The group testing problem is a canonical inference task where one seeks to identify $k$ infected individuals out of a population of $n$ people, based on the outcomes of $m$ group tests. Of particular interest is the case of Bernoulli group testing (BGT), where each individual participates in each test independently and with a fixed probability. BGT is known to be an ``information-theoretically'' optimal design, as there exists a decoder that can identify with high probability as $n$ grows the infected individuals using $m^*=\log_2 \binom{n}{k}$ BGT tests, which is the minimum required number of tests among \emph{all} group testing designs. An important open question in the field is if a polynomial-time decoder exists for BGT which succeeds also with $m^*$ samples. In a recent paper (Iliopoulos, Zadik COLT '21) some evidence was presented (but no proof) that a simple low-temperature MCMC method could succeed. The evidence was based on a first-moment (or ``annealed'') analysis of the landscape, as well as simulations that show the MCMC success for $n \approx 1000s$. In this work, we prove that, despite the intriguing success in simulations for small $n$, the class of MCMC methods proposed in previous work for BGT with $m^*$ samples takes super-polynomial-in-$n$ time to identify the infected individuals, when $k=n^{\alpha}$ for $\alpha \in (0,1)$ small enough. Towards obtaining our results, we establish the tight max-satisfiability thresholds of the random $k$-set cover problem, a result of potentially independent interest in the study of random constraint satisfaction problems.

翻译：群组检测问题是一个典型的推断任务，其目标是在对n个人进行m次群组测试的结果基础上，识别出k名感染者。特别值得关注的是Bernoulli群组检测（BGT）情形，其中每个个体以固定概率独立参与每次测试。BGT被认为是一种“信息论意义上”最优的设计方案，因为存在一种解码器，当n增长时，能够以高概率使用m* = log₂ C(n,k)次BGT测试识别感染者，这是所有群组检测方案中所需的最小测试次数。该领域一个重要悬而未决的问题是：是否存在一种多项式时间解码器，在仅使用m*个样本时仍能成功完成BGT解码。近期论文（Iliopoulos, Zadik COLT '21）提出了（但未证明）一种简单的低温MCMC方法可能成功的证据，该证据基于对解空间的一阶矩（或“退火”）分析，以及显示MCMC在n≈1000量级时成功的模拟实验。本研究中，我们证明尽管小规模n的模拟实验显示出令人瞩目的成功，但当k = n^α（其中α∈(0,1)足够小）时，先前工作中提出的针对m*样本BGT的MCMC方法类需要超多项式时间（以n为变量）才能识别感染者。为获得此结果，我们建立了随机k-集合覆盖问题的紧致最大可满足性阈值，这一成果对随机约束满足问题的研究可能具有独立价值。