Constraint Programming Solver Research Articles

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.

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/ .

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.

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.

On Solving the Set Orienteering Problem

In the Set Orienteering Problem, a single vehicle, leaving from and returning to a depot, has to serve some customers, each one associated with a given spacial location. Customers are grouped in clusters and a given prize is collected once a customer in a cluster is visited. The prize associated with a cluster can be collected at most once. Travel times among locations are provided, together with a maximum available mission time, which normally makes it impossible to visit all the clusters. The target is to design a route for the vehicle that maximizes the total prize collected within the given time limit. In this study, building on the recent literature, we present new preprocessing rules and a new constraint programming model for the problem. Thanks to the symmetry exploitation carried out by the constraint programming solver, new state-of-the-art results are established.

Automatic constraint programming solver selection method based on machine learning for the cable tree wiring problem

AbstractCable trees are primarily employed in industrial products to facilitate energy transfer and information exchange among various components. When utilizing machines for assembly, it is essential to convert the wiring plan into a sequence of cable insertion operations executed by the machine under various constraints. This poses a combinatorial optimization problem. In this domain, constraint programming (CP) solvers often exhibit outstanding performance by leveraging their robust problem‐modelling capabilities, excellent scalability, and precise solving capabilities. However, CP solvers may achieve various performances for different problem instances. Selecting the most suitable CP solver for each problem instance is crucial. This paper introduces an automatic selection algorithm for CP solvers to solve the cable tree wiring problem (CTW). Firstly, a scoring system is used to conduct an in‐depth analysis and compare four well‐known CP solvers: CPLEX, Chuffed, OR‐Tools, and Gurobi. The results indicate that OR‐Tools and CPLEX outperform other solvers in performance. Moreover, these two solvers exhibit complementary advantages in quickly finding optimal and feasible solutions within specified time limits. Therefore, CP and machine learning are ingeniously integrated, harnessing their complementary advantages. 4240 instances covering various scenarios are randomly generated to form the problem space. This method incorporates decision trees, random forests, K‐nearest neighbours, and naive Bayes, utilizing these four machine learning techniques. The proposed method can achieve better results than traditional single CP solvers. Among all the evaluated machining learning techniques, the automatic solver selection methods based on decision trees and random forests can achieve accuracy rates of 91.29% and 84.15%, respectively.

Fast and Robust Resource-Constrained Scheduling with Graph Neural Networks

Resource-Constrained Project Scheduling Problems (RCPSPs) are NP-complete, which makes it challenging to efficiently solve large instances and robustify solutions in the presence of uncertainty. To remedy this, we learn to efficiently mimic the solutions produced by Constraint Programming (CP) solver, using a Graph Neural Network (GNN) architecture designed to capture the structure of RCPSPs. Since the GNN solution may violate constraints, we ensure schedule feasibility at inference time by extracting the task ordering from the GNN schedule and post-processing it with the well-known Schedule Generation Scheme (SGS). We find that SIREN, the resulting algorithm, produces schedules that are of higher quality than those produced by the CP solver within the same computation time budget. The speed and solution quality of SIREN make it suitable as a component of an on-line scenario-based optimisation procedure for RCPSPs with stochastic durations. This leads to the SERENE system, which robustly selects, in real-time, the best next tasks to start in order to minimise the average makespan over the scenarios. Empirically, SERENE achieves better average makespan over different realisations of uncertainty than deterministic algorithms that continuously reschedule on the basis of either the worst, best or average task durations.

An End-to-End Reinforcement Learning Approach for Job-Shop Scheduling Problems Based on Constraint Programming

Constraint Programming (CP) is a declarative programming paradigm that allows for modeling and solving combinatorial optimization problems, such as the Job-Shop Scheduling Problem (JSSP). While CP solvers manage to find optimal or near-optimal solutions for small instances, they do not scale well to large ones, i.e., they require long computation times or yield low-quality solutions. Therefore, real-world scheduling applications often resort to fast, handcrafted, priority-based dispatching heuristics to find a good initial solution and then refine it using optimization methods. This paper proposes a novel end-to-end approach to solve scheduling problems by means of CP and Reinforcement Learning (RL). In contrast to previous RL methods, tailored for a given problem by including procedural simulation algorithms, complex feature engineering, or handcrafted reward functions, our neural-network architecture and training algorithm merely require a generic CP encoding of some scheduling problem along with a set of small instances. Our approach leverages existing CP solvers to train an agent learning a Priority Dispatching Rule (PDR) that generalizes well to large instances, even from separate datasets. We evaluate our method on seven JSSP datasets from the literature, showing its ability to find higher-quality solutions for very large instances than obtained by static PDRs and by a CP solver within the same time limit.

