和$1/\epsilon$的多项式查询复杂度进行测试。此外，我们的测试算法可用于通过额外的$d(n+m)$次查询找到$M$的近似二元分解。也就是说，在保证$M$的二元秩至多为$d$的前提下，我们展示了如何以至少$5/6$的概率找到两个$0,1$矩阵$A',B'$，使得$M' = A' \cdot B'$是一个$0,1$矩阵，并且$M'$与$M$在至多$O(\epsilon)$比例的条目上不同。
英文摘要
We design an adaptive two-sided error testing algorithm for the binary rank of a $0,1$ matrix $M$ with query complexity $O(d^3\log(d+1)/ε^2)$, where $d$ is the tested binary rank bound and $ε$ is the distance parameter. This answers an open question posed by Parnas, Ron and Shraibman~\cite{parnas2021property}, who asked if the binary rank can be tested with query complexity polynomial in $d$ and $1/ε$. Furthermore, our testing algorithm can be used to find an approximate binary decomposition of $M$ with an additional $d(n+m)$ queries. That is, under the promise that the binary rank of $M$ is at most $d$, we show how to find, with probability at least $5/6$, two $0,1$ matrices $A',B'$ such that $M' = A' \cdot B'$ is a $0,1$ matrix which differs from $M$ on at most an $O(ε)$ fraction of its entries. Our results also imply a testing algorithm with polynomial query complexity for the equivalent problem of testing if the edges of a bipartite graph can be partitioned into at most $d$ bicliques.
