Constraint Programming Research Articles

Experimental investigation of the polyhedral structure of BIBDs

We examine the polyhedral structure of Balanced Incomplete Block Designs (BIBDs) in an effort to improve runtimes for finding optimal solutions via constraint or integer program. For example, an optimal BIBD is one that minimizes the range of blocksums. This range, which we refer to as diffsum, has applications in drug trials and Redundant Arrays of Independent Disks (RAID). The polytope we explore is the convex hull of all lexicographically-ordered BIBD solutions satisfying a given set of parameters, ( v , b , r , k , λ ) , with solutions represented as a vector in R kb . The dimension-reducing equations and the facets for some of the smaller polytopes were found by using the Double-Description Method on a algebraic program modelling these solutions. Because of the presence of the lexicographic ordering, none of these polytopes are full dimensional. After describing some of these smaller BIBD polytopes, we generalize some dimension-reducing equations, and evaluate the efficiency of these equations to optimization models.

Counting and hardness-of-finding fixed points in cellular automata on random graphs

Abstract We study the fixed points of outer-totalistic cellular automata on sparse random regular graphs. These can be seen as constraint satisfaction problems, where each variable must adhere to the same local constraint, which depends solely on its state and the total number of its neighbors in each possible state. Examples of this setting include classical problems such as independent sets or assortative/dissasortative partitions. We analyse the existence and number of fixed points in the large system limit using the cavity method, under both the replica symmetric (RS) and one-step replica symmetry breaking (1RSB) assumption. This method allows us to characterize the structure of the space of solutions, in particular, if the solutions are clustered and whether the clusters contain frozen variables. This last property is conjectured to be linked to the typical algorithmic hardness of the problem. We bring experimental evidence for this claim by studying the performance of the belief-propagation reinforcement algorithm, a message-passing-based solver for these constraint satisfaction problems.

A Novel Approach for Solving the N-Queen Problem Using a Non-Sequential Conflict Resolution Algorithm

The N-Queens problem is a fundamental challenge in combinatorial optimization, commonly used as a benchmark for assessing the efficiency of algorithms. Traditional algorithms, such as Backtracking with Forward Checking (BFC), constraint satisfaction problem (CSP) techniques, Lookahead algorithms, and heuristic-based methods, often face challenges with exponential time complexity, making them less practical for large-scale instances. This paper introduces a novel algorithm, non-sequential conflict resolution (NSCR), which improves performance over traditional algorithms through dynamic conflict resolution. The NSCR algorithm iteratively resolves conflicts among queens by adjusting their positions, aiming to optimize both time complexity and memory usage. While NSCR also operates within exponential time bounds, it demonstrates improved scalability and efficiency compared to traditional methods. A significant strength of the NSCR algorithm lies in its space complexity, which is O(n), and a time complexity that, while typically lower than traditional methods, can reach O(n3) in the worst-case scenario. This linear space complexity is highly advantageous, particularly when dealing with large problem sizes, as it ensures efficient use of memory resources. Comparative analysis with the aforementioned algorithms shows that NSCR offers superior resource management, using up to 60% less memory and reducing runtime by approximately 50%, making it an efficient option for large-scale instances of the N-Queens problem. The algorithm’s performance, evaluated on problem sizes ranging from 8 to 1000 queens, highlights its ability to manage computational resources effectively, despite the inherent challenges of exponential time complexity.

Investigating constraint programming and hybrid methods for real world industrial test laboratory scheduling

AbstractIn this paper we deal with a complex real world scheduling problem closely related to the well-known Resource-Constrained Project Scheduling Problem (RCPSP). The problem concerns industrial test laboratories in which a large number of tests are performed by qualified personnel using specialised equipment, while respecting deadlines and other constraints. We present different constraint programming models and search strategies for this problem. Furthermore, we propose a Very Large Neighborhood Search approach based on our CP methods. Our models are evaluated using CP solvers and a MIP solver both on real-world test laboratory data and on a set of generated instances of different sizes based on the real-world data. Further, we compare the exact approaches with VLNS and a Simulated Annealing heuristic. We could find feasible solutions for all instances and several optimal solutions and we show that using VLNS we can improve upon the results of the other approaches.

