arXiv subjects

Philipp Loick

Publications and source records attributed to Philipp Loick.

11 recordsLinked to original sources

arXiv2025

Optimization of Next-Day Delivery Coverage using Constraint Programming and Random Key Optimizers

We consider the logistics network of an e-commerce retailer, specifically the so-called "middle mile" network, that routes inventory from supply warehouses to distribution stations to be ingested into the terminal ("last mile") delivery network. The speed of packages through this middle mile network is a key determinant for the ultimate delivery speed to the end user. An important target for a retailer is to maximize the fraction of user orders that can be serviced within one day, i.e., next-day delivery. As such, we formulate the maximization of expected next-day delivery coverage within the middle-mile network as an optimization problem, involving a set of temporal and capacity-based constraints on the network and requiring the use of a black-box model to evaluate the objective function. We design both exact constraint programming (CP) and heuristic random-key optimizer (RKO) approaches, the former of which uses a proxy objective function. We perform experiments on large-scale, real-world problem instances and show that both approaches have merit, in that they can match or outperform the baseline solution, a bespoke greedy solver with integrated local search, in expected next-day delivery coverage. Our experiments focus on two high-level problem definitions, starting with a base problem and then adding more complexity, and also explore the generalization of the solvers across a range of problem instance sizes. We find that a hybrid model using RKO and a bespoke local search protocol performs best on the full problem definition with respect to expected next-day delivery (increase of +50 basis points [bps] over baseline) but can take days to run, whereas the hybrid model using CP and local search is slightly less competitive (+20 bps) but takes only hours to run.

math.OC↗

arXiv2021

Efficient and accurate group testing via Belief Propagation: an empirical study

The group testing problem asks for efficient pooling schemes and algorithms that allow to screen moderately large numbers of samples for rare infections. The goal is to accurately identify the infected samples while conducting the least possible number of tests. Exploring the use of techniques centred around the Belief Propagation message passing algorithm, we suggest a new test design that significantly increases the accuracy of the results. The new design comes with Belief Propagation as an efficient inference algorithm. Aiming for results on practical rather than asymptotic problem sizes, we conduct an experimental study.

cs.AI↗

arXiv2021

Note on the offspring distribution for group testing in the linear regime

The group testing problem is concerned with identifying a small set of $k$ infected individuals in a large population of $n$ people. At our disposal is a testing scheme that can test groups of individuals. A test comes back positive if and only if at least one individual is infected. In this note, we lay groundwork for analysing belief propagation for group testing when $k$ scales linearly in $n$. To this end, we derive the offspring distribution for different types of individuals. With these distributions at hand, one can employ the population dynamics algorithm to simulate the posterior marginal distribution resulting from belief propagation.

cs.IT↗

arXiv2021

The Ising antiferromagnet in the replica symmetric phase

Partition functions are an important research object in combinatorics and mathematical physics [Barvinok, 2016]. In this work, we consider the partition function of the Ising antiferromagnet on random regular graphs and characterize its limiting distribution in the replica symmetric phase up to the Kesten-Stigum bound. Our proof relies on a careful execution of the method of moments, spatial mixing arguments and small subgraph conditioning.

math.CO↗

arXiv2020

The Ising antiferromagnet and max cut on random regular graphs

The Ising antiferromagnet is an important statistical physics model with close connections to the {\sc Max Cut} problem. Combining spatial mixing arguments with the method of moments and the interpolation method, we pinpoint the replica symmetry breaking phase transition predicted by physicists. Additionally, we rigorously establish upper bounds on the {\sc Max Cut} of random regular graphs predicted by Zdeborov\'a and Boettcher [Journal of Statistical Mechanics 2010]. As an application we prove that the information-theoretic threshold of the disassortative stochastic block model on random regular graphs coincides with the Kesten-Stigum bound.

math.CO↗

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

Improved bounds for noisy group testing with constant tests per item

The group testing problem is concerned with identifying a small set of infected individuals in a large population. At our disposal is a testing procedure that allows us to test several individuals together. In an idealized setting, a test is positive if and only if at least one infected individual is included and negative otherwise. Significant progress was made in recent years towards understanding the information-theoretic and algorithmic properties in this noiseless setting. In this paper, we consider a noisy variant of group testing where test results are flipped with certain probability, including the realistic scenario where sensitivity and specificity can take arbitrary values. Using a test design where each individual is assigned to a fixed number of tests, we derive explicit algorithmic bounds for two commonly considered inference algorithms and thereby naturally extend the results of Scarlett \& Cevher (2016) and Scarlett \& Johnson (2020). We provide improved performance guarantees for the efficient algorithms in these noisy group testing models -- indeed, for a large set of parameter choices the bounds provided in the paper are the strongest currently proved.

cs.IT↗

arXiv2019

Optimal adaptive group testing

The group testing problem is concerned with identifying a small number $k \sim n^\theta$ for $\theta \in (0,1)$ 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 [M\'ezard \& Toninelli 2011] and resolves open problems prominently posed in [Aldridge, Johnson \& Scarlett 2019, Berger \& Levenshtein 2002, Damaschke \& Muhammad 2012].

cs.DM↗

arXiv2019

Optimal group testing

In the group testing problem the aim is to identify a small set of $k\sim n^\theta$ infected individuals out of a population size $n$, $0<\theta<1$. 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↗

arXiv2019

On the Parallel Reconstruction from Pooled Data

In the pooled data problem the goal is to efficiently reconstruct a binary signal from additive measurements. Given a signal $\sigma \in \{ 0,1 \}^n$, we can query multiple entries at once and get the total number of non-zero entries in the query as a result. We assume that queries are time-consuming and therefore focus on the setting where all queries are executed in parallel. For the regime where the signal is sparse such that $ || \sigma ||_1 = o(n)$ our results are twofold: First, we propose and analyze a simple and efficient greedy reconstruction algorithm. Secondly, we derive a sharp information-theoretic threshold for the minimum number of queries required to reconstruct $\sigma$ with high probability. Our first result matches the performance guarantees of much more involved constructions (Karimi et al. 2019). Our second result extends a result of Alaoui et al. (2014) and Scarlett & Cevher (2017) who studied the pooled data problem for dense signals. Finally, our theoretical findings are complemented with empirical simulations. Our data not only confirm the information-theoretic thresholds but also hint at the practical applicability of our pooling scheme and the simple greedy reconstruction algorithm.

cs.DM↗

arXiv2019

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↗