Enumeration Scheme Research Articles

The Click-Based MNL Model: A Framework for Modeling Click Data in Assortment Optimization

We introduce the click-based MNL choice model, a framework for capturing customer purchasing decisions in e-commerce settings. Specifically, we augment the classical Multinomial Logit choice model by assuming that customers only consider the items they have clicked on before they proceed to compare their random utilities. In this context, we study the resulting assortment optimization problem, where the objective is to select a subset of products, made available for purchase, to maximize the expected revenue. Our main algorithmic contribution comes in the form of a polynomial-time approximation scheme (PTAS) for this problem, showing that the optimal expected revenue can be efficiently approached within any degree of accuracy. To establish this result, we develop several technical ideas, including enumeration schemes and stochastic inequalities, which may be of broader interest. Using data from Alibaba’s online marketplace, we fit click-based MNL and latent class MNL models to historical sales and click data in a setting where the online platform recommends a personalized six-product display to each user. We propose an estimation methodology for the click-based MNL model that leverages clickstream data and machine learning classification algorithms. Our numerical results suggest that clickstream data are valuable for predicting choices and that the click-based MNL model can outperform standard logit-based models in certain settings. This paper was accepted by Chung Piaw Teo, optimization. Funding: D. Segev was supported by the Israel Science Foundation [Grant 1407/20]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/mnsc.2021.00281 .

Total variation distance and compound poisson approximations for random sums

We derive new upper bounds for the total variation distance between compound Poisson distributions as well as between a random sum and a compound Poisson distribution, and as a result we also obtain upper bounds for the total variation distance between compound Poisson distributions and a sum of independent random variables. These bounds are generalizations and refinements of some well-known bounds in the literature. We also derive upper bounds for the total variation distance between negative binomial distributions of order k and between the negative binomial distributions of order k and compound Poisson distributions. Upper bounds for the total variation distances between the number of success runs of length k in binary Markovian trials and its limiting distributions for several enumeration schemes, are also given.

Crystal-Structure Matches in Solid-Solid Phase Transitions.

The exploration of solid-solid phase transition suffers from the uncertainty of how atoms in two crystal structures match. We devised a theoretical framework to describe and classify crystal-structure matches (CSM). Such description fully exploits the translational and rotational symmetries and is independent of the choice of supercells. This is enabled by the use of the Hermite normal form, an analog of reduced echelon form for integer matrices. With its help, exhausting all CSMs is made possible, which goes beyond the conventional optimization schemes. In an example study of the martensitic transformation of steel, our enumeration algorithm finds many candidate CSMs with lower strains than known mechanisms. Two long-sought CSMs accounting for the most commonly observed Kurdjumov-Sachs orientation relationship and the Nishiyama-Wassermann orientation relationship are unveiled. Given the comprehensiveness and efficiency, our enumeration scheme provide a promising strategy for solid-solid phase transition mechanism research.

Optimal and heuristic solutions for placing multiple finite-size rectangular facilities in an existing layout

In a companion paper (Date, K., and R. Nagi. [(2023) Optimal Placement of Multiple Finite-size Rectangular Facilities in an Existing Layout. International Journal of Production Research, THIS ISSUE]) we investigated a new problem of optimal placement of multiple finite-size rectangular facilities with known dimensions in the presence of existing rectangular facilities. We introduced the requisite theory to arrive at a solution by dividing the feasible region into sub-regions whose boundaries provide the candidates for the optimal placement. In this paper, we focus on developing optimal and heuristic solution methods for solving problem instances with various facility counts. To solve small instances with fewer facilities, we develop two optimal methods: (1) efficient explicit enumeration based on a tree state-space representation and (2) an implicit enumeration scheme that uses a flow-decomposition-based lower bound to reduce the search space and time. To solve industrial-sized instances, we have to resort to a family of construction and improvement heuristics given that the problem is NP-hard. The heuristic procedures perform well for non-pathological cases with an acceptable optimality gap. The main contribution of this paper is a range of optimal to heuristic methods that can be applied by the practitioner depending on their problem characteristics and desired efficiency.

UPPER BOUND FOR THE COMPETITIVE FACILITY LOCATION PROBLEM WITH DEMAND UNCERTAINTY

We consider a competitive facility location problem with two competing parties operating in a situation of uncertain demand scenario. The problem to find the best solutions for the parties is formulated as a discrete bi-level mathematical programming problem. In the paper, we suggest a procedure to compute an upper bound for the objective function on subsets. The procedure could be employed in implicit enumeration schemes capable to compute an optimal solution for the problem under study. Within the procedure, additional constraints iteratively augment the high-point relaxation of the initial bi-level problem, what strengthens the relaxation and improves the upper bound’s quality. New procedure to generate such cuts allows to construct the strongest cuts without enumerating the parameters encoding them.

