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Constraint

A constraint is a limitation or restriction that controls what can be done within particular bounds, often arising from external conditions, rules, or inherent properties of a system.^[1] In everyday usage, constraints manifest as factors like budget limitations or time pressures that shape decisions and actions.^[2] In mathematics and optimization, a constraint is a condition that restricts the values of variables in a problem, defining the feasible region of possible solutions, such as inequalities in linear programming that model real-world limits like production capacity or resource availability.^[3] These are essential in fields like operations research, where they ensure solutions remain practical and bounded.^[4] In physics, particularly classical mechanics, constraints describe parameters that a system must obey, such as the requirement for a particle to remain on a surface, thereby reducing degrees of freedom and influencing motion through forces like tension or normal reactions.^[5] Constraints can be holonomic (expressible as equations of position) or non-holonomic (involving velocities), and they simplify analysis by eliminating unnecessary variables.^[6] In computer science, constraints form the basis of constraint programming, a paradigm where problems are solved by declaratively specifying logical relations among variables and their domains, allowing automated solvers to find satisfying assignments for tasks like scheduling or configuration.^[7] This approach, rooted in artificial intelligence and logic, contrasts with traditional procedural programming by focusing on what must hold true rather than how to compute step-by-step.^[8] Beyond technical domains, constraints also appear in social and psychological contexts, where they refer to the repression or control of natural impulses and behaviors, influencing individual and group dynamics.^[9] Across disciplines, understanding and managing constraints is crucial for modeling complex systems and achieving optimal outcomes.

Definition and General Concepts

Core Definition

A constraint is a condition, rule, or limitation that restricts the feasible solutions or behaviors within a given system or problem, often expressed as an equation, inequality, or logical relation.^[10] In essence, it acts as a boundary that delineates acceptable outcomes from infeasible ones, ensuring that only viable options are considered in decision-making or analysis processes.^[11] The concept of constraints traces its origins to early philosophical inquiries into logic, where Aristotle (384–322 BCE) imposed restrictions on syllogistic reasoning to ensure valid inferences, viewing syllogisms as acts of inferring conclusions constrained by specific rules of form and content.^[12] In mathematics, the formal treatment of constraints emerged in the 18th century through Joseph-Louis Lagrange's work on optimization, where he introduced multipliers to handle restrictions in variational problems, as detailed in his 1788 treatise Mécanique Analytique.^[13] Real-world examples illustrate constraints clearly: a budget limit imposes a numerical upper bound on spending, restricting purchases to available funds, while a physical barrier, such as a wall, prevents motion beyond a certain point by enforcing a spatial restriction.^[10] These cases highlight how constraints operate across domains to shape possible actions or states. Key properties of constraints include their classification as binding or non-binding: a binding constraint actively limits the solution at its boundary, such that altering it changes the optimal outcome, whereas a non-binding constraint leaves slack and does not influence the result.^[14] Collectively, constraints define the feasible region or set, comprising all points satisfying the imposed conditions.^[11] Constraints are often mathematically represented as equations or inequalities that define the boundaries of possible solutions.

Classification of Constraints

Constraints are broadly classified into hard and soft based on the degree of flexibility in their satisfaction. Hard constraints must be strictly satisfied for a solution to be feasible, representing non-negotiable conditions inherent to the problem, such as physical laws or resource limits that cannot be violated without invalidating the outcome.^[15] In contrast, soft constraints allow violations but incur penalties in the objective function, often modeling preferences or secondary requirements, like scheduling tolerances where minor deviations are acceptable at a cost.^[15] This distinction is crucial in optimization, where hard constraints define the feasible region, while soft ones enable trade-offs via methods like penalty functions.^[16] Structurally, constraints are categorized by their mathematical form into linear and nonlinear types. Linear constraints involve affine relationships between variables, such as a^T x \leq b, forming polyhedral feasible sets that are computationally tractable in linear programming.^[15] Nonlinear constraints, involving curved boundaries like x^2 + y^2 = 1, complicate the feasible region and often require advanced techniques, including constraint qualifications to ensure well-behaved optimization.^[15] Additionally, constraints can be explicit, directly formulated as equations or inequalities (e.g., g(x) = 0), or implicit, embedded within the problem's definition through set specifications (e.g., variable bounds incorporated into the domain).^[15] Explicit forms facilitate direct handling in algorithms, whereas implicit ones may arise from system properties and necessitate reformulation for analysis.^[17] From a dimensional perspective, constraints vary in scope: point constraints fix specific variables or states at particular instances (e.g., initial conditions in dynamic systems), path constraints restrict trajectories or paths over a continuum (e.g., bounds on state evolution during optimization), and global constraints influence the entire solution space (e.g., total resource allocation across all variables).^[18] Point constraints localize enforcement, path constraints ensure ongoing compliance along sequences, and global ones enforce aggregate properties, impacting scalability in high-dimensional problems.^[19] Regarding their impact on systems, constraints are reducible or irreducible depending on whether they can be eliminated without loss of information. Reducible constraints, such as redundant inequalities derivable from others, can be removed via substitution or preprocessing, simplifying the problem while preserving equivalence.^[15] Irreducible constraints are fundamental and cannot be simplified further, remaining essential to the problem's structure, often requiring specialized handling in solvers.^[20] This classification aids in preprocessing for efficiency, particularly in large-scale optimization where reducing dimensionality enhances solvability.^[21]

