3,473 research outputs found
Parameter Setting in Quantum Approximate Optimization of Weighted Problems
Quantum Approximate Optimization Algorithm (QAOA) is a leading candidate algorithm for solving combinatorial optimization problems on quantum computers. However, in many cases QAOA requires computationally intensive parameter optimization. The challenge of parameter optimization is particularly acute in the case of weighted problems, for which the eigenvalues of the phase operator are non-integer and the QAOA energy landscape is not periodic. In this work, we develop parameter setting heuristics for QAOA applied to a general class of weighted problems. First, we derive optimal parameters for QAOA with depth applied to the weighted MaxCut problem under different assumptions on the weights. In particular, we rigorously prove the conventional wisdom that in the average case the first local optimum near zero gives globally-optimal QAOA parameters. Second, for we prove that the QAOA energy landscape for weighted MaxCut approaches that for the unweighted case under a simple rescaling of parameters. Therefore, we can use parameters previously obtained for unweighted MaxCut for weighted problems. Finally, we prove that for the QAOA objective sharply concentrates around its expectation, which means that our parameter setting rules hold with high probability for a random weighted instance. We numerically validate this approach on general weighted graphs and show that on average the QAOA energy with the proposed fixed parameters is only percentage points away from that with optimized parameters. Third, we propose a general heuristic rescaling scheme inspired by the analytical results for weighted MaxCut and demonstrate its effectiveness using QAOA with the XY Hamming-weight-preserving mixer applied to the portfolio optimization problem. Our heuristic improves the convergence of local optimizers, reducing the number of iterations by 7.4x on average
Dynamic Connectivity in Disk Graphs
Let S ⊆ R2 be a set of n sites in the plane, so that every site s ∈ S has an associated
radius rs > 0. Let D(S) be the disk intersection graph defined by S, i.e., the graph
with vertex set S and an edge between two distinct sites s, t ∈ S if and only if the
disks with centers s, t and radii rs , rt intersect. Our goal is to design data structures
that maintain the connectivity structure of D(S) as sites are inserted and/or deleted
in S. First, we consider unit disk graphs, i.e., we fix rs = 1, for all sites s ∈ S.
For this case, we describe a data structure that has O(log2 n) amortized update time
and O(log n/ log log n) query time. Second, we look at disk graphs with bounded
radius ratio Ψ, i.e., for all s ∈ S, we have 1 ≤ rs ≤ Ψ, for a parameter Ψ that is
known in advance. Here, we not only investigate the fully dynamic case, but also the
incremental and the decremental scenario, where only insertions or only deletions of
sites are allowed. In the fully dynamic case, we achieve amortized expected update
time O(Ψ log4 n) and query time O(log n/ log log n). This improves the currently
best update time by a factor of Ψ. In the incremental case, we achieve logarithmic
dependency on Ψ, with a data structure that has O(α(n)) amortized query time and
O(log Ψ log4 n) amortized expected update time, where α(n) denotes the inverse Ackermann
function. For the decremental setting, we first develop an efficient decremental
disk revealing data structure: given two sets R and B of disks in the plane, we can delete
disks from B, and upon each deletion, we receive a list of all disks in R that no longer
intersect the union of B. Using this data structure, we get decremental data structures
with a query time of O(log n/ log log n) that supports deletions in O(n log Ψ log4 n)
overall expected time for disk graphs with bounded radius ratio Ψ and O(n log5 n)
overall expected time for disk graphs with arbitrary radii, assuming that the deletion
sequence is oblivious of the internal random choices of the data structures
A revisited branch-and-cut algorithm for large-scale orienteering problems
The orienteering problem is a route optimization problem which consists of finding a simple cycle that maximizes the total collected profit subject to a maximum distance limitation. In the last few decades, the occurrence of this problem in real-life applications has boosted the development of many heuristic algorithms to solve it. However, during the same period, not much research has been devoted to the field of exact algorithms for the orienteering problem. The aim of this work is to develop an exact method which is able to obtain the optimum in a wider set of instances than with previous methods, or to improve the lower and upper bounds in its disability. We propose a revisited version of the branch-and-cut algorithm for the orienteering problem which includes new contributions in the separation algorithms of inequalities stemming from the cycle problem, in the separation loop, in the variables pricing, and in the calculation of the lower and upper bounds of the problem. Our proposal is compared to three state-of-the-art algorithms on 258 benchmark instances with up to 7397 nodes. The computational experiments show the relevance of the designed components where 18 new optima, 76 new best-known solutions and 85 new upper-bound values were obtained