Discovering optimal fermion-qubit mappings through algorithmic enumeration

Simulating fermionic systems on a quantum computer requires a high-performing mapping of fermionic states to qubits. A characteristic of an efficient mapping is its ability to translate local fermionic interactions into local qubit interactions, leading to easy-to-simulate qubit Hamiltonians.All fermion-qubit mappings must use a numbering scheme for the fermionic modes in order for translation to qubit operations. We make a distinction between the unordered labelling of fermions and the ordered labelling of the qubits. This separation shines light on a new way to design fermion-qubit mappings by making use of the enumeration scheme for the fermionic modes. The purpose of this paper is to demonstrate that this concept permits notions of fermion-qubit mappings that are optimal with regard to any cost function one might choose. Our main example is the minimisation of the average number of Pauli matrices in the Jordan-Wigner transformations of Hamiltonians for fermions interacting in square lattice arrangements. In choosing the best ordering of fermionic modes for the Jordan-Wigner transformation, and unlike other popular modifications, our prescription does not cost additional resources such as ancilla qubits.We demonstrate how Mitchison and Durbin's enumeration pattern minimises the average Pauli weight of Jordan-Wigner transformations of systems interacting in square lattices. This leads to qubit Hamiltonians consisting of terms with average Pauli weights 13.9% shorter than previously known. By adding only two ancilla qubits we introduce a new class of fermion-qubit mappings, and reduce the average Pauli weight of Hamiltonian terms by 37.9% compared to previous methods. For n-mode fermionic systems in cellular arrangements, we find enumeration patterns which result in n1/4 improvement in average Pauli weight over naïve schemes.

Compression with wildcards: All exact or all minimal hitting sets

Abstract Our objective is the compressed enumeration (based on wildcards) of all minimal hitting sets of general hypergraphs. To the author’s best knowledge, the only previous attempt towards compression, due to Toda, is based on binary decision diagrams and much different from our techniques. Traditional one-by-one enumeration schemes cannot compete when the number of minimal hitting sets is large and the degree of compression is high. Our method works particularly well in these two cases: either compressing all minimum cardinality hitting sets, or compressing all exact hitting sets.

Multitarget Enumeration and Localization in Distributed MIMO Radar Based on Energy Modeling and Compressive Sensing

Target localization is one of the most important research topics in the field of radar signal processing. In this paper, the problem of multitarget enumeration and localization in the distributed multiple-input multiple-output (MIMO) radar with noncoherent processing mode is investigated. We first analyze the theoretical bound of the multitarget localization accuracy under the discrete time signal model and the Swerling 1 target model. It is determined by the Cram'r-Rao lower bound (CRLB) at low signal-to-noise ratio (SNR) and the sampling lower bound (SLB) when the SNR is high. Furthermore, an innovative multitarget enumeration and localization scheme is developed, which is based on the energy modeling of the multiple transmitter-receiver paths and the compressive sensing theory. To solve the sparse vector recovery issue, we design a lightweight iterative greedy pursuit algorithm including the similarity evaluation strategy. In addition, an iterative-based target position refinement process is designed to alleviate the off-grid problem caused by the spatial discretization. The proposal utilizes the samples of the raw signals and belongs to the category of the direct localization. Nevertheless, it has significantly higher computational efficiency and lower data communication burden than the conventional direct localization methods, while avoiding the complex data association encountered by the indirect localization methods. Finally, the simulation results validate the effectiveness and robustness of the proposed method.

Distributions of runs and scans in multistate Markov exchangeable sequences

In this paper, we look at the distributions of runs and scans in multistate Markov exchangeable sequences. The joint distributions of runs of several lengths under four types of enumeration schemes are analyzed. We evaluate the upper tail probabilities for ratchet scan statistics exactly. By utilizing the expansion of the generating functions, we propose effective computational tools for the derivation of probability functions. The results presented here provide approaches for the evaluation of the exact distributions of runs and scans in a wide class of practical problems. Finally, we discuss several applications and numerical examples to show how our theoretical results are applied to the investigation of runs and scans, as well as a parameter estimation problem.