Constraints in Mathematics

Equality and Inequality Constraints

In mathematics, equality constraints impose exact conditions on variables, typically formulated as g(\mathbf{x}) = 0, where \mathbf{x} is a vector of decision variables and g is a function mapping to the appropriate dimension. These constraints define hypersurfaces in the variable space, restricting solutions to lie precisely on them, as seen in models enforcing conservation principles or fixed relationships between quantities. Inequality constraints, in contrast, establish boundaries for feasible values, expressed as h(\mathbf{x}) \leq 0 or h(\mathbf{x}) \geq 0, where h similarly maps \mathbf{x} to a vector. They delimit half-spaces or regions, such as upper limits on resource usage in allocation scenarios, allowing solutions within or on the boundary but excluding those beyond. Constraints may be linear (affine functions of \mathbf{x}) or nonlinear, influencing the geometry of the resulting space. The feasible set S comprises all points satisfying both types:

S = \{ \mathbf{x} \mid g(\mathbf{x}) = 0, \, h(\mathbf{x}) \leq 0 \}.

This set represents the solution space, often convex when constraints are convex. In two dimensions with linear inequalities, S forms a convex polygon; for example, the constraints x_1 \geq 0, x_2 \geq 0, and x_1 + x_2 \leq 1 yield a right triangle with vertices at (0,0), (1,0), and (0,1).^[22] A key technique for handling inequalities involves slack variables, nonnegative auxiliaries s \geq 0 that convert them to equalities: h(\mathbf{x}) + s = 0. This reformulation aids in applying equality-based methods while preserving the original bounds. Degeneracy arises when more constraints than the problem's dimension intersect at a point, such as three lines meeting at one vertex in 2D, which can signal redundancy or complicate boundary analysis without altering feasibility.^[23]^[22]

