Constraint Programming Research Articles

Mathematical Models for the Single-Channel and Multi-Channel PMU Allocation Problem and Their Solution Algorithms

Phasor measurement units (PMUs) are deployed at power grid nodes around the transmission grid, determining precise power system monitoring conditions. In real life, it is not realistic to place a PMU at every power grid node; thus, the lowest PMU number is optimally selected for the full observation of the entire network. In this study, the PMU placement model is reconsidered, taking into account single- and multi-capacity placement models rather than the well-studied PMU placement model with an unrestricted number of channels. A restricted number of channels per monitoring device is used, instead of supposing that a PMU is able to observe all incident buses through the transmission connectivity lines. The optimization models are declared closely to the power dominating set and minimum edge cover problem in graph theory. These discrete optimization problems are directly related with the minimum set covering problem. Initially, the allocation model is declared as a constrained mixed-integer linear program implemented by mathematical and stochastic algorithms. Then, the integer linear problem is reformulated into a non-convex constraint program to find optimality. The mathematical models are solved either in binary form or in the continuous domain using specialized optimization libraries, and are all implemented in YALMIP software in conjunction with MATLAB. Mixed-integer linear solvers, nonlinear programming solvers, and heuristic algorithms are utilized in the aforementioned software packages to locate the global solution for each instance solved in this application, which considers the transformation of the existing power grids to smart grids.

A Constraint Solving Approach to Parikh Images of Regular Languages

A common problem in string constraint solvers is computing the Parikh image, a linear arithmetic formula that describes all possible combinations of character counts in strings of a given language. Automata-based string solvers frequently need to compute the Parikh image of products (or intersections) of finite-state automata, in particular when solving string constraints that also include the integer data-type due to operations like string length and indexing. In this context, the computation of Parikh images often turns out to be both prohibitively slow and memory-intensive. This paper contributes a new understanding of how the reasoning about Parikh images can be cast as a constraint solving problem, and questions about Parikh images be answered without explicitly computing the product automaton or the exact Parikh image. The paper shows how this formulation can be efficiently implemented as a calculus, PC*, embedded in an automated theorem prover supporting Presburger logic. The resulting standalone tool Catra is evaluate on constraints produced by the Ostrich+ string solver when solving standard string constraint benchmarks involving integer operations. The experiments show that PC* strictly outperforms the standard approach by Verma et al. to extract Parikh images from finite-state automata, as well as the over-approximating method recently described by Janků and Turoňová by a wide margin, and for realistic timeouts (under 60 s) also the nuXmv model checker. When added as the Parikh image backend of Ostrich+ to the Ostrich string constraint solver’s portfolio, it boosts its results on the quantifier-free strings with linear integer algebra track of SMT-COMP 2023 (QF_SLIA) enough to solve the most Unsat instances in that track of all competitors.

Short-term underground mine planning with uncertain activity durations using constraint programming

Short-term underground mine planning with uncertain activity durations using constraint programming

On Classifying Continuous Constraint Satisfaction Problems

A continuous constraint satisfaction problem (CCSP) is a constraint satisfaction problem (CSP) with an interval domain $U \subset \mathbb{R}$. We engage in a systematic study to classify CCSPs that are complete of the Existential Theory of the Reals, i.e., ER-complete. To define this class, we first consider the problem ETR, which also stands for Existential Theory of the Reals. In an instance of this problem we are given some sentence of the form $\exists x_1, \ldots, x_n \in \mathbb{R} : \Phi(x_1, \ldots, x_n)$, where $\Phi$ is a well-formed quantifier-free formula consisting of the symbols $\{0, 1, +, \cdot, \geq, >, \wedge, \vee, \neg\}$, the goal is to check whether this sentence is true. Now the class ER is the family of all problems that admit a polynomial-time many-one reduction to ETR. It is known that NP $\subseteq$ ER $\subseteq$ PSPACE. We restrict our attention on CCSPs with addition constraints ($x + y = z$) and some other mild technical conditions. Previously, it was shown that multiplication constraints ($x \cdot y = z$), squaring constraints ($x^2 = y$), or inversion constraints ($x\cdot y = 1$) are sufficient to establish ER-completeness. We extend this in the strongest possible sense for equality constraints as follows. We show that CCSPs (with addition constraints and some other mild technical conditions) that have any one well-behaved curved equality constraint ($f(x,y) = 0$) are ER-complete. We further extend our results to inequality constraints. We show that any well-behaved convexly curved and any well-behaved concavely curved inequality constraint ($f(x,y) \geq 0$ and $g(x,y) \geq 0$) imply ER-completeness on the class of such CCSPs.