List coloring based algorithm for the Futoshiki puzzle

Given a graph G=(V, E) and a list of available colors L(v) for each vertex v\in V, where L(v) \subseteq {1, 2, ..., k}, List k-Coloring refers to the problem of assigning colors to the vertices of $G$ so that each vertex receives a color from its own list and no two neighboring vertices receive the same color. The decision version of the problem, List k-Coloring, is NP-complete even for bipartite graphs. As an application of list coloring problem we are interested in the Futoshiki Problem. Futoshiki is an NP-complete Latin Square Completion Type Puzzle. Considering Futoshiki puzzle as a constraint satisfaction problem, we first give a list coloring based algorithm for it which is efficient for small boards of fixed size. To thoroughly investigate the efficiency of our algorithm in comparison with a proposed backtracking-based algorithm, we conducted a substantial number of computational experiments at different difficulty levels, considering varying numbers of inequality constraints and given values. Our results from the extensive range of experiments indicate that the list coloring-based algorithm is much more efficient.

Scaling Abstraction Refinement for Program Analyses in Datalog using Graph Neural Networks

Counterexample-guided abstraction refinement (CEGAR) is a popular approach for automatically selecting abstractions with high precision and low time costs. Existing works cast abstraction refinements as constraint-solving problems. Due to the complexity of these problems, they cannot be scaled to large programs or complex analyses. We propose a novel approach that applies graph neural networks to improve the scalability of CEGAR for Datalog-based program analyses. By constructing graphs directly from the Datalog solver’s calculations, our method then uses a neural network to score abstraction parameters based on the information in these graphs. Then we reform the constraint problems such that the constraint solver ignores parameters with low scores. This in turn reduces the solution space and the size of the constraint problems. Since our graphs are directly constructed from Datalog computation without human effort, our approach can be applied to a broad range of parametric static analyses implemented in Datalog. We evaluate our approach on a pointer analysis and a typestate analysis and our approach can answer 2.83× and 1.5× as many queries as the baseline approach on large programs for the pointer analysis and the typestate analysis, respectively.

Perception-based constraint solving for sudoku images

AbstractWe consider the problem of perception-based constraint solving, where part of the problem specification is provided indirectly through an image provided by a user. As a pedagogical example, we use the complete image of a Sudoku grid. While the rules of the puzzle are assumed to be known, the image must be interpreted by a neural network to extract the values in the grid. In this paper, we investigate (1) a hybrid modeling approach combining machine learning and constraint solving for joint inference, knowing that blank cells need to be both predicted as being blank and filled-in to obtain a full solution; (2) the effect of classifier calibration on joint inference; and (3) how to deal with cases where the constraints of the reasoning system are not satisfied. More specifically, in the case of handwritten user errors in the image, a naive approach fails to obtain a feasible solution even if the interpretation is correct. Our framework identifies human mistakes by using a constraint solver and helps the user to correct these mistakes. We evaluate the performance of the proposed techniques on images taken through the Sudoku Assistant Android app, among other datasets. Our experiments show that (1) joint inference can correct classifier mistakes, (2) overall calibration improves the solution quality on all datasets, and (3) estimating and discriminating between user-written and original visual input while reasoning makes for a more robust system, even in the presence of user errors.

Domain-Independent Dynamic Programming and Constraint Programming Approaches for Assembly Line Balancing Problems with Setups

Domain-Independent Dynamic Programming and Constraint Programming Approaches for Assembly Line Balancing Problems with Setups

Code and Data Repository for Domain-Independent Dynamic Programming and Constraint Programming Approaches for Assembly Line Balancing Problems with Setups

The goal of this repository is to demonstrate the effect of DIDP and CP for SUALBP.