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Tropical medians by transportation
Fermat-Weber points with respect to an asymmetric tropical distance function
are studied. It turns out that they correspond to the optimal solutions of a
transportation problem. The results are applied to obtain a new method for
computing consensus trees in phylogenetics. This method has several desirable
properties; e.g., it is Pareto and co-Pareto on rooted triplets.Comment: 23 pages, 9 figures, computational experiments adde
Automated identification and behaviour classification for modelling social dynamics in group-housed mice
Mice are often used in biology as exploratory models of human conditions, due to their similar genetics and physiology. Unfortunately, research on behaviour has traditionally been limited to studying individuals in isolated environments and over short periods of time. This can miss critical time-effects, and, since mice are social creatures, bias results.
This work addresses this gap in research by developing tools to analyse the individual behaviour of group-housed mice in the home-cage over several days and with minimal disruption. Using data provided by the Mary Lyon Centre at MRC Harwell we designed an end-to-end system that (a) tracks and identifies mice in a cage, (b) infers their behaviour, and subsequently (c) models the group dynamics as functions of individual activities. In support of the above, we also curated and made available a large dataset of mouse localisation and behaviour classifications (IMADGE), as well as two smaller annotated datasets for training/evaluating the identification (TIDe) and behaviour inference (ABODe) systems. This research constitutes the first of its kind in terms of the scale and challenges addressed. The data source (side-view single-channel video with clutter and no identification markers for mice) presents challenging conditions for analysis, but has the potential to give richer information while using industry standard housing.
A Tracking and Identification module was developed to automatically detect, track and identify the (visually similar) mice in the cluttered home-cage using only single-channel IR video and coarse position from RFID readings. Existing detectors and trackers were combined with a novel Integer Linear Programming formulation to assign anonymous tracks to mouse identities. This utilised a probabilistic weight model of affinity between detections and RFID pickups.
The next task necessitated the implementation of the Activity Labelling module that classifies the behaviour of each mouse, handling occlusion to avoid giving unreliable classifications when the mice cannot be observed. Two key aspects of this were (a) careful feature-selection, and (b) judicious balancing of the errors of the system in line with the repercussions for our setup.
Given these sequences of individual behaviours, we analysed the interaction dynamics between mice in the same cage by collapsing the group behaviour into a sequence of interpretable latent regimes using both static and temporal (Markov) models. Using a permutation matrix, we were able to automatically assign mice to roles in the HMM, fit a global model to a group of cages and analyse abnormalities in data from a different demographic
Quantum Alternating Operator Ansatz (QAOA) beyond low depth with gradually changing unitaries
The Quantum Approximate Optimization Algorithm and its generalization to
Quantum Alternating Operator Ansatz (QAOA) is a promising approach for applying
quantum computers to challenging problems such as combinatorial optimization
and computational chemistry. In this paper, we study the underlying mechanisms
governing the behavior of QAOA circuits beyond shallow depth in the practically
relevant setting of gradually varying unitaries. We use the discrete adiabatic
theorem, which complements and generalizes the insights obtained from the
continuous-time adiabatic theorem primarily considered in prior work. Our
analysis explains some general properties that are conspicuously depicted in
the recently introduced QAOA performance diagrams. For parameter sequences
derived from continuous schedules (e.g. linear ramps), these diagrams capture
the algorithm's performance over different parameter sizes and circuit depths.