Domain-Independent Dynamic Programming: Generic State Space Search for Combinatorial Optimization

For combinatorial optimization problems, model-based approaches such as mixed-integer programming (MIP) and constraint programming (CP) aim to decouple modeling and solving a problem: the `holy grail' of declarative problem solving. We propose domain-independent dynamic programming (DIDP), a new model-based paradigm based on dynamic programming (DP). While DP is not new, it has typically been implemented as a problem-specific method. We propose Dynamic Programming Description Language (DyPDL), a formalism to define DP models, and develop Cost-Algebraic A* Solver for DyPDL (CAASDy), a generic solver for DyPDL using state space search. We formalize existing problem-specific DP and state space search methods for combinatorial optimization problems as DP models in DyPDL. Using CAASDy and commercial MIP and CP solvers, we experimentally compare the DP models with existing MIP and CP models, showing that, despite its nascent nature, CAASDy outperforms MIP and CP on a number of common problem classes.

Automated streamliner portfolios for constraint satisfaction problems

Constraint Programming (CP) is a powerful technique for solving large-scale combinatorial problems. Solving a problem proceeds in two distinct phases: modelling and solving. Effective modelling has a huge impact on the performance of the solving process. Even with the advance of modern automated modelling tools, search spaces involved can be so vast that problems can still be difficult to solve. To further constrain the model, a more aggressive step that can be taken is the addition of streamliner constraints, which are not guaranteed to be sound but are designed to focus effort on a highly restricted but promising portion of the search space. Previously, producing effective streamlined models was a manual, difficult and time-consuming task. This paper presents a completely automated process to the generation, search and selection of streamliner portfolios to produce a substantial reduction in search effort across a diverse range of problems. The results demonstrate a marked improvement in performance for both Chuffed, a CP solver with clause learning, and lingeling, a modern SAT solver.

Mixed-Integer Programming vs. Constraint Programming for Shop Scheduling Problems: New Results and Outlook

Constraint programming (CP) has been recently in the spotlight after new CP-based procedures have been incorporated into state-of-the-art solvers, most notably the CP Optimizer from IBM. Classical CP solvers were only capable of guaranteeing the optimality of a solution, but they could not provide bounds for the integer feasible solutions found if interrupted prematurely due to, say, time limits. New versions, however, provide bounds and optimality guarantees, effectively making CP a viable alternative to more traditional mixed-integer programming (MIP) models and solvers. We capitalize on these developments and conduct a computational evaluation of MIP and CP models on 12 select scheduling problems. 1 We carefully chose these 12 problems to represent a wide variety of scheduling problems that occur in different service and manufacturing settings. We also consider basic and well-studied simplified problems. These scheduling settings range from pure sequencing (e.g., flow shop and open shop) or joint assignment-sequencing (e.g., distributed flow shop and hybrid flow shop) to pure assignment (i.e., parallel machine) scheduling problems. We present MIP and CP models for each variant of these problems and evaluate their performance over 17 relevant and standard benchmarks that we identified in the literature. The computational campaign encompasses almost 6,623 experiments and evaluates the MIP and CP models along five dimensions of problem characteristics, objective function, decision variables, input parameters, and quality of bounds. We establish the areas in which each one of these models performs well and recognize their conceivable reasons. The obtained results indicate that CP sets new limits concerning the maximum problem size that can be solved using off-the-shelf exact techniques. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. 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.2023.1287 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2021.0326 ) at ( http://dx.doi.org/10.5281/zenodo.7541223 ).

Answer-Set Programming for Lexicographical Makespan Optimisation in Parallel Machine Scheduling

AbstractWe deal with a challenging scheduling problem on parallel machines with sequence-dependent setup times and release dates from a real-world application of semiconductor work-shop production. There, jobs can only be processed by dedicated machines, thus few machines can determine the makespan almost regardless of how jobs are scheduled on the remaining ones. This causes problems when machines fail and jobs need to be rescheduled. Instead of optimising only the makespan, we put the individual machine spans in non-ascending order and lexicographically minimise the resulting tuples. This achieves that all machines complete as early as possible and increases the robustness of the schedule. We study the application of answer-set programming (ASP) to solve this problem. While ASP eases modelling, the combination of timing constraints and the considered objective function challenges current solving technology. The former issue is addressed by using an extension of ASP by difference logic. For the latter, we devise different algorithms that use multi-shot solving. To tackle industrial-sized instances, we study different approximations and heuristics. Our experimental results show that ASP is indeed a promising knowledge representation and reasoning (KRR) paradigm for this problem and is competitive with state-of-the-art constraint programming (CP) and Mixed-Integer Programming (MIP) solvers.

