arXiv subjects

Max Hahn-Klimroth

Publications and source records attributed to Max Hahn-Klimroth.

24 records · Page 2Linked to original sources

arXiv2020

Information-theoretic and algorithmic thresholds for group testing

In the group testing problem we aim to identify a small number of infected individuals within a large population. We avail ourselves to a procedure that can test a group of multiple individuals, with the test result coming out positive iff at least one individual in the group is infected. With all tests conducted in parallel, what is the least number of tests required to identify the status of all individuals? In a recent test design [Aldridge et al.\ 2016] the individuals are assigned to test groups randomly, with every individual joining an equal number of groups. We pinpoint the sharp threshold for the number of tests required in this randomised design so that it is information-theoretically possible to infer the infection status of every individual. Moreover, we analyse two efficient inference algorithms. These results settle conjectures from [Aldridge et al.\ 2014, Johnson et al.\ 2019].

cs.DM↗

arXiv2020

Inference and mutual information on random factor graphs

Random factor graphs provide a powerful framework for the study of inference problems such as decoding problems or the stochastic block model. Information-theoretically the key quantity of interest is the mutual information between the observed factor graph and the underlying ground truth around which the factor graph was created; in the stochastic block model, this would be the planted partition. The mutual information gauges whether and how well the ground truth can be inferred from the observable data. For a very general model of random factor graphs we verify a formula for the mutual information predicted by physics techniques. As an application we prove a conjecture about low-density generator matrix codes from [Montanari: IEEE Transactions on Information Theory 2005]. Further applications include phase transitions of the stochastic block model and the mixed $k$-spin model from physics.

cs.DM↗

arXiv2020

Optimal group testing

In the group testing problem the aim is to identify a small set of $k\sim n^θ$ infected individuals out of a population size $n$, $0<θ<1$. We avail ourselves of a test procedure capable of testing groups of individuals, with the test returning a positive result iff at least one individual in the group is infected. The aim is to devise a test design with as few tests as possible so that the set of infected individuals can be identified correctly with high probability. We establish an explicit sharp information-theoretic/algorithmic phase transition $\minf$ for non-adaptive group testing, where all tests are conducted in parallel. Thus, with more than $\minf$ tests the infected individuals can be identified in polynomial time \whp, while learning the set of infected individuals is information-theoretically impossible with fewer tests. In addition, we develop an optimal adaptive scheme where the tests are conducted in two stages.

cs.DM↗

arXiv2020

Random perturbation of sparse graphs

In the model of randomly perturbed graphs we consider the union of a deterministic graph $\mathcal{G}_α$ with minimum degree $αn$ and the binomial random graph $\mathbb{G}(n,p)$. This model was introduced by Bohman, Frieze, and Martin and for Hamilton cycles their result bridges the gap between Dirac's theorem and the results by Posá and Koršunov on the threshold in $\mathbb{G}(n,p)$. In this note we extend this result in $\mathcal{G}_α\cup \mathbb{G}(n,p)$ to sparser graphs with $α=o(1)$. More precisely, for any $\varepsilon>0$ and $α\colon \mathbb{N} \mapsto (0,1)$ we show that a.a.s. $\mathcal{G}_α\cup \mathbb{G}(n,β/n)$ is Hamiltonian, where $β= -(6 + \varepsilon) \log(α)$. If $α>0$ is a fixed constant this gives the aforementioned result by Bohman, Frieze, and Martin and if $α=O(1/n)$ the random part $\mathbb{G}(n,p)$ is sufficient for a Hamilton cycle. We also discuss embeddings of bounded degree trees and other spanning structures in this model, which lead to interesting questions on almost spanning embeddings into $\mathbb{G}(n,p)$.

math.CO↗

arXiv2020

The random 2-SAT partition function

We show that throughout the satisfiable phase the normalised number of satisfying assignments of a random $2$-SAT formula converges in probability to an expression predicted by the cavity method from statistical physics. The proof is based on showing that the Belief Propagation algorithm renders the correct marginal probability that a variable is set to `true' under a uniformly random satisfying assignment.

math.PR↗

arXiv2019

Optimal adaptive group testing

The group testing problem is concerned with identifying a small number $k \sim n^θ$ for $θ\in (0,1)$ of infected individuals in a large population of size $n$. At our disposal is a testing procedure that allows us to test groups of individuals. This paper considers two-stage designs where the test results of the first stage can inform the design of the second stage. We are interested in the minimum number of tests to recover the set of infected individuals w.h.p. Equipped with a novel algorithm for one-stage group testing from [Coja-Oghlan, Gebhard, Hahn-Klimroth \& Loick 2019], we propose a polynomial-time two-stage algorithm that matches the universal information-theoretic lower bound of group testing. This result improves on results from [Mézard \& Toninelli 2011] and resolves open problems prominently posed in [Aldridge, Johnson \& Scarlett 2019, Berger \& Levenshtein 2002, Damaschke \& Muhammad 2012].

cs.DM↗