Sketching Approximability of All Finite CSPs

A constraint satisfaction problem (CSP), \(\textsf {Max-CSP}(\mathcal {F})\) , is specified by a finite set of constraints \(\mathcal {F}\subseteq \lbrace [q]^k \rightarrow \lbrace 0,1\rbrace \rbrace\) for positive integers q and k . An instance of the problem on n variables is given by m applications of constraints from \(\mathcal {F}\) to subsequences of the n variables, and the goal is to find an assignment to the variables that satisfies the maximum number of constraints. In the (γ ,β)-approximation version of the problem for parameters 0 ≤ β ≤ γ ≤ 1, the goal is to distinguish instances where at least γ fraction of the constraints can be satisfied from instances where at most β fraction of the constraints can be satisfied. In this work, we consider the approximability of this problem in the context of sketching algorithms and give a dichotomy result. Specifically, for every family \(\mathcal {F}\) and every β < γ, we show that either a linear sketching algorithm solves the problem in polylogarithmic space or the problem is not solvable by any sketching algorithm in \(o(\sqrt {n})\) space. In particular, we give non-trivial approximation algorithms using polylogarithmic space for infinitely many constraint satisfaction problems. We also extend previously known lower bounds for general streaming algorithms to a wide variety of problems, and in particular the case of q = k =2, where we get a dichotomy, and the case when the satisfying assignments of the constraints of \(\mathcal {F}\) support a distribution on \([q]^k\) with uniform marginals. Prior to this work, other than sporadic examples, the only systematic classes of CSPs that were analyzed considered the setting of Boolean variables q = 2, binary constraints k =2, and singleton families \(|\mathcal {F}|=1\) and only considered the setting where constraints are placed on literals rather than variables. Our positive results show wide applicability of bias-based algorithms used previously by [ 47 ] and [ 41 ], which we extend to include richer norm estimation algorithms, by giving a systematic way to discover biases. Our negative results combine the Fourier analytic methods of [ 56 ], which we extend to a wider class of CSPs, with a rich collection of reductions among communication complexity problems that lie at the heart of the negative results. In particular, previous works used Fourier analysis over the Boolean cube to initiate their results and the results seemed particularly tailored to functions on Boolean literals (i.e., with negations). Our techniques surprisingly allow us to get to general q -ary CSPs without negations by appealing to the same Fourier analytic starting point over Boolean hypercubes.

Learning to Branch: Generalization Guarantees and Limits of Data-Independent Discretization

Tree search algorithms, such as branch-and-bound, are the most widely used tools for solving combinatorial and non-convex problems. For example, they are the foremost method for solving (mixed) integer programs and constraint satisfaction problems. Tree search algorithms come with a variety of tunable parameters that are notoriously challenging to tune by hand. A growing body of research has demonstrated the power of using a data-driven approach to automatically optimize the parameters of tree search algorithms. These techniques use a training set of integer programs sampled from an application-specific instance distribution to find a parameter setting that has strong average performance over the training set. However, with too few samples, a parameter setting may have strong average performance on the training set but poor expected performance on future integer programs from the same application. Our main contribution is to provide the first sample complexity guarantees for tree search parameter tuning. These guarantees bound the number of samples sufficient to ensure that the average performance of tree search over the samples nearly matches its future expected performance on the unknown instance distribution. In particular, the parameters we analyze weight scoring rules used for variable selection. Proving these guarantees is challenging because tree size is a volatile function of these parameters: we prove that, for any discretization (uniform or not) of the parameter space, there exists a distribution over integer programs such that every parameter setting in the discretization results in a tree with exponential expected size, yet there exist parameter settings between the discretized points that result in trees of constant size. In addition, we provide data-dependent guarantees that depend on the volatility of these tree-size functions: our guarantees improve if the tree-size functions can be well approximated by simpler functions. Finally, via experiments, we illustrate that learning an optimal weighting of scoring rules reduces tree size.