Mining diverse sets of patterns with constraint programming using the pairwise Jaccard similarity relaxation

Mining diverse sets of patterns with constraint programming using the pairwise Jaccard similarity relaxation

A novel two-stage network data envelopment analysis model for kidney allocation problem under medical and logistical uncertainty: a real case study.

Organ transplantation is one of the most complicated and challenging treatments in healthcare systems. Despite the significant medical advancements, many patients die while waiting for organ transplants because of the noticeable differences between organ supply and demand. In the organ transplantation supply chain, organ allocation is the most significant decision during the organ transplantation procedure, and kidney is the most widely transplanted organ. This research presents a novel method for assessing the efficiency and ranking of qualified organ-patient pairs as decision-making units (DMUs) for kidney allocation problem in the existence of COVID-19 pandemic and uncertain medical and logistical data. To achieve this goal, two-stage network data envelopment analysis (DEA) and credibility-based chance constraint programming (CCP) are utilized to develop a novel two-stage fuzzy network data envelopment analysis (TSFNDEA) method. The main benefits of the developed method can be summarized as follows: considering internal structures in kidney allocation system, investigating both medical and logistical aspects of the problem, the capability of expanding to other network structures, and unique efficiency decomposition under uncertainty. Moreover, in order to evaluate the validity and applicability of the proposed approach, a validation algorithm utilizing a real case study and different confidence levels is used. Finally, the numerical results indicate that the developed approach outperforms the existing kidney allocation system.

A branch-and-price approach for the nurse rostering problem with multiple units

In this paper, we study the nurse rostering problem that considers multiple units and many soft time-related constraints. An efficient branch and price solution approach that relies on a fast algorithm to solve the pricing subproblem of the column generation process is presented. For the nurse rostering problem, its pricing subproblem can be formulated as a shortest path problem with resource constraints, which has been the backbone of several solutions for several classical problems like vehicle routing problems. However, approaches that perform well on these problems cannot be used since most constraints in the nurse rostering problem are soft. Based on ideas borrowed from global constraints in constraint programming to model rostering problems, an efficient dynamic programming algorithm with novel label definitions and dominating rules specific to soft time-related constraints is proposed. In addition, several acceleration strategies are employed to improve the branch and price algorithm. Computational results on two sets of instances ranging from 2 to 4 units over a horizon of 2 or 4 weeks demonstrate the effectiveness of the proposed algorithms. The accelerated dynamic programming algorithm is able to solve pricing subproblems to optimality with the fewest extended labels. When contrasted with Cplex, the branch and price approach yields solutions that are either equivalent or superior across all instances.

Scheduling as a Constraint Satisfaction Problem (Using the Example of Open-Pit Minе Production Scheduling Problem)

The research described in the work is aimed at developing methods for Scheduling. The fundamental disadvantage of the existing methods of Mixed-Integer Linear Programming in application to the problems under consideration is the fact that they are too demanding on the amount of RAM. The difficulty of applying local search procedures to such high-dimensional problems is to develop an effective way to find an acceptable initial approximation and determine the neighboring state transition function, which would allow achieving the optimum fast enough. In the Operations Research Theory, adding additional conditions to a problem can lead to a fundamental change in the problem-solving scheme. The methods proposed in the study are implemented within the framework of the Constraint Programming Paradigm which makes it possible to represent the subject domain dependencies saving RAM, as well as to provide the ability to step-by-step take into account heterogeneous problem conditions without essentially changing the scheme of finding solutions. A significant part of the research deals with methods of logical inference on constraints to reduce the search space and speed up the computational process. The approach to scheduling is illustrated by the Open-Pit Mine Production Scheduling Problem, which was first proposed to be solved as a Constraint Satisfaction Problem. In order to find the first feasible solution, a «greedy» search method is proposed, the result of which can be improved using the developed hybrid method. Both methods rely on original procedures of inference on constraints. The proposed approach has proven its efficiency for block models with sizes of tens and hundreds of thousands of blocks.

