On The MCMC Performance In Bernoulli Group Testing And The Random Max Set-Cover Problem

TL;DR

It is proved 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.

Abstract

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 $k=n^α$ for $α\in (0,1)$ 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.

Figures (8)

• Figure 1: A realization for an instance of Bernoulli group testing.
• Figure 2: $\Phi_k$, the maximal proportion of covered sets for some size $k$ set of elements, as a function of $C$ for the random MAX k-set cover problem, where $C$ control the number of "target" sets (or constraints).
• Figure 3: A realization for an instance of Bernoulli group testing, now with the COMP post-processing applied.
• Figure 4: Solutions to two differing first moment functions, iliopoulos2021group's unconditional first moment function in blue and our conditional first moment function in black. These plots were made with parameters $\alpha = .01$, $a = 1.17$ and varying $C$ and $n$. The unconditional first moment function is monotonic for all of our chosen values of $C$ and $n$, confirming the analysis done by iliopoulos2021group. Our conditional first moment function is non-monotonic for $C$ sufficiently small and $n$ sufficiently large, confirming our Theorem \ref{['lem:FMFnonMono_0']}.
• Figure 5: The green and orange regions in the above plot represent the values of $\alpha$ and $C$ for which conditions \ref{['eq:FMFExists1']} and \ref{['eq:FMFExists2']} from Assumption \ref{['as:FMFExists']} are satisfied under the choice of $a$ from the lower boundary of the set \ref{['eq:a:set']} and setting $C_{34}, C_{35} = 0$. Note that the region in green is a subset of the region in orange.

Theorems & Definitions (93)

• Theorem 1.1: Informal theorem, see Theorem \ref{['thm:QualInc_0']}
• Corollary 1.2
• Theorem 1.3
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 3.1
• Definition 3.2
• Lemma 3.3: cojaoghlan2022statistical, Section 9.2.1 (arxiv version)
• Definition 3.4