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Solving constraint systems

Constraint solving systems are an algorithmic approach to solving Linear Programming problems. A linear programming problem is a mathematical problem where you have a set of non- negative, real valued variables (x[1], x[2], ... x[n]), and a series of linear constraints (i.e, no exponential terms) on those variables. These constraints are expressed as a set of equations of the form:

a[1]x[1] + ... + a[n]x[n] = b,

a[1]x[1] + ... + a[n]x[n] <= b, or

a[1]x[1] + ... + a[n]x[n] >= b,

Given these constraints, the problem is to find the values of x[i] that minimize or maximize the value of an objective function:

c + d[1]x[1] + ... + d[n]x[n]

Cassowary is an algorithm designed to solve linear programming problems of this type. Published in 1997, it now forms the basis for the UI layout tools in OS X Lion, and iOS 6+ (the approach known as Auto Layout). The Cassowary algorithm (and this implementation of it) provides the tools to describe a set of constraints, and then find an optimal solution for that set of constraints.

Variables

At the core of the constraint problem are the variables in the system. In the formal mathematical system, these are the x[n] terms; in Python, these are rendered as instances of the :class:`~cassowary.Variable` class.

Each variable is named, and can accept a default value. To create a variable, instantiate an instance of :class:`~cassowary.Variable`:

from cassowary import Variable

# Create a variable with a default value.
x1 = Variable('x1')

# Create a variable with a specific value
x2 = Variable('x1', 42.0)

Any value provided for the variable is just a starting point. When constraints are imposed, this value can and will change, subject to the requirements of the constraints. However, providing an initial value may affect the search process; if there's an ambiguity in the constraints (i.e., there's more than one possible solution), the initial value provided to variables will affect the solution on which the system converges.

Constraints

A constraint is a mathematical equality or inequality that defines the linear programming system.

A constraint is declared by providing the Python expression that encompasses the logic described by the constraint. The syntax looks essentially the same as the raw mathematical expression:

from cassowary import Variable

# Create a variable with a default value.
x1 = Variable('x1')
x2 = Variable('x2')
x3 = Variable('x4')

# Define the constraint
constraint = x1 + 3 * x2 <= 4 * x3 + 2

In this example, constraint holds the definition for the constraint system. Although the statement uses the Python comparison operator <=, the result is not a boolean. The comparison operators <=, <, >=, >, and == have been overridden for instances of :class:`~cassowary.Variable` to enable you to easily define constraints.

Solvers

The solver is the engine that resolves the linear constraints into a solution. There are many approaches to this problem, and the development of algorithmic approaches has been the subject of math and computer science research for over 70 years. Cassowary provides one implementation -- a :class:`~cassowary.SimplexSolver`, implementing the Simplex algorithm defined by Dantzig in the 1940s.

The solver takes no arguments during constructions; once constructed, you simply add constraints to the system.

As a simple example, let's solve the problem posed in Section 2 of the Badros & Borning's paper on Cassowary. In this problem, we have a 1 dimensional number line spanning from 0 to 100. There are three points on it (left, middle and right), with the following constraints:

  • The middle point must be halfway between the left and right point;
  • The left point must be at least 10 to the left of the right point;
  • All points must fall in the range 0-100.

This system can be defined in Python as follows:

from cassowary import SimplexSolver, Variable

solver = SimplexSolver()

left = Variable('left')
middle = Variable('middle')
right = Variable('right')

solver.add_constraint(middle == (left + right) / 2)
solver.add_constraint(right == left + 10)
solver.add_constraint(right <= 100)
solver.add_constraint(left >= 0)

There are an infinite number of possible solutions to this system; if we interrogate the variables, you'll see that the solver has provided one possible solution:

>>> left.value
90.0
>>> middle.value
95.0
>>> right.value
100.0

Stay constraints

If we want a particular solution to our left/right/middle problem, we need to fix a value somewhere. To do this, we add a Stay - a special constraint that says that the value should not be altered.

For example, we might want to enforce the fact that the middle value should stay at a value of 45. We construct the system as before, but add:

middle.value = 45.0
solver.add_stay(middle)

Now when we interrogate the solver, we'll get values that reflect this fixed point:

>>> left.value
40.0
>>> middle.value
45.0
>>> right.value
50.0

Constraint strength

Not all constraints are equal. Some are absolute requirements - for example, a requirement that all values remain in a specific range. However, other constraints may be suggestions, rather than hard requirements.

To accommodate this, Cassowary allows all constraints to have a strength. Strength can be one of:

  • REQUIRED
  • STRONG
  • MEDIUM
  • WEAK

REQUIRED constraints must be satisfied; the remaining strengths will be satisfied with declining priority.

To define a strength, provide the strength value as an argument when adding the constraint (or stay):

from cassowary import SimplexSolver, Variable, STRONG, WEAK

solver = SimplexSolver()
x = Variable('x')

# Define some non-required constraints
solver.add_constraint(x <= 100, strength=STRONG)
solver.add_stay(x, strength=WEAK)

Unless otherwise specified, all constraints are REQUIRED.

Constraint weight

If you have multiple constraints of the same strength, you may want to have a tie-breaker between them. To do this, you can set a weight, in addition to a strength:

from cassowary import SimplexSolver, Variable, STRONG

solver = SimplexSolver()
x = Variable('x')

# Define some non-required constraints
solver.add_constraint(x <= 100, strength=STRONG, weight=10)
solver.add_constraint(x >= 50, strength=STRONG, weight=20)

Editing constraints

Any constraint can be removed from a system; just retain the reference provided when you add the constraint:

from cassowary import SimplexSolver, Variable

solver = SimplexSolver()
x = Variable('x')

# Define a constraint
constraint = solver.add_constraint(x <= 100)

# Remove it again
solver.remove_constraint(constraint)

Once a constraint is removed, the system will be automatically re-evaluated, with the possible side effect that the values in the system will change.

But what if you want to change a variable's value without introducing a new constraint? In this case, you can use an edit context.

Here's an example of an edit context in practice:

from cassowary import SimplexSolver, Variable

solver = SimplexSolver()
x = Variable('x')

# Add a stay to x - that is, don't change the value.
solver.add_stay(x)

# Now, mark x as being editable...
solver.add_edit_variable(x)

# ... start and edit context...
with solver.edit():
    # ... and suggest a new value for the variable.

    solver.suggest_value(x, 42.0)

When the edit context exits, the system will re-evaluate itself, and the variable will have the new value. However, the variable isn't guaranteed to have the value you suggested - in this case it will, but if your constraint system has other constraints, they may affect the value of the variable after the suggestion has been applied.

All variables in the system will be re-evaluated when you leave the edit context; however, if you need to force a re-evaluation in the middle of an edit context, you can do so by calling :meth:`~cassowary.Solver.resolve()`.