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In this post I will try to do a high level explanation of the CP-SAT solver.

References:

  • Laurent Perron and FrΓ©dΓ©ric Didier, CPAIOR 2020: https://youtu.be/lmy1ddn4cyw
  • Peter J. Stuckey, Search is Dead: https://people.eng.unimelb.edu.au/pstuckey/PPDP2013.pdf

Model Building

The first step is building the model using the CPModel class. This class is actually a wrapper around the cp_model protobuf.

Let’s see an example (source):

from ortools.sat.python import cp_model

"""Minimal CP-SAT example to showcase calling the solver."""
# Creates the model.
# [START model]
model = cp_model.CpModel()
# [END model]

# Creates the variables.
# [START variables]
num_vals = 3
x = model.NewIntVar(0, num_vals - 1, 'x')
y = model.NewIntVar(0, num_vals - 1, 'y')
z = model.NewIntVar(0, num_vals - 1, 'z')
# [END variables]

# Creates the constraints.
# [START constraints]
model.Add(x != y)
# [END constraints]

# Creates a solver and solves the model.
# [START solve]
solver = cp_model.CpSolver()
status = solver.Solve(model)
# [END solve]

if status == cp_model.FEASIBLE:
    print('x = %i' % solver.Value(x))
    print('y = %i' % solver.Value(y))
    print('z = %i' % solver.Value(z))

This model creates the following proto print(str(model)):

variables {
  name: "x"
  domain: 0
  domain: 2
}
variables {
  name: "y"
  domain: 0
  domain: 2
}
variables {
  name: "z"
  domain: 0
  domain: 2
}
constraints {
  linear {
    vars: 1
    vars: 0
    coeffs: -1
    coeffs: 1
    domain: -9223372036854775808
    domain: -1
    domain: 1
    domain: 9223372036854775807
  }
}

Note: int64 is [-9223372036854775808, 9223372036854775807]

Presolve Loop

First stage: We will process all active constraints until a fix point is reached. During this stage:

  • Variable will never be deleted, but their domain will be reduced.
  • Constraint will never be deleted (they will be marked as empty if needed).
  • New variables and new constraints can be added after the existing ones.
  • Constraints are added only when needed to the mapping_problem if they are needed during the postsolve.

Second stage:

  • All the variables domain will be copied to the mapping_model.
  • Everything will be remapped so that only the variables appearing in some constraints will be kept and their index will be in [0, num_new_variables). - source
  1. Presolve: Domain reduction, constraint simplification/rewrite
  2. Constraint expansion/decomposition: Similar to Minizinc -> Flatzinc (element constraint, table, automaton, inverse, product, modulo, reservoir)
  3. Detect variable equivalence and affine relations.
  4. Substitute by canonical representation.
  5. Probing: Fix variables and see what is propagated.

This produces 2 new models, the inner model that will be solved and a channeling model used to populate the solution of the initial model. - source

Solver

The CP-SAT solver uses a lazy clause generation solver on top of an SAT solver. The best description is a presentation from Peter Stuckey called Search is Dead - Laurent Perron

In Lazy clause generation (LCG), integer variables are encoded as booleans, ortools creates 2 booleans for each variable and value:

  • var == value
  • var <= value

Note: var >= value (represented as ![x <= value-1])

  • (var == value) <=> (var >= value) and (var <= value)
  • (var <= value) => (var <= value+1)

Propagation is clause generation:

  • e.g. [x <= 2] and x >= y means that [y <= 2]
  • clause [x <= 2] => [y <= 2]

Linear relaxation

TODO

Default search (single thread)

VSIDS on the Boolean problem, when it reaches a fixed point, it asks the heuristic to select an integer variable, a value and a braching direction.

Multithreading

The solver uses the first X threads to generic methods, and use all the remaining ones on LNS (Large Neighborhood Search). -Laurent Perron

  • Default SAT Search
  • Fixed search or LP Branching
  • PSEUDO_COST_SEARCH (follow last best solution when branching)
  • No linear relaxation (good for big models where CP-SAT propagation is enough)
  • Max linear relaxation (gives good lower bounds)
  • Core Based search
  • LNS for remaining workers

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