组测试通过有效地汇集项目组来解决从一组$n$项目中识别$K$缺陷项目的问题，以减少所需的测试数量。如果一组项目中的任何项目有缺陷，则该组项目的测试结果为阳性，否则为阴性。我们的目标是明智地分组项目的子集，使得有缺陷的项目可以使用最少的测试次数可靠地恢复，同时还具有低复杂度的解码过程。我们描述SAFFRON（稀疏grAph代码框架gROuptestNg），一个非自适应组测试范式，恢复至少$(1-ε)$ -分数（对于任何任意小的$ε> 0$）的$K$缺陷项目具有高概率与$m=6C(ε)K\log_2{n}$测试，其中$C(ε)$是一个精确的特征常数，只依赖于$ε$。例如，它可以通过$m \simeq 68 K \log_2{n}$测试至少可证明地恢复$(1-10^{-6})K$缺陷项。SAFFRON译码算法的计算复杂度为$\mathcal{O}(K\log n)$，是阶数最优的。此外，我们对SAFFRON进行了鲁棒化，使其即使在存在错误或噪声测试结果的情况下也能可靠地恢复$K$缺陷项集。我们还提出了Singleton-Only-SAFFRON，SAFFRON的一个变体，它可以用概率为$1-\mathcal{O}{\left(\frac{1}{K^α}\right)}$的$m=2e(1+α)K\log K \log_2 n$测试来恢复所有$K$缺陷项，其中$α>0$是一个常数。通过利用现代稀疏图编码理论中强大的设计和分析工具，SAFFRON是可靠的大规模概率组测试的第一种方法，它提供了所需的精确表征的测试数量（下至常数）以及阶数最优的解码复杂性。
英文摘要：
Group testing tackles the problem of identifying a population of $K$ defective items from a set of $n$ items by pooling groups of items efficiently in order to cut down the number of tests needed. The result of a test for a group of items is positive if any of the items in the group is defective and negative otherwise. The goal is to judiciously group subsets of items such that defective items can be reliably recovered using the minimum number of tests, while also having a low-complexity decoding procedure.
We describe SAFFRON (Sparse-grAph codes Framework For gROup testiNg), a non-adaptive group testing paradigm that recovers at least a $(1-ε)$-fraction (for any arbitrarily small $ε> 0$) of $K$ defective items with high probability with $m=6C(ε)K\log_2{n}$ tests, where $C(ε)$ is a precisely characterized constant that depends only on $ε$. For instance, it can provably recover at least $(1-10^{-6})K$ defective items with $m \simeq 68 K \log_2{n}$ tests. The computational complexity of the decoding algorithm of SAFFRON is $\mathcal{O}(K\log n)$, which is order-optimal. Further, we robustify SAFFRON such that it can reliably recover the set of $K$ defective items even in the presence of erroneous or noisy test results. We also propose Singleton-Only-SAFFRON, a variant of SAFFRON, that recovers all the $K$ defective items with $m=2e(1+α)K\log K \log_2 n$ tests with probability $1-\mathcal{O}{\left(\frac{1}{K^α}\right)}$, where $α>0$ is a constant. By leveraging powerful design and analysis tools from modern sparse-graph coding theory, SAFFRON is the first approach to reliable, large-scale probabilistic group testing that offers both precisely characterizable number of tests needed (down to the constants) together with order-optimal decoding complexity.