Constraints in Optimization Problems

In optimization, constraints define the feasible region within which an objective function f(\mathbf{x}) is to be minimized or maximized, distinguishing constrained optimization from unconstrained optimization, where the search space is unrestricted and local optima are typically found at points where the gradient vanishes. Constrained problems arise in numerous applications, such as resource allocation and engineering design, where variables must satisfy equality or inequality conditions to ensure practicality and feasibility. The presence of constraints complicates the search for global optima, often requiring specialized algorithms that respect the boundaries of the feasible set. A prominent class of constrained optimization problems is linear programming (LP), where both the objective function and constraints are linear. The standard form of an LP is to minimize \mathbf{c}^T \mathbf{x} subject to A\mathbf{x} \leq \mathbf{b} and \mathbf{x} \geq \mathbf{0}, where A is an m \times n matrix, \mathbf{b} \in \mathbb{R}^m, \mathbf{c} \in \mathbb{R}^n, and \mathbf{x} \in \mathbb{R}^n; this formulation assumes non-negativity for variables, with equalities convertible to inequalities via slack variables. The feasible region forms a convex polyhedron, and optimal solutions occur at vertices. The simplex method, introduced by George Dantzig in 1947, efficiently solves LPs by iteratively pivoting from one basic feasible solution (vertex) to an adjacent one, improving the objective until optimality is reached, with a worst-case exponential complexity but strong average-case performance. For nonlinear optimization problems, where the objective or constraints involve nonlinear functions, handling constraints requires more advanced theoretical tools. The Karush-Kuhn-Tucker (KKT) conditions serve as first-order necessary optimality criteria for a local minimum in problems of the form minimize f(\mathbf{x}) subject to g_i(\mathbf{x}) \leq 0 (inequalities) and h_j(\mathbf{x}) = 0 (equalities), assuming constraint qualifications like linear independence of gradients hold. These conditions consist of four components: stationarity, where \nabla f(\mathbf{x}^*) + \sum \lambda_i \nabla g_i(\mathbf{x}^*) + \sum \mu_j \nabla h_j(\mathbf{x}^*) = \mathbf{0} with multipliers \lambda_i \geq 0; primal feasibility, ensuring g_i(\mathbf{x}^*) \leq 0 and h_j(\mathbf{x}^*) = 0; dual feasibility, \lambda_i \geq 0; and complementary slackness, \lambda_i g_i(\mathbf{x}^*) = 0 for all i. Originally derived by William Karush in his 1939 master's thesis and independently published by Harold W. Kuhn and Albert W. Tucker in 1951, the KKT conditions generalize Lagrange multipliers to include inequalities and underpin algorithms like sequential quadratic programming. Duality provides a powerful framework for understanding constrained optimization, associating a primal problem with a dual problem that offers bounds and insights into optimality. In the primal LP minimize \mathbf{c}^T \mathbf{x} subject to A\mathbf{x} \geq \mathbf{b}, \mathbf{x} \geq \mathbf{0}, the dual is maximize \mathbf{b}^T \mathbf{y} subject to A^T \mathbf{y} \leq \mathbf{c}, \mathbf{y} \geq \mathbf{0}; weak duality ensures the primal minimum exceeds or equals the dual maximum. For convex problems, including all LPs and nonlinear convex programs, the strong duality theorem asserts that under constraint qualifications (e.g., Slater's condition), the optimal primal and dual values coincide, with zero duality gap, enabling dual methods for verification and sensitivity analysis. This result, formalized in R. Tyrrell Rockafellar's 1970 convex analysis framework, extends earlier LP duality from Gale, Kuhn, and Tucker (1951) and facilitates efficient computation in large-scale convex optimization.

Constraints in Physics

Holonomic Constraints

Holonomic constraints in classical mechanics are restrictions on the motion of a system that can be expressed as algebraic equations involving the generalized coordinates q_1, \dots, q_n and possibly time t, typically in the form f(q_1, \dots, q_n, t) = 0.^[24] These constraints are integrable, meaning they derive from exact differential forms, allowing the system's configuration space to be reduced to a lower-dimensional manifold.^[25] A classic example is the simple pendulum, where the bob is constrained to move on a circle of fixed radius l, given by the equation x^2 + y^2 = l^2 in Cartesian coordinates or equivalently f(x, y) = x^2 + y^2 - l^2 = 0.^[26] For a rigid body composed of N particles, holonomic constraints enforce fixed distances between all pairs of particles, imposing \frac{N(N-1)}{2} distance constraints, of which 3N - 6 are independent for N ≥ 3, of the form |\mathbf{r}_i - \mathbf{r}_j| = d_{ij} for constants d_{ij}.^[27] Holonomic constraints reduce the number of degrees of freedom in a system of N particles from $3N to $3N - k, where k is the number of independent constraints, enabling the selection of $3N - k independent generalized coordinates. To derive the equations of motion under such constraints, Lagrange's method of undetermined multipliers is employed: the Lagrangian \mathcal{L} = T - V is augmented with terms \sum \lambda_m f_m, where \lambda_m are the multipliers, leading to the modified Euler-Lagrange equations \frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{q}_i} \right) - \frac{\partial \mathcal{L}}{\partial q_i} = \sum_m \lambda_m \frac{\partial f_m}{\partial q_i}.^[28] Holonomic constraints are further classified as scleronomic if they do not explicitly depend on time (f(q_1, \dots, q_n) = 0), such as fixed joints in a mechanical linkage, or rheonomic if they do (f(q_1, \dots, q_n, t) = 0), as in a pendulum with a support oscillating vertically with time.^[29] Scleronomic constraints preserve certain symmetries, like time invariance in the Lagrangian, while rheonomic ones introduce explicit time dependence, potentially affecting energy conservation.^[30]