Surprisingly, they have been observed to be qualitatively similar across
different performance metrics and application domains. Our analysis explains
this behavior as well as entails some unexpected results, such as connections
between the eigenstates of the cost and mixer QAOA Hamiltonians changing based
on parameter size and the possibility of reducing circuit depth without
sacrificing performance
Exploration autonome et efficiente de chantiers miniers souterrains inconnus avec un drone filaire
Abstract: Underground mining stopes are often mapped using a sensor located at the end of a pole that the operator introduces into the stope from a secure area. The sensor emits laser beams that provide the distance to a detected wall, thus creating a 3D map. This produces shadow zones and a low point density on the distant walls. To address these challenges, a research team from the Université de Sherbrooke is designing a tethered drone equipped with a rotating LiDAR for this mission, thus benefiting from several points of view. The wired transmission allows for unlimited flight time, shared computing, and real-time communication. For compatibility with the movement of the drone after tether entanglements, the excess length is integrated into an onboard spool, contributing to the drone payload. During manual piloting, the human factor causes problems in the perception and comprehension of a virtual 3D environment, as well as the execution of an optimal mission. This thesis focuses on autonomous navigation in two aspects: path planning and exploration. The system must compute a trajectory that maps the entire environment, minimizing the mission time and respecting the maximum onboard tether length. Path planning using a Rapidly-exploring Random Tree (RRT) quickly finds a feasible path, but the optimization is computationally expensive and the performance is variable and unpredictable. Exploration by the frontier method is representative of the space to be explored and the path can be optimized by solving a Traveling Salesman Problem (TSP) but existing techniques for a tethered drone only consider the 2D case and do not optimize the global path. To meet these challenges, this thesis presents two new algorithms. The first one, RRT-Rope, produces an equal or shorter path than existing algorithms in a significantly shorter computation time, up to 70% faster than the next best algorithm in a representative environment. A modified version of RRT-connect computes a feasible path, shortened with a deterministic technique that takes advantage of previously added intermediate nodes. The second algorithm, TAPE, is the first 3D cavity exploration method that focuses on minimizing mission time and unwound tether length. On average, the overall path is 4% longer than the method that solves the TSP, but the tether remains under the allowed length in 100% of the simulated cases, compared to 53% with the initial method. The approach uses a 2-level hierarchical architecture: global planning solves a TSP after frontier extraction, and local planning minimizes the path cost and tether length via a decision function. The integration of these two tools in the NetherDrone produces an intelligent system for autonomous exploration, with semi-autonomous features for operator interaction. This work opens the door to new navigation approaches in the field of inspection, mapping, and Search and Rescue missions.La cartographie des chantiers miniers souterrains est souvent réalisée à l’aide d’un capteur situé au bout d’une perche que l’opérateur introduit dans le chantier, depuis une zone sécurisée. Le capteur émet des faisceaux laser qui fournissent la distance à un mur détecté, créant ainsi une carte en 3D. Ceci produit des zones d’ombres et une faible densité de points sur les parois éloignées. Pour relever ces défis, une équipe de recherche de l’Université de Sherbrooke conçoit un drone filaire équipé d’un LiDAR rotatif pour cette mission, bénéficiant ainsi de plusieurs points de vue. La transmission filaire permet un temps de vol illimité, un partage de calcul et une communication en temps réel. Pour une compatibilité avec le mouvement du drone lors des coincements du fil, la longueur excédante est intégrée dans une bobine embarquée, qui contribue à la charge utile du drone. Lors d’un pilotage manuel, le facteur humain entraîne des problèmes de perception et compréhension d’un environnement 3D virtuel, et d’exécution d’une mission