Implications of Including Tumor Presence Probabilities as Weights in Radiotherapy Optimization

<h3>Purpose/Objective(s)</h3> Recently, the fixed nature of the clinical target volume has been challenged in several studies that choose to incorporate microscopic tumor infiltration as a source of uncertainty in radiation therapy treatment planning. The approach to handle this uncertainty typically involves voxel-wise objective function weights that are proportional to each voxel's marginal probability of tumor presence. The main argument for doing so is to balance the probability of target coverage against the increased dose to organs at risk (OARs). We study the implications of basing the optimization on such probability weights, given an isotropic tumor infiltration model used for both optimization and evaluation, in which the infiltration distance is taken from a truncated gaussian probability distribution. We hypothesize that the low voxel weights far from the gross tumor volume (GTV) will cause the dose to fall off unevenly between directions that have high and low conflict with OARs, leading to a suboptimal trade-off between the probability of target coverage and OAR sparing. <h3>Materials/Methods</h3> We implement the probability weight method on a phantom geometry where suspected target voxels are weighted with probabilities from the tumor infiltration model, decreasing away from the GTV. This probability weighted plan is then compared against plans that are optimized using target volumes from an enumeration over a discrete set of scenarios from the tumor infiltration model. For both methods, plans are optimized using a range of scaling factors which scale the (uniform dose) objectives on the target in relation to those on OARs (max EUD), allowing comparison of the approximate Pareto fronts of the plans in the enumeration scheme to that of the probability weighted plan. The plans are evaluated on target samples from the tumor infiltration model which are used to compute target dose statistics. For each plan, the probability of satisfying a target coverage criterion (D98 > 0.95*D_pres.) is then plotted against the OAR objective to evaluate whether the probability weight method can produce Pareto optimal plans with respect to these criteria. In evaluation, we also include a tumor control probability (TCP) model to investigate how the methods compare on a smooth target coverage criterion. <h3>Results</h3> For a given scaling factor, the probability weighted plan is usually close to the corresponding set of plans from the enumeration scheme. However, the non-dominated subset of all generated plans consists almost entirely of plans from the enumeration scheme. Although the result holds also for TCP, there is less difference between the non-dominated subset of plans and the set of probability weighted plans. <h3>Conclusion</h3> The results indicate that finding the right combination of scaling factor and target expansion distance gives a better trade-off than the probability weight method, although efficiently doing so remains a future challenge. Meanwhile, using probability weights may serve as a practical alternative.

Globally optimal design of Minimal WHEN systems using enumeration

AbstractThe process synthesis problem referred to as work and heat exchange networks (WHENs) asks to determine the optimal heat exchanger network intertwined with compressors, expanders, and valves, all integrated into one single network, such that streams with certain initial temperature and pressure attain their temperature and pressure targets at minimum cost. This article presents a procedure that obtains the globally optimal solution of Minimal WHEN systems using an enumeration scheme, using only a few subproblems, each one solved using mathematical programming to global optimality. Minimal structures feature one compression/decompression task and one heat exchange task per stream. In addition, we depart from the ideal gas assumption and use a cubic equation of state for stream properties. Finally, the approach allows for the use of turboexpanders. Four examples are presented to show the effectiveness and accuracy of the proposed method.

CALDERA: finding all significant de Bruijn subgraphs for bacterial GWAS.

MotivationGenome-wide association studies (GWAS), aiming to find genetic variants associated with a trait, have widely been used on bacteria to identify genetic determinants of drug resistance or hypervirulence. Recent bacterial GWAS methods usually rely on k-mers, whose presence in a genome can denote variants ranging from single-nucleotide polymorphisms to mobile genetic elements. This approach does not require a reference genome, making it easier to account for accessory genes. However, a same gene can exist in slightly different versions across different strains, leading to diluted effects.ResultsHere, we overcome this issue by testing covariates built from closed connected subgraphs (CCSs) of the de Bruijn graph defined over genomic k-mers. These covariates capture polymorphic genes as a single entity, improving k-mer-based GWAS both in terms of power and interpretability. However, a method naively testing all possible subgraphs would be powerless due to multiple testing corrections, and the mere exploration of these subgraphs would quickly become computationally intractable. The concept of testable hypothesis has successfully been used to address both problems in similar contexts. We leverage this concept to test all CCSs by proposing a novel enumeration scheme for these objects which fully exploits the pruning opportunity offered by testability, resulting in drastic improvements in computational efficiency. Our method integrates with existing visual tools to facilitate interpretation.Availability and implementationWe provide an implementation of our method, as well as code to reproduce all results at https://github.com/HectorRDB/Caldera_ISMB.Supplementary information Supplementary data are available at Bioinformatics online.