Non-Holonomic Constraints

Non-holonomic constraints in mechanics are velocity-dependent restrictions on the motion of a system that cannot be expressed solely in terms of position coordinates. They take the form of differential equations known as Pfaffian constraints, \sum_{i=1}^n a_i(\mathbf{q}) \, dq_i + a_t \, dt = 0, where \mathbf{q} represents the generalized coordinates, and the differential form is not exact, meaning it lacks an integrating factor that would allow reduction to a holonomic (position-based) constraint.^[31] These constraints arise in systems where the allowable paths in configuration space are unrestricted, but the instantaneous velocities are limited, distinguishing them from holonomic constraints, which integrable to equality relations on positions alone. Classic examples include the rolling of a wheel without slipping, where the linear velocity v of the center equals the product of the angular velocity \omega and radius r, v = r \omega, enforcing a relation between translational and rotational velocities that cannot be integrated to a position constraint.^[32] Another illustrative case is a skate gliding on ice, subject to a no-sideways-slip condition that prohibits velocity components perpendicular to the blade, again a non-integrable velocity restriction.^[33] The presence of non-holonomic constraints prevents full reduction of the system's degrees of freedom in the configuration space, as the constraints act only on velocities, leaving the accessible configurations unchanged while constraining the distribution of allowable velocities.^[34] To derive the equations of motion, specialized methods are required, such as the introduction of quasi-coordinates—linear combinations of generalized velocities that simplify the constraint enforcement—or Appell's equations, which extend the Lagrangian formalism by incorporating the work done against constraint forces in an auxiliary function.^[35] These approaches allow systematic treatment without eliminating coordinates, unlike in holonomic cases.^[36] The term "non-holonomic" was coined by Heinrich Hertz in his 1894 treatise The Principles of Mechanics, where he distinguished such systems from holonomic ones to address constraints beyond simple positional ties.^[37] Non-holonomic systems present particular challenges in Hamiltonian mechanics, as the velocity constraints disrupt the symplectic structure of phase space, necessitating non-standard almost-Hamiltonian or vakonomic formulations to capture the dynamics accurately.^[38]

Constraints in Computer Science

Constraint Satisfaction Problems

A constraint satisfaction problem (CSP) is formally defined as a triple (X, D, C), where X is a finite set of variables, D is a set of finite domains with one domain D_x for each variable x \in X, and C is a finite set of constraints, each specifying a relation over a subset of variables indicating the allowable tuples of values.^[39] The goal of solving a CSP is to determine whether there exists an assignment of a value from its domain to each variable in X such that all constraints in C are satisfied.^[39] Constraints in CSPs are typically binary (relating two variables) or of higher arity, and they model hard constraints that must be fully satisfied without violation.^[39] To solve CSPs, backtracking search is a fundamental strategy that proceeds by assigning values to variables one at a time in a depth-first manner, checking constraints as variables are instantiated and backtracking to previous choices upon encountering inconsistency.^[40] Efficiency is enhanced by preprocessing techniques such as enforcing arc consistency, which removes values from domains that cannot participate in any complete solution due to lack of support from neighboring variables' domains.^[41] The AC-3 algorithm achieves this by maintaining a queue of arcs (directed constraint pairs) and iteratively revising domains until no further pruning is possible, with a time complexity of O(ed^3) where e is the number of constraints and d is the maximum domain size.^[41] In general, deciding the satisfiability of a CSP is NP-complete, as it encompasses problems like 3-SAT through appropriate reductions.^[42] However, certain tractable subclasses exist; for instance, CSPs whose constraint graphs are trees can be solved in polynomial time, specifically O(nd^2) where n is the number of variables, using dynamic programming to propagate constraints from leaves to root.^[43] Classic examples of CSPs include the map coloring problem, where variables represent regions on a map, domains are a set of colors (e.g., red, green, blue), and binary constraints require adjacent regions to receive different colors.^[44] Another is the n-queens puzzle, with variables as positions in columns on an n×n chessboard, domains as row numbers (1 to n), and constraints ensuring no two queens share the same row, diagonal, or column.^[45]