“I Want It That Way”: Enabling Interactive Decision Support Using Large Language Models and Constraint Programming

A critical factor in the success of many decision support systems is the accurate modeling of user preferences. Psychology research has demonstrated that users often develop their preferences during the elicitation process, highlighting the pivotal role of system-user interaction in developing personalized systems. This paper introduces a novel approach, combining Large Language Models (LLMs) with Constraint Programming to facilitate interactive decision support. We study this hybrid framework through the lens of meeting scheduling, a time-consuming daily activity faced by a multitude of information workers. We conduct three studies to evaluate the novel framework, including a diary study to characterize contextual scheduling preferences, a quantitative evaluation of the system’s performance, and a user study to elicit insights with a technology probe that encapsulates our framework. Our work highlights the potential for a hybrid LLM and optimization approach for iterative preference elicitation, and suggests design considerations for building systems that support human-system collaborative decision-making processes.

A constraint programming approach for resource-constrained flexible assembly flow shop scheduling problem with batch direct delivery

Production and distribution are both crucial components of supply chains. Integrated production and distribution scheduling (IPDS) in the context of flexible assembly flow shop scheduling and batch delivery problems is often overlooked. A realistic problem inspired by the production and distribution processes of a dishwasher factory can be modeled as a resource-constrained flexible assembly flow shop scheduling problem with batch direct delivery (RCFAFSP-BDD). In this problem, order requirements are decomposed into several production tasks processed at different stages of the workshop, and are then delivered in batches via a third-party logistics provider to regional distributors at various locations. Auxiliary resource restrictions, hierarchical coupling constraints, machine eligibility restrictions and sequence-dependent setup times, are incorporated into the problem as operational constraints. To the best of our knowledge, this is the first attempt to solve this problem. This work formulates a mixed-integer linear programming (MIP) model to minimize total costs, including tardiness, inventory, and delivery costs. Given the problem’s strong NP-hard nature, the focus is on developing an efficient solution approach using constraint programming (CP). A CP model is proposed and enhanced with multiple redundant constraints. To reduce runtime, two branching strategies are designed for the CP model. Numerical experiments with varying instance scales reveal that the proposed CP model outperforms the MIP model in accuracy and efficiency within the given time limit. The redundant constraints and search strategy can reduce CP model runtime by up to 263.83%. Compared to manual scheduling at the studied factory, the CP model can cut costs by up to 26.59% for real data, offering viable alternatives for factory planners.

DT-DDQNR: A digital twin assisted direct-to-cell satellite network intelligent routing algorithm

This paper presents a novel routing algorithm designed for direct-to-cell satellite constellations in satellite networks. The algorithm aims to ensure stable Quality of Service (QoS) for users despite the highly dynamic nature of satellite links and network conditions. It comprises two key components: a terrestrial-satellite link selection model and an inter-satellite multi-hop routing decision model. To enhance decision-making, the algorithm incorporates a mechanism that interacts with large-scale digital twin constellations, offering agents with high-precision state data inputs and feedback for training and application support. The terrestrial-satellite link selection process is modeled as a constraint satisfaction problem, maximizing the Link Quality Indicator (LQI). Meanwhile, inter-satellite data forwarding decisions are determined utilizing an enhanced dynamic reward mechanism and a re-designed neural network in the DDQN model. Simulation results demonstrate the superiority of the Digital Twin-Assisted Double Deep Q-Network Routing (DT-DDQNR) algorithm over traditional routing algorithms in terms of overall network QoS performance, offering a promising solution for optimizing data transmission in integrated terrestrial-satellite networks.