Architectural design space exploration of complex engineered systems with management constraints and preferences

During the architectural design phase, the decisions made have a crucial impact on developing complex systems. These decisions often limit the performance bounds and steer subsequent research directions. However, the multitude of architectural components and intricate interrelationships cause component combination explosion. Additionally, the diverse requirements of different stakeholders add complexity to determining the weight of indicators, posing challenges to complex system architecture generation and trade-off. This paper proposes a method for exploring the architectural design space in complex systems realisation, which aims to resolve issues related to the formal representation of architectures, selection within the architectural design space, and trade-offs. Firstly, the architectural design space is represented by formulating and mathematically describing a morphological matrix. Then, the construction of the Constraint Satisfaction Problem model aims to achieve a reduction in the scale of the design space. Moreover, through the evolutionary algorithm, the Pareto front is identified. Weight sensitivity analysis and the improved Technique for Order Preference by Similarity to the Ideal Solution method are combined to assist designers in finding a satisfactory architecture from the design space with many combinations. Finally, the proposed method is verified by the design of the launch vehicle’s primary and secondary separation system.

The Physical Cell Identity Assignment Problem: A Practical Optimization Approach

The Physical Cell Identity (PCI) is a critical parameter to radio access networks used by the user equipment, like a smartphone, to identify the cell to connect. The PCI misassignment leads to call drops, no voice/data connections, and other sorts of impairments. Many works in the literature only treat the PCI assignment problem from the technological constraints viewpoint. In this work, we take a realistic approach and incorporate several operational constraints found daily in carrier-grade networks. For that, we model the problem with multiple objectives with a strict lexicographical dominance relation. We solve it using a hybrid memetic algorithm with warm start solutions obtained by a constraint programming solver and intensification with path-relinking and custom local search procedures. The experimental results suggest that our approach is superior to commercial solver packages in several scenarios.

Solving the three-dimensional open-dimension rectangular packing problem: A constraint programming model

This paper addresses the three-dimensional open-dimension rectangular packing problem (3D-ODRPP). This problem addresses a set of rectangular boxes of given dimensions and a rectangular container of open dimensions. The objective is to pack all boxes orthogonally into the container while minimizing the container volume. Real-world applications of the 3D-ODRPP arise in production systems with operations of shipping or moving. The literature has presented mainly mixed-integer programming (MIP) formulations and their linearization techniques for the problem allied with general-purpose optimization solvers. To model and solve the 3D-ODRPP, we propose a constraint programming model based on a position-free modeling approach with logic operators. We ran computational experiments to assess the performance of the proposed model compared to the benchmark MIP models from instances of the literature. The results show our approach is competitive in different sets of problem instances in terms of reaching optimality as well as providing satisfactory feasible solutions quickly.

Computing Bipath Multicommodity Flows with Constraint Programming–Based Branch-and-Price-and-Cut

We propose a constraint programming (CP)–based branch-and-price-and-cut framework to exactly solve bipath multicommodity flow (MCF): an MCF problem with two paths for each demand. The goal is to route demands in a capacitated network under the minimum cost. The two paths must have disjoint arcs, and the delays accumulated along the two paths must be within a small deviation of each other. CP is used at multiple points in this framework: for solving pricing problems, for cut generation, and for primal and branching node heuristics. These modules use a CP solver designed for network routing problems and can be adapted to other combinatorial optimization problems. We also develop a novel, complete, two-level branching scheme. On a set of diverse bipath MCF instances, experimental results show that our algorithm significantly outperforms monolithic CP and mixed integer linear programming models and demonstrate the efficiency and flexibility brought by the tailored integration of linear programming and CP methodologies. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This work was supported by the Natural Sciences and Engineering Research Council of Canada; Huawei Technologies. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0128 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0128 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

The container premarshalling problem under limited crane time: A constraint programming approach

This study addresses the premarshalling problem when the crane time is limited and a complete rearrangement of the bay is not possible. This issue has been neglected in the literature, but it is very common in practice. We show that the use of standard approaches often fails to yield good solutions and develop a model for the problem in two steps. First, a constraint programming model is proposed for premarshalling with crane time minimization objective. Then, this model is adapted to the new problem. An extensive computational study shows that the first model improves on the performance of the existing state-of-the-art integer programming model and that the model of the new problem obtains high quality solutions even in short running times.