Constraint Programming Languages

Constraint programming languages facilitate the declarative modeling of problems by allowing users to specify constraints on variables, while underlying solvers manage the inference and search processes to find solutions. This paradigm shifts the focus from algorithmic control to problem description, enabling concise representations of complex combinatorial tasks. For instance, the Oz programming language, implemented in the Mozart system, exemplifies this approach by integrating constraint handling as a core feature alongside logic, functional, and concurrent paradigms, allowing constraints to be solved through automated propagation and enumeration.^[46]^[47] Central to these languages are key techniques for efficient solving. Constraint propagation reduces the search space by iteratively pruning inconsistent values from variable domains based on the constraints; forward checking, a fundamental method, eliminates values from unassigned variables that conflict with recently assigned ones, while domain splitting—often used in branching—bisects continuous or large discrete domains to facilitate further propagation. For optimization problems, branch-and-bound algorithms extend this by maintaining upper and lower bounds on objective functions during search, pruning branches that cannot yield better solutions. These mechanisms, rooted in early constraint logic programming systems, ensure soundness and completeness in solution finding.^[48]^[49] Prominent tools and libraries support this paradigm across modeling and solving. MiniZinc serves as a high-level, solver-independent modeling language, using intuitive syntax to define variables, constraints, and objectives, which are then translated to backend solvers for execution. Solvers like Choco, a Java-based open-source library, implement advanced propagation and search heuristics for integer and Boolean constraints, while Gecode, a C++ toolkit, offers state-of-the-art performance with support for over 70 global constraints and parallel search engines. Integration with logic programming is common, as seen in extensions to Prolog such as SWI-Prolog's CLP modules, which embed constraint solving within declarative rules for hybrid applications.^[50]^[51]^[52] In practice, constraint programming languages excel in applications like scheduling and configuration. For scheduling, they model job-shop problems—where tasks must be sequenced on machines without overlaps—using resource and precedence constraints, often outperforming traditional methods on industrial-scale instances. Similarly, in product configuration, such as assembling customizable items, constraints ensure compatibility of components, enabling interactive solvers to generate valid assemblies efficiently.^[53]^[54]

Constraints in Other Disciplines

Constraints in Economics

In economics, constraints represent the fundamental limitations imposed by scarcity on the choices of individuals, firms, and markets, shaping decision-making processes in resource allocation and production. These constraints arise from limited income, production capacities, and market conditions, forcing economic agents to prioritize among competing options to achieve objectives such as utility maximization or profit optimization. Unlike abstract mathematical inequalities, economic constraints are embedded in behavioral and institutional contexts, influencing how agents respond to prices, technology, and external factors.^[55] A primary example is the budget constraint faced by consumers, which delineates the feasible set of goods and services they can afford given their income and prevailing prices. Formally, for a vector of prices \mathbf{p} and quantities \mathbf{x} of goods, with income m, the constraint is \mathbf{p} \cdot \mathbf{x} \leq m, representing a linear boundary in the commodity space. Consumers aim to maximize utility subject to this constraint, often visualized through indifference curves—contours of equal utility levels that are convex to the origin due to diminishing marginal rates of substitution. The optimal choice occurs where the budget line is tangent to the highest attainable indifference curve, balancing marginal utility per dollar across goods. This framework, central to neoclassical consumer theory, underscores how price changes or income variations shift the constraint, altering consumption patterns.^[56]^[57] Firms encounter production constraints, which limit output based on available inputs like labor and capital, often illustrated by isoquants—curves showing combinations of inputs yielding the same output level—and isocost lines representing equal-cost input bundles. The goal is to minimize costs for a target output, achieved at the tangency point where the marginal rate of technical substitution equals the input price ratio. A seminal illustration is the Cobb-Douglas production function, Q = A L^{\alpha} K^{1-\alpha}, where Q is output, L labor, K capital, A a technology parameter, and \alpha (between 0 and 1) the output elasticity of labor; this form exhibits constant returns to scale and has been empirically validated for manufacturing sectors, highlighting input substitutability under technological limits.^[58]^[59] Market constraints emerge from the interplay of supply and demand under scarcity, culminating in equilibrium where the quantity supplied equals quantity demanded at a clearing price. This intersection resolves excess supply or demand pressures, with scarcity amplifying price signals to ration resources efficiently. For instance, in competitive markets, binding resource limits prevent unlimited expansion, enforcing trade-offs that align individual optimizations with aggregate outcomes.^[60] Behavioral economics introduces nuances to these constraints by challenging the assumption of perfect rationality, positing that agents operate under bounded rationality due to limited information, cognitive capacity, and time. Pioneered by Herbert Simon, this concept describes how decision-makers "satisfice" rather than optimize, selecting satisfactory options within perceived constraints rather than exhaustively searching for the global best. Simon's framework, developed in the 1950s, explains deviations from ideal models in real-world choices, such as heuristic-based responses to budget or production limits, and has influenced analyses of economic behavior under uncertainty.

