Numberjack
Modelling, reformulation, and solving
Barry Hurley
barry.hurley@insight-centre.org
ModRef 2014
Outline
Introduction to Numberjack
Modelling Constructs
solve() and friends
Example Application and Demos
Summary
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September 8, 2014
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Numberjack Overview
What is Numberjack?
• A platform for combinatorial optimization
• Open source project (Github, LGPL)
• Python
• Common platform for diverse paradigms (CP, SAT, MIP)
• Fast backend solvers
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September 8, 2014
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CP, SAT, MIP: choice is good!
Numberjack
• Write a single model in Python
• Numberjack constructs: union of SAT, MIP, and CP
• Encoded depending on the choice of solver
Operationally different
• Algorithms are different
• CP: Constraint propagation + Search
• SAT: Unit propagation + Clause Learning + Search
• MIP: Linear relaxation + Cutting planes + Branch & Bound
• Encodings matter
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Backend Solvers Available
CP Solvers
MIP Solvers
• Mistral v1 & v2
• IBM ILOG CPLEX
• Toulbar2
• Gurobi
SAT Solvers
• MiniSat
• Walksat
• Lingeling
• CryptoMiniSat
• Glucose
• SCIP
• Through COIN-OR Open
Solver Interface:
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COIN-OR CBC
COIN-OR CLP
COIN-OR DyLP
COIN-OR SYMPHONY
COIN-OR Volume Algorithm
GNU LP Toolkit
Soplex
• . . . any CNF solver
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Installation
Python Package Index (PyPI)
pip install Numberjack
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Section 2
Modelling Constructs
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Variables in Numberjack
Different Variable constructors
Constructor
Variable()
Variable(’x’)
Variable(N)
Variable(N, ’x’)
Variable(l,u)
Variable(l,u, ’x’)
Variable(list)
Variable(list, ’x’)
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Description
Boolean variable
Boolean variable called ’x’
Variable in the domain of [0 . . . N − 1]
Variable in the domain of [0 . . . N − 1] called ’x’
Variable in the domain of [l . . . u]
Variable in the domain of [l . . . u] called ’x’
Variable with domain specified as a list
Variable with domain as a list called ’x’
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More Variables in Numberjack
Similarily for arrays and matrices
Constructor
VarArray(N)
VarArray(N, u)
VarArray(N, l, u)
..
.
Description
Array of N Boolean variables
Array of N variables with domains [0 . . . u − 1]
Array of N variables with domains [l . . . u]
Matrix(N, M)
Matrix(N, M, u)
Matrix(N, M, l, u)
N × M matrix of Boolean variables
N × M matrix of variables with domains [0 . . . u − 1]
N × M matrix of variables with domains [l . . . u]
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Concise Models
VarArray and Matrix allow for concise modelling.
Slices
Given a 9 × 9 Sudoku matrix, we
could get a 3 × 3 cell:
AllDiff( matrix[x:x+3, y:y+3] )
Rows and Columns
for row in matrix.row:
m += AllDiff(row)
for col in matrix.col:
m += AllDiff(col)
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Constraints in Numberjack
Constraints can be specified in a number of ways
Simply by arithmetic operators on variables:
x
y
x
x
> y
<= z
!= z
+ y == z
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x + 4 > z * 3
z == (x < y)
z <= (x < y)
(x == y) != (a == b)
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Global Constraints
v == Max([x,y,z])
Disjunction([x < y, y > z, a != b])
AllDiff([x,y,z])
Sum([a,b,c,d]) >= e
Sum([a,b,c,d], [2, 1, 0.5, 3]) == e # Weighted Sum
Sum([2∗a, b, 0.5∗c, 3∗d]) == e # Weighted Sum
LessLex(VarArray(5, 5), VarArray(5, 5))
LeqLex(VarArray(5, 5), VarArray(5, 5))
Gcc(VarArray(4, 1, 5), {1: [1, 3], 2: [1, 2]})
Element(VarArray(10, 1, 10), Variable(10))
myarray[myvariable] # Element
Minimize(2∗x + 3∗y)
Maximize(objective)
PostUnary, PostBinary, PostNary, ... # Cost functions
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Custom Constraints
Easy to define your own custom constraints for modelling purposes
and add custom decompositions for solvers which don’t support it.
Example
class MyAllDiff(Expression):
def decompose(self):
return [var1 != var2 for var1, var2 in pair_of(self.children)]
class NoOverlap(Predicate):
def decompose(self):
task_i, task_j = self.children
return [((task_i + task_i.duration) <= task_j) |
((task_j + task_j.duration) <= task_i)]
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Custom Decompositions
Override the default decomposition
simply by defining your own method:
Example
def decompose_AllDiff(self):
return [var1 != var2 for var1, var2 in pair_of(self.children)]
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Section 3
solve() and friends
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One Model, Many Solvers
Constraint Solvers
mistral = model.load("Mistral")
toulbar = model.load("Toulbar2")
MIP Solvers
gurobi = model.load("Gurobi")
cplex = model.load("CPLEX")
SAT Solvers
minisat = model.load("MiniSat")
walksat = model.load("Walksat")
lingeling = model.load("Lingeling")
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Customizable SAT Encoding
Change the default encoding globally
model.load(solver, encoding=...)
direct
support
p cnf
1 2 3
-1 -2
-1 -3
-2 -3
4 5 6
-4 -5
-4 -6
-5 -6
7 8 9
-7 -8
-7 -9
-8 -9
-1 -4
-2 -5
-3 -6
-1 -7
-2 -8
-3 -9
-4 -7
-5 -8
-6 -9
p cnf 9 30
1 2 3 0
-1 -2 0
-1 -3 0
-2 -3 0
4 5 6 0
-4 -5 0
-4 -6 0
-5 -6 0
7 8 9 0
-7 -8 0
-7 -9 0
-8 -9 0
5 6 -1 0
4 6 -2 0
4 5 -3 0
2 3 -4 0
1 3 -5 0
1 2 -6 0
8 9 -1 0
7 9 -2 0
...