Enhancing constraint programming via supervised learning for job shop scheduling

Constraint programming (CP) is a powerful technique for solving constraint satisfaction and optimization problems. In CP solvers, the variable ordering strategy used to select which variable to explore first in the solving process has a significant impact on solver effectiveness. To address this issue, we propose a novel variable ordering strategy based on supervised learning, which we evaluate in the context of job shop scheduling problems. Our learning-based methods predict the optimal solution of a problem instance and use the predicted solution to order variables for CP solvers. Unlike traditional variable ordering methods, our methods can learn from the characteristics of each problem instance and customize the variable ordering strategy accordingly, leading to improved solver performance. Our experiments demonstrate that training machine learning models is highly efficient and can achieve high accuracy. Furthermore, our learned variable ordering methods perform competitively compared to four existing methods. Finally, we showcase the benefits of integrating machine learning-based variable ordering methods with conventional domain-based approaches through tie-breaking.

What Are the Rules? Discovering Constraints from Data

Constraint programming and AI planning are powerful tools for solving assignment, optimization, and scheduling problems. They require, however, the rarely available combination of domain knowledge and mathematical modeling expertise. Learning constraints from exemplary solutions can close this gap and alleviate the effort of modeling. Existing approaches either require extensive user interaction, need exemplary invalid solutions that must be generated by experts at great expense, or show high noise-sensitivity. We aim to find constraints from potentially noisy solutions, without the need of user interaction. To this end, we formalize the problem in terms of the Minimum Description Length (MDL) principle, by which we select the model with the best lossless compression of the data. Solving the problem involves model counting, which is #P-hard to approximate. We therefore propose the greedy URPILS algorithm to find high-quality constraints in practice. Extensive experiments on constraint programming and AI planning benchmark data show URPILS not only finds more accurate and succinct constraints, but also is more robust to noise, and has lower sample complexity than the state of the art.

DeepSaDe: Learning Neural Networks That Guarantee Domain Constraint Satisfaction

As machine learning models, specifically neural networks, are becoming increasingly popular, there are concerns regarding their trustworthiness, specially in safety-critical applications, e.g. actions of an autonomous vehicle must be safe. There are approaches that can train neural networks where such domain requirements are enforced as constraints, but they either cannot guarantee that the constraint will be satisfied by all possible predictions (even on unseen data) or they are limited in the type of constraints that can be enforced. In this paper, we present an approach to train neural networks which can enforce a wide variety of constraints and guarantee that the constraint is satisfied by all possible predictions. The approach builds on earlier work where learning linear models is formulated as a constraint satisfaction problem (CSP). To make this idea applicable to neural networks, two crucial new elements are added: constraint propagation over the network layers, and weight updates based on a mix of gradient descent and CSP solving. Evaluation on various machine learning tasks demonstrates that our approach is flexible enough to enforce a wide variety of domain constraints and is able to guarantee them in neural networks.

Conversational Modeling for Constraint Satisfaction

Many problems, from Sudoku to factory scheduling, can be regarded as constraint satisfaction problems. A key component of real world problem solving is a conversation between a constraint programming expert and a problem domain expert to specify the problem to be solved. This presentation argues that the time is ripe for progress in automating the constraint programmer side of this conversation and suggests promising avenues for this pursuit.

