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Group testing tackles the problem of identifying a population of $K$defective items from a set of $n$ items by pooling groups of items efficientlyin order to cut down the number of tests needed. The result of a test for agroup of items is positive if any of the items in the group is defective andnegative otherwise. The goal is to judiciously group subsets of items such thatdefective 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), anon-adaptive group testing paradigm that recovers at least a$(1-\epsilon)$-fraction (for any arbitrarily small $\epsilon > 0$) of $K$defective items with high probability with $m=6C(\epsilon)K\log_2{n}$ tests,where $C(\epsilon)$ is a precisely characterized constant that depends only on$\epsilon$. For instance, it can provably recover at least $(1-10^{-6})K$defective items with $m \simeq 68 K \log_2{n}$ tests. The computationalcomplexity of the decoding algorithm of SAFFRON is $\mathcal{O}(K\log n)$,which is order-optimal. Further, we robustify SAFFRON such that it can reliablyrecover the set of $K$ defective items even in the presence of erroneous ornoisy test results. We also propose Singleton-Only-SAFFRON, a variant ofSAFFRON, that recovers all the $K$ defective items with $m=2e(1+\alpha)K\log K\log_2 n$ tests with probability$1-\mathcal{O}{\left(\frac{1}{K^\alpha}\right)}$, where $\alpha>0$ is aconstant. By leveraging powerful design and analysis tools from modernsparse-graph coding theory, SAFFRON is the first approach to reliable,large-scale probabilistic group testing that offers both preciselycharacterizable number of tests needed (down to the constants) together withorder-optimal decoding complexity.

SAFFRON: A Fast, Efficient, and Robust Framework for Group Testing based on Sparse-Graph Codes