An exact constraint programming method for the multi-manned assembly line balancing problem with assignment restrictions

This paper proposes an exact Constraint Programming (CP) method with an extensive focus on real-world constraints for the Multi-manned Assembly Line Balancing Problem with Assignment Restrictions (MALBP-AR). We perform an in-depth literature review to gather examples from real assembly lines and organize the AR regarding tasks, stations, workers, and mounting positions. Our study classifies the AR related to transformed resources and provides a general and unified model. We explore the concept of variable workplaces to dynamically assign workers to mounting positions and aggregate no overlap restrictions to avoid interference between workers. The classic MALBP model is extended by gradually incrementing the number of restrictions. The model variant found in the literature is herein called Partial MALBP-AR. Compared to the previous state-of-the-art Tabu Search Algorithm (TSA) for this problem, the Partial MALBP-AR found twelve additional optimality proofs. Besides the relevant results regarding solution quality, the CP method also has a satisfactory CPU performance. We also propose an entirely new set of AR and test these practical conditions with the so-called Extended MALBP-AR. Such an extended model, which covers all the AR presented here, reached optimality within the computational time limit for 36 out of 38 instances. The worst-case gap for an open instance is 8.20%. The results show a trade-off between the number of deemed restrictions and the computational performance. However, considering the detailed set of AR, we can obtain more representative solutions regarding the final balancing implementation compared to theoretical cases. The method can be used to design experiments, turning certain constraints on and off and allowing managers to evaluate different resource allocation scenarios.

Turret-index optimisation with mathematical programming and metaheuristic approaches

This paper addresses the turret-index optimisation problem by proposing two mathematical programming approaches based on quadratic programming (QP) and constraint programming (CP). In addition, a metaheuristic algorithm based on weighted superposition attraction (WSA) is developed due to the combinatorial complexity of the problem. Despite attempting to linearise the QP formulation for problem resolution, both the QP and Linearized QP models prove ineffective for medium and larger-sized instances, providing solutions only for small-sized problems. In the second mathematical programming approach, a novel CP model is introduced in the literature. While CP can rapidly offer optimal solutions for small-sized problems, it only provides satisfactory solutions for medium and larger-sized problems within the predetermined computational time limits and does not guarantee optimal results. On the other hand, through computational analysis and relevant statistical tests, the proposed WSA algorithm provides the best solutions for all test problems within a reasonable computational time.

Solving Boolean Satisfiability Problems With The Quantum Approximate Optimization Algorithm

One of the most prominent application areas for quantum computers is solving hard constraint satisfaction and optimization problems. However, detailed analyses of the complexity of standard quantum algorithms have suggested that outperforming classical methods for these problems would require extremely large and powerful quantum computers. The quantum approximate optimization algorithm (QAOA) is designed for near-term quantum computers, yet previous work has shown strong limitations on the ability of QAOA to outperform classical algorithms for optimization problems. Here we instead apply QAOA to hard constraint satisfaction problems, where both classical and quantum algorithms are expected to require exponential time. We analytically characterize the average success probability of QAOA on a constraint satisfaction problem commonly studied using statistical physics methods: random k-SAT at the threshold for satisfiability, as the number of variables n goes to infinity. We complement these theoretical results with numerical experiments on the performance of QAOA for small n, which match the limiting theoretical bounds closely. We then compare QAOA with leading classical solvers. For random 8-SAT, we find that for more than 14 quantum circuit layers, QAOA achieves more efficient scaling than the highest-performance classical solver we tested, WalkSATlm. Our results suggest that near-term quantum algorithms for solving constraint satisfaction problems may outperform their classical counterparts. Published by the American Physical Society 2024

Finite algebras with Hom-sets of polynomial size

We provide an internal characterization of those finite algebras (i.e., algebraic structures) A \mathbf {A} such that the number of homomorphisms from any finite algebra X \mathbf {X} to A \mathbf {A} is bounded from above by a polynomial in the size of X \mathbf {X} . Namely, an algebra A \mathbf {A} has this property if, and only if, no subalgebra of A \mathbf {A} has a nontrivial strongly abelian congruence. We also show that the property can be decided in polynomial time for algebras in finite signatures. Moreover, if A \mathbf {A} is such an algebra, the set of all homomorphisms from X \mathbf {X} to A \mathbf {A} can be computed in polynomial time given X \mathbf {X} as input. As an application of our results to the field of computational complexity, we characterize inherently tractable constraint satisfaction problems over fixed finite structures, i.e., those that are tractable and remain tractable after expanding the fixed structure by arbitrary relations or functions.