Constraints in Engineering Design

In engineering design, constraints serve as critical specifications that define the boundaries of feasible solutions, ensuring that systems meet requirements for functionality, safety, and manufacturability while operating within physical and practical limits. These constraints guide the iterative process of conceptualizing, analyzing, and refining designs to balance competing demands such as performance and cost. By imposing limits on variables like dimensions, materials, and operational parameters, engineers mitigate risks associated with failure modes, including structural collapse or inadequate performance under load.^[61] Design constraints are categorized into functional, geometric, and material types, each addressing specific aspects of the design space. Functional constraints pertain to performance requirements, such as maximum acceleration time or power output thresholds, ensuring the system fulfills its intended purpose without exceeding operational limits like battery discharge rates. Geometric constraints impose limits on size and shape, for instance, restricting component dimensions to fit within an assembly envelope or to comply with spatial restrictions in applications like aerospace structures. Material constraints focus on properties such as strength, durability, and compatibility, bounding selections to materials that withstand specified loads or environmental conditions, such as corrosion resistance in marine environments. These categories often intersect, requiring engineers to evaluate interactions that could amplify variations in assembly fit or stress distribution.^[61]^[62]^[63] A key challenge in engineering design involves trade-offs through multi-objective optimization under these constraints, where objectives like minimizing weight conflict with maintaining structural integrity. For example, in truss or beam designs, weight minimization is pursued subject to stress limits (σ ≤ σ_yield), where σ represents the induced stress and σ_yield the material's yield strength, often incorporating a safety margin to prevent yielding under peak loads. This approach generates Pareto-optimal solutions, allowing designers to select configurations that prioritize certain goals, such as reducing mass in automotive components while ensuring deflection stays below allowable thresholds. Such optimizations highlight the need for iterative evaluation to resolve conflicts, as lighter materials may necessitate thicker sections to satisfy stress constraints, impacting overall feasibility.^[64]^[65] Methodologies for handling constraints include tolerance analysis in manufacturing and finite element methods (FEM) for structural evaluation. Tolerance analysis assesses the cumulative effects of dimensional variations in assemblies, using techniques like worst-case or statistical stack-up to predict fit and function, ensuring geometric and material constraints are met without excessive scrap rates or rework. For instance, it quantifies how part tolerances propagate to assembly gaps, guiding tighter specifications where needed to maintain performance. FEM simulates structural behavior by discretizing components into elements, enabling prediction of stresses, deformations, and failure risks under load, thus verifying constraints like maximum allowable stress or buckling limits in complex geometries such as aircraft frames. These methods integrate with design software to iterate rapidly, refining parameters to satisfy multiple constraints simultaneously.^[66]^[65] Engineering standards from organizations like ASME and ISO formalize constraint application through safety factors in load-bearing designs, providing verifiable guidelines for reliability. The ASME Boiler and Pressure Vessel Code (BPVC) Section VIII mandates safety factors, such as 3.5 on ultimate tensile strength for pressure-retaining components, to ensure vessels withstand operational loads without rupture, incorporating margins for material variability and fabrication imperfections. Similarly, ISO 19902 for fixed steel offshore structures employs limit state design with material partial factors γ_m of approximately 1.0 to 1.15 on yield strength for ultimate limit states, combined with load factors such as 1.3 for environmental actions like waves, ensuring global stability in load-bearing elements.^[67] These standards promote uniformity, with safety factors calibrated to risk levels, enabling designs that exceed expected demands while minimizing over-conservatism.