9 21
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
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order
regular
p cnf 6 12
-1 2 0
-3 4 0
-5 6 0
-1 -3 0
1 3 -2 -4 0
2 4 0
-1 -5 0
1 5 -2 -6 0
2 6 0
-3 -5 0
3 5 -4 -6 0
4 6 0
p cnf 15 27
1 2 3 0
-4 5 0
-1 4 0
-2 -4 0
-2 5 0
-3 -5 0
6 7 8 0
-9 10 0
-6 9 0
-7 -9 0
-7 10 0
-8 -10 0
11 12 13 0
-14 15 0
-11 14 0
-12 -14 0
-12 15 0
-13 -15 0
-1 -6 0
-2 -7 0
...
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Customizable SAT Encoding
Pure direct/sparse domain encoding
x = Variable(3)
x.encoding = EncodingConfiguration(direct=True, order=False)
x0 ∨ x1 ∨ x2
¬x0 ∨ ¬x1
¬x0 ∨ ¬x2
¬x1 ∨ ¬x2
Regular domain encoding
y0 ∨ y1 ∨ y 2
¬y≤0 ∨ y≤1
y = Variable(3)
y.encoding = EncodingConfiguration(direct=True, order=True)
¬y0 ∨ y≤0
¬y1 ∨ ¬y≤0
¬y1 ∨ y≤1
¬y2 ∨ ¬y≤1
Full regular domain encoding
z = Variable(3)
z.encoding = EncodingConfiguration(direct=False, order=True)
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¬z≤0 ∨ z≤1
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Customizable SAT Encoding
Per constraint
neq.encoding = EncodingConfiguration(
conflict=False, support=True, order=False)
¬x0 ∨ y1 ∨ y2
¬x1 ∨ y0 ∨ y2
¬x2 ∨ y0 ∨ y1
...
Multiple encodings
con = AllDiff(variables)
con.encoding = EncodingConfiguration(alldiff_encoding=
AllDiffEncoding.PairwiseDecomp |
AllDiffEncoding.LadderAMO |
AllDiffEncoding.PigeonHole)
Named presets
con.encoding = NJEncodings[’pairwise_ladder_pigeon’]
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Configuring the Solver
# Search strategies (solver specific)
mistral.setHeuristic("DomainOverWDegree", "Lex")
mistral.setHeuristic("Impact")
# Set Limits
solver.setTimeLimit(60) # Seconds
solver.setNodeLimit(1000000)
# Solve
solver.solve()
# Solve using custom restart parameters
mistral.solveAndRestart(GEOMETRIC, 64, 1.3)
mistral.solveAndRestart(LUBY, 1000)
# Find all solutions
solver.getNextSolution()
# Feasibility?
solver.is_sat()
solver.is_opt()
solver.is_unsat()
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Custom Search: Full binary DFS
solver.startNewSearch()
while not proven_infeasibility:
if solver.propagate(): # left branch
x = variableselection(variables)
if x is None:
# Complete, no more variables to assign
return solver.get_solution()
v = x.get_min()
solver.save()
solver.post(x == v)
else: # right branch
proven_infeasibility = not solver.undo()
if not proven_infeasibility:
# Invert the previous decision, i.e. x != v
solver.deduce()
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Model Output
Output LP model
cplex = model.load("CPLEX")
cplex.output_lp("mymodel.lp")
Output MPS model
gurobi = model.load("Gurobi")
gurobi.output_mps("mymodel.mps")
Output CNF model
minisat = model.load("MiniSat")
minisat.output_cnf("mymodel.cnf")
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External Input Formats
XCSP
from Numberjack.XCSP import XCSPParser
parser = XCSPParser(filename)
model, variables = parser.model, parser.variables
mzn/fzn
# Just run Numberjack solvers directly
mzn_numberjack model.mzn data.dzn
fzn_numberjack flattened.fzn
# Translate to a Numberjack/Python file
mzn2py model.mzn data.dzn > model.py
Other custom parsers are easy in Python
cnf, pls, uai, . . .
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‘import antigravity’
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Section 4
Example Application and Demos
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Optical Network Design
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Demo: Map Colouring
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Demo: Controlling Search
Demo: controlling search
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Summary
What is Numberjack?
•
•
•
•
•
•
A Python platform for combinatorial optimization
Common platform for diverse paradigms (CP, SAT, MIP)
Practical, easy to use, and intuitive
Highly customizable encodings
Fast backend solvers
Open source (Github, LGPL)
http://numberjack.ucc.ie
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Acknowledgements
This work is supported by Science Foundation Ireland (SFI) Grant
10/IN.1/I3032 and FP7 FET-Open Grant 284715.
The Insight Centre for Data Analytics is supported by SFI Grant
SFI/12/RC/2289.
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September 8, 2014
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