A constructive branch-and-bound algorithm for the project duration problem with partially renewable resources and general temporal constraints

This paper deals with the resource-constrained project scheduling problem with partially renewable resources and general temporal constraints with the objective to minimize the project duration. The consideration of partially renewable resources allows to integrate the decision about the availability of a resource for a specific time period into the scheduling process. Together with general temporal constraints, which permit to establish minimum and maximum time lags between activities, even more aspects of real-life projects can be taken into account. We present a branch-and-bound algorithm for the stated problem that is based on a serial schedule-generation scheme. For the first time it is shown how a dominance criterion can be applied on the associated generation scheme to reduce the start times in each scheduling step. To improve the performance of the solution procedure, we integrate consistency tests and lower bounds from the literature and devise new pruning techniques. In a comprehensive experimental performance analysis we compare our exact solution procedure with all available branch-and-bound algorithms for partially renewable resources. Additionally, we investigate a directly derived priority rule-based approximation method from our new enumeration scheme. The results of the computational study demonstrate the efficiency of our branch-and-bound algorithm and reveal that the derived approximation method is only suited to solve small- and medium-sized instances.

МЕТОД И АЛГОРИТМ ПЛАНИРОВАНИЯ ОПЕРАЦИЙ НА ОСНОВЕ МОДЕЛИ НЕЧЕТКОГО КОНЕЧНОГО АВТОМАТА

In this paper the planning and scheduling problem as an important optimization problem inmany transportation and robotic applications is discussed. To solve planning problems, the mainapproaches are based on optimization methods, sampling-based methods, and usually such kindsof problems are NP-hard and high dimensional. In this work, the method for planning and schedulingbased on the fuzzy finite state machine model is developed. Fuzzy graph presentation of thescheduling problem and operation planning is given. The paper presents two approaches to theformulation of the planning problem with limited resources and temporal variables: state-oriented(with transitions between states), temporal ordering-oriented (on a time scale). Temporal modelingfor planning problems implies a qualitative approach to managing the distribution of operationsor topological ordering, as well as a quantitative approach to handling imprecise durationsrelationships between operations in multiple parameters. The concepts of fuzzy intervals and fuzzyrelations are introduced for planning operations on a graph. A planning algorithm based on thetheory of automata and temporal modeling under uncertainty has been developed. Using this formalism,a path planning problem is solved by successively altering a state using various operationsuntil a solution is found. The idea of temporal-ordered partial schedule associated with theplanning state of the system is discussed. A model of a finite automaton for a planning system underconditions of uncertainty is proposed. A method and algorithm for scheduling operationsbased on a non-deterministic finite automaton and an enumeration scheme have been developed.The non-deterministic computation for a scheduling problem is a decision tree whose root correspondsto the beginning of the scheduling process, and each branch point in the tree correspondsto a computation point at which the machine has multiple choices. And the finite state machinemodel (automata) for the planning system under uncertainty is suggested.

A Catalog of Formulations for the Network Pricing Problem

We study the network pricing problem where the leader maximizes revenue by determining the optimal amounts of tolls to charge on a set of arcs, under the assumption that the followers will react rationally and choose the shortest paths to travel. Many distinct single-level reformulations of this bilevel optimization program have been proposed; however, their relationship has not been established. In this paper, we aim to build a connection between those reformulations and explore the combination of the path representation with various modeling options, allowing us to generate 12 different reformulations of the problem. Moreover, we propose a new path enumeration scheme, path-based preprocessing, and hybrid framework to further improve performance and robustness when solving the final model. We provide numerical results, comparing all the derived reformulations and confirming the efficiency of the novel dimensionality reduction procedures.

Modular enumeration - a new method for solving discrete programming problems

A new approach resulting in a new group of algorithms for finding globally optimal solutions to discrete programming problems with Boolean variables is proposed. It is proved that the effectiveness of this method depends only on the number of variables of the problem to be solved and of the amount of RAM of used computer. The main difference from the existing methods of implicit enumeration is in the ability for some of the modular enumeration procedures to a priori predict, solving any extreme problem with Boolean variables, the number of iterations, the gain in running time in comparison with the traditional enumeration scheme, as well as the amount of RAM used. The experimental results confirm high efficiency of the proposed approaches.