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Although AC-2 appears more complex than AC-3 it is just a special case of the latter algorithm corresponding to the choice of a particular ordering of AC-3's.
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[PDF] Complexity of conservative Constraint Satisfaction Problems
The CSP is known to be NP-complete in general. However, certain restrictions on the form of the allowed constraints can lead to problems solvable in polynomial ...
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[PDF] Constraint Satisfaction - Washington
Tree-Structured CSPs. ▫ Theorem: if the constraint graph has no loops, the CSP can be solved in O(n d2) time! ▫ Compare to general CSPs, where worst-case ...
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[PDF] Map-Coloring Example - People @EECS
Example: Map-Coloring contd. General-purpose CSP algorithms use the graph structure to speed up search. E.g., Tasmania is an independent subproblem! Variables: ...
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Constraint Satisfaction and the N-Queens Problem
Nov 1, 2011 · Other examples are map coloring, where the variables are regions, the values are colors, and the constraint is that no adjacent regions can ...
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Mozart Programming System
The Mozart Programming System combines ongoing research in programming language design and implementation, constraint logic programming, distributed computing,Documentation · Mozart v1 · Publications · History
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8 Constraint Logic Programming - SWI-Prolog
This chapter describes the extensions primarily designed to support constraint logic programming (CLP), an important declarative programming paradigm.
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[PDF] CONSTRAINT LOGIC PROGRAMMING: A SURVEY
Constraint Logic Programming (CLP) is a merger of two declarative paradigms: constraint solving and logic programming. Although a relatively new field, CLP.<|control11|><|separator|>
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Constraint Propagation - ktiml
Forward checking checks only the constraints between the current variable and the future variables. So why not to perform full arc consistency that will further ...Missing: splitting | Show results with:splitting
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MiniZinc
MiniZinc is a high-level constraint modelling language that allows you to easily express and solve discrete optimisation problems.Downloads · Resources · Team · Publications
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Choco-solver
Choco is a Free Open-Source Java library dedicated to Constraint Programming. ... that need to be satisfied in every solution. Then, the problem is solved by ...Handling constraintsDocumentation
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GECODE - An open, free, efficient constraint solving toolkit
Gecode is an open source C++ toolkit for developing constraint-based systems and applications. Gecode provides a constraint solver with state-of-the-art ...
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Industrial-size job shop scheduling with constraint programming
A new (publicly available) job shop scheduling benchmark resembling the size of current industrial scheduling problems.
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FdConfig: A Constraint-Based Interactive Product Configurator - arXiv
Aug 29, 2011 · We present a constraint-based approach to interactive product configuration. Our configurator tool FdConfig is based on feature models for the representation ...
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[PDF] 14.01 F23 Lecture Summary 3: Budget Constraints
Budget over two goods X and Y is defined to be. I = pX X + pY Y. • The slope of budget constraint is defined as marginal rate of trans- formation (MRT): rate ...Missing: economics formula
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How Individuals Make Choices Based on Their Budget Constraint
The budget constraint framework suggests that when people make choices in a world of scarcity, they will use marginal analysis and think about whether they ...Missing: formula | Show results with:formula
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6.1 Consumption Choices – Principles of Economics - UH Pressbooks
In a budget constraint line, the quantity of one good is measured on the horizontal axis and the quantity of the other good is measured on the vertical axis.<|control11|><|separator|>
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[PDF] Chapter 2 - Cost Minimization
The production constraint, then pins down the factor input by indicating which point of tangency is the one in which we are interested. The production.
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A Theory of Production - jstor
Douglas has been responsible for sections 1-5 and 8-10, of this paper and. Mr. Cobb for sections 6 and 7. Page 2. 140 American Economic Association to whether ...
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3.1 Demand, Supply, and Equilibrium in Markets for Goods and ...
Demand is what consumers want to buy, supply is what producers want to sell. Equilibrium is where demand and supply are equal.
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Design Constraint - an overview | ScienceDirect Topics
Design constraints are defined as limitations imposed on the solution space of a system, affecting elements such as hardware, software, data, ...
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Functional Constraints - EngineeringTechnology.org
Functional constraints are those limitations that directly affect how a system operates, ensuring that the design fulfills its intended purpose.
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Constraints – Introduction to Mechanical Design and Manufacturing
Functional Analysis ... Force, strength, or stress constraints refer to the capacity of the material to resist deformation or mechanical failure under load.
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[PDF] Optimization Methods for Engineering Design - APMonitor
These criteria take two forms: objectives and constraints. Objectives represent goals we wish to maximize or minimize. Constraints represent limits we must stay ...
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[PDF] STRUCTURAL DESIGN USING FINITE ELEMENTS
Engineering design: a process of synthesis in which parts are put together to build a structure that will perform a given set of functions satisfactorily.
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What is Tolerance Analysis | Dimensional Control Systems - 3DCS
Tolerance analysis is the name given to a number of processes used to determine the overall variation and effect of variation on products.