optimale. Cette thèse se concentre sur la navigation autonome sous deux aspects : la planification de trajectoire et l’exploration. Le système doit calculer une trajectoire qui cartographie l’environnement complet, en minimisant le temps de mission et en respectant la longueur maximale de fil embarquée. La planification de trajectoire à l’aide d’un Rapidly-exploring Random Tree (RRT) trouve rapidement un chemin réalisable, mais l’optimisation est coûteuse en calcul et la performance est variable et imprévisible. L’exploration par la méthode des frontières est représentative de l’espace à explorer et le chemin peut être optimisé en résolvant un Traveling Salesman Problem (TSP), mais les techniques existantes pour un drone filaire ne considèrent que le cas 2D et n’optimisent pas le chemin global. Pour relever ces défis, cette thèse présente deux nouveaux algorithmes. Le premier, RRT-Rope, produit un chemin égal ou plus court que les algorithmes existants en un temps de calcul jusqu’à 70% plus court que le deuxième meilleur algorithme dans un environnement représentatif. Une version modifiée de RRT-connect calcule un chemin réalisable, raccourci avec une technique déterministe qui tire profit des noeuds intermédiaires préalablement ajoutés. Le deuxième algorithme, TAPE, est la première méthode d’exploration de cavités en 3D qui minimise le temps de mission et la longueur du fil déroulé. En moyenne, le trajet global est 4% plus long que la méthode qui résout le TSP, mais le fil reste sous la longueur autorisée dans 100% des cas simulés, contre 53% avec la méthode initiale. L’approche utilise une architecture hiérarchique à 2 niveaux : la planification globale résout un TSP après extraction des frontières, et la planification locale minimise le coût du chemin et la longueur de fil via une fonction de décision. L’intégration de ces deux outils dans le NetherDrone produit un système intelligent pour l’exploration autonome, doté de fonctionnalités semi-autonomes pour une interaction avec l’opérateur. Les travaux réalisés ouvrent la porte à de nouvelles approches de navigation dans le domaine des missions d’inspection, de cartographie et de recherche et sauvetage
When Deep Learning Meets Polyhedral Theory: A Survey
In the past decade, deep learning became the prevalent methodology for
predictive modeling thanks to the remarkable accuracy of deep neural networks
in tasks such as computer vision and natural language processing. Meanwhile,
the structure of neural networks converged back to simpler representations
based on piecewise constant and piecewise linear functions such as the
Rectified Linear Unit (ReLU), which became the most commonly used type of
activation function in neural networks. That made certain types of network
structure \unicode{x2014}such as the typical fully-connected feedforward
neural network\unicode{x2014} amenable to analysis through polyhedral theory
and to the application of methodologies such as Linear Programming (LP) and
Mixed-Integer Linear Programming (MILP) for a variety of purposes. In this
paper, we survey the main topics emerging from this fast-paced area of work,
which bring a fresh perspective to understanding neural networks in more detail
as well as to applying linear optimization techniques to train, verify, and
reduce the size of such networks
A proof of the Ryser-Brualdi-Stein conjecture for large even
A Latin square of order is an by grid filled using symbols so
that each symbol appears exactly once in each row and column. A transversal in
a Latin square is a collection of cells which share no symbol, row or column.
The Ryser-Brualdi-Stein conjecture, with origins from 1967, states that every
Latin square of order contains a transversal with cells, and a
transversal with cells if is odd. Keevash, Pokrovskiy, Sudakov and
Yepremyan recently improved the long-standing best known bounds towards this
conjecture by showing that every Latin square of order has a transversal
with cells. Here, we show, for sufficiently large ,
that every Latin square of order has a transversal with cells.
We also apply our methods to show that, for sufficiently large , every
Steiner triple system of order has a matching containing at least
edges. This improves a recent result of Keevash, Pokrovskiy, Sudakov and
Yepremyan, who found such matchings with edges, and
proves a conjecture of Brouwer from 1981 for large .Comment: 71 pages, 13 figure
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