Learning to Learn in Interactive Constraint Acquisition

Constraint Programming (CP) has been successfully used to model and solve complex combinatorial problems. However, modeling is often not trivial and requires expertise, which is a bottleneck to wider adoption. In Constraint Acquisition (CA), the goal is to assist the user by automatically learning the model. In (inter)active CA, this is done by interactively posting queries to the user, e.g. does this partial solution satisfy your (unspecified) constraints or not. While interactive CA methods learn the constraints, the learning is related to symbolic concept learning, as the goal is to learn an exact representation. However, a large number of queries is required to learn the model, which is a major limitation. In this paper, we aim to alleviate this limitation by tightening the connection of CA and Machine Learning (ML), by, for the first time in interactive CA, exploiting statistical ML methods. We propose to use probabilistic classification models to guide interactive CA queries to the most promising parts. We discuss how to train classifiers to predict whether a candidate expression from the bias is a constraint of the problem or not, using both relation-based and scope-based features. We then show how the predictions can be used in all layers of interactive CA: the query generation, the scope finding, and the lowest-level constraint finding. We experimentally evaluate our proposed methods using different classifiers and show that our methods greatly outperform the state of the art, decreasing the number of queries needed to converge by up to 72%.

Parallel Beam Search Algorithms for Domain-Independent Dynamic Programming

Domain-independent dynamic programming (DIDP), a model-based paradigm based on dynamic programming, has shown promising performance on multiple combinatorial optimization problems compared with mixed integer programming (MIP) and constraint programming (CP). The current DIDP solvers are based on heuristic search, and the state-of-the-art solver, complete anytime beam search (CABS), uses beam search. However, the current DIDP solvers cannot utilize multiple threads, unlike state-of-the-art MIP and CP solvers. In this paper, we propose three parallel beam search algorithms and develop multi-thread implementations of CABS. With 32 threads, our multi-thread DIDP solvers achieve 9 to 39 times speedup on average and significant performance improvement over the sequential solver, finding the new best solutions for two instances of the traveling salesperson problem with time windows. In addition, our solvers outperform multi-thread MIP and CP solvers in four of the six combinatorial optimization problems evaluated.

Check-In Desk Scheduling Optimisation at CDG International Airport

More than ever, air transport players (i.e., airline and airport companies) in an intensely competitive climate need to benefit from a carefully optimized management of airport resources to improve the quality of service and control the induced costs. In this paper, we investigate the Airport Check-in Desk Assignment Problem. We propose a Constraint Programming (CP) model for this problem, and present some promising experimental results from data coming from ADP (Aéroport de Paris). Our works are deployed in a preprod environment since 1 year.

Stable Matchings in Practice: A Constraint Programming Approach

We study a practical two-sided matching problem of allocating children to daycare centers, which has significant social implications. We are cooperating with several municipalities in Japan and our goal is to devise a reliable and trustworthy clearing algorithm to deal with the problem. In this paper, we describe the design of our new algorithm that minimizes the number of unmatched children while ensuring stability. We evaluate our algorithm using real-life data sets, and experimental results demonstrate that our algorithm surpasses the commercial software that currently dominates the market in terms of both the number of matched children and the number of blocking coalitions (measuring stability). Our findings have been reported to local governments, and some are considering adopting our proposed algorithm in the near future, instead of the existing solution. Moreover, our model and algorithm have broader applicability to other important matching markets, such as hospital-doctor matching with couples and school choice with siblings.

Advancing signal integrity and application analysis through data-flow mapping

Data-flow mapping is a crucial method in signal processing and optimization, managing data flow within systems. Its essential in signal compensation, particularly in telecommunications, audio processing, and biomedical signal processing. Four main algorithm categories underpin data-flow mapping: heuristics, meta-heuristics, Integer Linear Programming (ILP), and Constraint Satisfaction Problems (CSP). Heuristic and meta-heuristic methods like Genetic Algorithms (GA) and Ant Colony Optimization (ACO) provide approximate solutions, crucial for complex problems. ILP and Branch and Bound (B&B) methods offer precise solutions by exhaustive searches under constraints. CSP focuses on satisfying imposed conditions. These methodologies have practical applications, such as signal compensation in communication systems and improving medical imaging like MRI and ultrasound. Theyre also integrated with machine learning, quantum computing, and specialized hardware for 5G/6G communications and IoT. Real-time processing and noise reduction advancements enhance consumer audio and diverse sectors. In summary, data-flow mapping and its algorithms drive signal processing innovations across domains, with evolving technology integration ensuring their lasting importance.