A task to connect counting processes to lists of outcomes in combinatorics

Research has shown that solving counting problems correctly can be difficult for students at all levels, and mathematics educators have sought to identify strategies and interventions to help students reason conceptually about combinatorial tasks. A set-oriented perspective (Lockwood, 2014) is a way of thinking about counting problems that emphasizes the importance of reasoning about the set of outcomes being counted. From a set-oriented perspective, one possible type of intervention is to have students focus on the sets of outcomes rather than formulas and expressions, and specifically to reason about the structure of the set of outcomes. Yet, reasoning about sets of outcomes is not sufficient for students to make connections between outcomes and counting processes. In this paper, we investigate tasks where students wrote computer code to enumerate the set of outcomes in a specific order by implementing listing processes, and they were then asked to determine a specific numbered outcome in their list by using the structure of their enumeration scheme. We clarify particular aspects of a set-oriented perspective that were productive for students, and we demonstrate that tasks that asked students to name a specific outcome in their list elicited meaningful connections between counting processes and sets of outcomes. Further, such tasks reinforce desirable mathematical practices such as leveraging structure and connecting representations.

The Referenced Vertex Ordering Problem: Theory, Applications, and Solution Methods

We introduce the referenced vertex ordering problem (REVORDER) as a combinatorial decision problem generalizing several vertex ordering problems that already appeared in the scientific literature under different guises. In other words, REVORDER is a generic problem with several possible extensions corresponding to various real-life applications. Previous works show that REVORDER has a polynomial complexity, whereas its optimization version, denoted in our work as MIN REVORDER, is NP-hard. We give a survey of methods and algorithms that can be applied to the solution of MIN REVORDER, and we develop a new enumeration scheme for its solution. Our theoretical analysis of this scheme yields several pruning techniques aimed at the reduction of the number of enumeration nodes. We then discuss how upper and lower bounds can be computed during the enumeration to design a branch-and-bound algorithm. Finally, we validate the branch-and-bound by conducting a large set of computational experiments on instances coming from various real-life applications. Our results highlight that the newly introduced pruning techniques allow for major reductions of computational cost, with a constant solution quality. In result, our branch-and-bound outperforms other existing solution methods: among 180 instances with 60 vertices, it solves 179 instances to optimality whereas the best existing method is only able to solve 109 of them. Moreover, our tests show that our algorithm can solve medium-scale instances including up to 500 vertices, which opens the perspective of handling new real-life problems. Our implementation of the branch-and-bound algorithm, together with all instances we have used, is publicly available on GitLab.

Probiotic Lactobacilli in Fermented Dairy Products: Selective Detection, Enumeration and Identification Scheme.

A selection of 36 commercial probiotic fermented dairy products from UK and Europe markets were evaluated for the numbers, types, and viability of Lactobacillus strains against the stated information on their packages. A comparative study was carried out on selectivity of MRS-Clindamycin, MRS-Sorbitol, and MRS-IM Maltose, to select the right medium for enumeration of probiotic Lactobacillus. Based on selectivity of medium for recovery of the targeted lactobacilli, and also simplicity of preparation, MRS-Clindamycin was chosen as the best medium for enumeration of probiotic Lactobacillus in fermented milks. The results of enumeration of lactobacilli showed that 22 out of a total 36 tested products contained more than 106 colony-forming units/g at the end of their shelf life, which comply with the recommended minimum therapeutic level for probiotics. Rep-PCR using primer GTG-5 was applied for initial discrimination of isolated strains, and isolates, which presented different band profile, were placed in different groups. The isolated Lactobacillus spp. were identified mainly as Lactobacillus acidophilus, Lactobacillus casei, and Lactobacillus paracasei by analysis of partial sequences of the 16S ribosomal RNA and rpoA genes.

Directed Projection Graph of N-Dimensional Hypercube and Subhypercube Decomposition of Balanced Linearly Separable Boolean Functions

A directed projection graph of the [Formula: see text]-dimensional hypercube on the two-dimensional plane is successfully created. Any [Formula: see text]-variable Boolean function can be easily transformed to an induced subgraph of the projection. Therefore, the discussions on [Formula: see text]-variable Boolean functions only need to focus on a two-dimensional planar graph. Some mathematical theories on the projection graph and the induced subgraph are established, and some properties and characteristics of a balanced linearly separable Boolean function (BLSBF) are uncovered. In particular, the sub-hypercube decompositions of BLSBF is easily represented on the projection, and meanwhile, the enumeration scheme for counting the number of [Formula: see text]-variable BLSBFs is developed by using equivalence classification and conformal transformation. With the aid of the directed projection grap constructed in this paper, one can further study many difficult problems in some fields such as Boolean functions and artificial neural networks.