A Hybrid Exact–Local Search Approach for One-Machine Scheduling with Time-Dependent Capacity

Machine scheduling is a hard combinatorial problem having many manifestations in real life. Due to the schedule followed, the possibility of installations of machines operating sub-optimally is high. In this work, we examine the problem of a single machine with time-dependent capacity that performs jobs of deterministic durations, while for each job, its due time is known in advance. The objective is to minimize the aggregated tardiness in all tasks. The problem was motivated by the need to schedule charging times of electric vehicles effectively. We formulate an integer programming model that clearly describes the problem and a constraint programming model capable of effectively solving it. Due to the usage of interval variables, global constraints, a powerful constraint programming solver, and a heuristic we have identified, which we call the “due times rule”, the constraint programming model can reach excellent solutions. Furthermore, we employ a hybrid approach that exploits three local search improvement procedures in a schema where the constraint programming part of the solver plays a central role. These improvement procedures exhaustively enumerate portions of the search space by exchanging consecutive jobs with a single job of the same duration, moving cost-incurring jobs to earlier times in a consecutive sequence of jobs or even exploiting periods where capacity is not fully utilized to rearrange jobs. On the other hand, subproblems are given to the exact constraint programming solver, allowing freedom of movement only to certain parts of the schedule, either in vertical ribbons of the time axis or in groups of consecutive sequences of jobs. Experiments on publicly available data show that our approach is highly competitive and achieves the new best results in many problem instances.

On the integration of diverging material flows into resource‐constrained project scheduling

This study deals with an extension of the resource-constrained project scheduling problem (RCPSP) by constraints on material flows released during the execution of project activities. These constraints arise from limited processing capacities for materials and maximum inventories of intermediate storage facilities. Production scheduling problems with converging material flows have been studied extensively. However, this is the first project scheduling problem integrating diverging material flows typically observed in dismantling projects, e.g., building deconstruction, power plant decommissioning, or battery/car decommissioning. Diverging material flows do not directly impact the project planning but only impose delays in the case of congestion. We model material flows by using operations that represent the processing of materials, and cumulative resources that represent storage facilities. As a method for efficiently generating starting solutions, we propose a schedule generation scheme tailored to the particular precedence structure of such problems. Furthermore, we extensively study the schedule generation scheme’s performance on generated test instances and compare it to the constraint programming solver IBM ILOG CP Optimizer. It turns out that the solution quality strongly depends on the employed model and that neither of the two solution methods is generally superior.

An algorithm selection approach for the flexible job shop scheduling problem: Choosing constraint programming solvers through machine learning

Constraint programming solvers are known to perform remarkably well for most scheduling problems. However, when comparing the performance of different available solvers, there is usually no clear winner over all relevant problem instances. This gives rise to the question of how to select a promising solver when knowing the concrete instance to be solved. In this article, we aim to provide first insights into this question for the flexible job shop scheduling problem. We investigate relative performance differences among five constraint programming solvers on problem instances taken from the literature as well as randomly generated problem instances. These solvers include commercial and non-commercial software and represent the state-of-the-art as identified in the relevant literature. We find that two solvers, the IBM ILOG CPLEX CP Optimizer and Google’s OR-Tools, outperform alternative solvers. These two solvers show complementary strengths regarding their ability to determine provably optimal solutions within practically reasonable time limits and their ability to quickly determine high quality feasible solutions across different test instances. Hence, we leverage the resulting performance complementarity by proposing algorithm selection approaches that predict the best solver for a given problem instance based on instance features or parameters. The approaches are based on two machine learning techniques, decision trees and deep neural networks, in various variants. In a computational study, we analyze the performance of the resulting algorithm selection models and show that our approaches outperform the use of a single solver and should thus be considered as a relevant tool by decision makers in practice.

A Novel Constraint Programming Decomposition Approach for the Total Flow Time Fixed Group Shop Scheduling Problem

This work addresses a particular case of the group shop scheduling problem (GSSP) which will be denoted as the fixed group shop scheduling problem (FGSSP). In a FGSSP, job operations are divided into stages and each stage has a set of machines associated to it which are not shared with the other stages. All jobs go through all the stages in a specific order, where the operations of the job at each stage need to be finished before the job advances to the following stage, but operations within a stage can be performed in any order. This setting is common in companies such as leaf spring manufacturers and other automotive companies. To solve the problem, we propose a novel heuristic procedure that combines a decomposition approach with a constraint programming (CP) solver and a restart mechanism both to avoid local optima and to diversify the search. The performance of our approach was tested on instances derived from other scheduling problems that the FGSSP subsumes, considering both the cases with and without anticipatory sequence-dependent setup times. The results of the proposed algorithm are compared with off-the-shelf CP and mixed integer linear programming (MILP) methods as well as with the lower bounds derived from the study of the problem. The experiments show that the proposed heuristic algorithm outperforms the other methods, specially on large-size instances with improvements of over 10% on average.

Towards a correct by construction design of complex systems: The MBSS approach

We present in this paper the Model Based System Synthesis (MBSS) approach for the design of complex systems that are correct by construction. Where the usual Model Based System Engineering (MBSE) approach offers formalisms and tools to represent a candidate system, to analyze it, to simulate it and even to optimize it, MBSS proposes to represent the global design problem using a problem representation language and then to solve it by using adapted synthesis tools producing one or several solutions necessarily satisfying the expressed requirements. The two approaches are therefore complementary; the MBSS being more adapted to the preliminary design and system integration stages. After presenting the different categories of problems encountered in system design (sizing, configuration, allocation, architecture generation), MBSS and MBSE will be positioned in relation to each other. The main concepts of MBSS will be detailed in order to understand the specific representation needs of the approach. The structural and behavioral notions related to the sub-definite systems will be explained as well as the links to be established with the functional and non-functional requirements. The approach is illustrated using the DEPS design problem specification language and the DEPS Studio modeling and solving tool on a system design case study. The DEPS language combines structural modeling features specific to object-oriented principles and ontology definition capabilities for engineers with problem specification features from constraint programming. DEPS Studio is an integrated modeling and solving environment designed to model and resolve system synthesis problems. It allows the engineer to edit, compile, debug and solve problems expressed in DEPS. It integrates a mixed constraint programming solver. The approach can be applied on physical systems, software intensive or mixed systems (embedded or cyber-physical).

Exact stochastic constraint optimisation with applications in network analysis

We present an extensive study of methods for exactly solving stochastic constraint (optimisation) problems (SCPs) in network analysis. These problems are prevalent in science, governance and industry. The first method we study is generic and decomposes stochastic constraints into a multitude of smaller local constraints that are solved using a constraint programming (CP) or mixed-integer programming (MIP) solver. However, many SCPs are formulated on probability distributions with a monotonic property, meaning that adding a positive decision to a partial solution to the problem cannot cause a decrease in solution quality. The second method is specifically designed for solving global stochastic constraints on monotonic probability distributions (SCMDs) in CP. Both methods use knowledge compilation to obtain a decision diagram encoding of the relevant probability distributions, where we focus on ordered binary decision diagrams (OBDDs). We discuss theoretical advantages and disadvantages of these methods and evaluate them experimentally. We observed that global approaches to solving SCMDs outperform decomposition approaches from CP, and perform complementarily to MIP-based decomposition approaches, while scaling much more favourably with instance size. Both methods have many alternative design choices, as both knowledge compilation and constraint solvers are used in a single pipeline. To identify which configurations work best, we apply programming by optimisation. Specifically, we show how an automated algorithm configurator can be used to find optimised configurations of our pipeline. After configuration, our global SCMD solving pipeline outperforms its closest competitor (a MIP-based decomposition pipeline) on all test sets we considered by up to two orders of magnitude in terms of PAR10 scores.

Predictive Machine Learning of Objective Boundaries for Solving COPs

Solving Constraint Optimization Problems (COPs) can be dramatically simplified by boundary estimation, that is providing tight boundaries of cost functions. By feeding a supervised Machine Learning (ML) model with data composed of the known boundaries and extracted features of COPs, it is possible to train the model to estimate the boundaries of a new COP instance. In this paper, we first give an overview of the existing body of knowledge on ML for Constraint Programming (CP), which learns from problem instances. Second, we introduce a boundary estimation framework that is applied as a tool to support a CP solver. Within this framework, different ML models are discussed and evaluated regarding their suitability for boundary estimation, and countermeasures to avoid unfeasible estimations that avoid the solver finding an optimal solution are shown. Third, we present an experimental study with distinct CP solvers on seven COPs. Our results show that near-optimal boundaries can be learned for these COPs with only little overhead. These estimated boundaries reduce the objective domain size by 60-88% and can help the solver find near-optimal solutions early during the search.

A new branch-and-filter exact algorithm for binary constraint satisfaction problems

A binary constraint satisfaction problem (BCSP) consists in determining an assignment of values to variables that is compatible with a set of constraints. The problem is called binary because the constraints involve only pairs of variables. The BCSP is a cornerstone problem in Constraint Programming (CP), appearing in a very wide range of real-world applications. In this work, we develop a new exact algorithm which effectively solves the BCSP by reformulating it as a k-clique problem on the underlying microstructure graph representation. Our new algorithm exploits the cutting-edge branching scheme of the state-of-the-art maximum clique algorithms combined with two filtering phases in which the domains of the variables are reduced. Our filtering phases are based on colouring techniques and on heuristically solving an associated boolean satisfiability (SAT) problem. In addition, the algorithm initialization phase performs a reordering of the microstructure graph vertices that produces an often easier reformulation to solve. We carry out an extensive computational campaign on a benchmark of almost 2000 instances, encompassing numerous real and synthetic problems from the literature. The performance of the new algorithm is compared against four SAT-based solvers and three general purpose CP solvers. Our tests reveal that the new algorithm significantly outperforms all the others in several classes of BCSP instances.