constraint programming peter van beek university of waterloo
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Constraint Programming
Peter van Beek
University of Waterloo
Applications
Reasoning tasks: abductive, diagnostic, temporal, spatial
Cognitive tasks: machine vision, natural language processing
Combinatorial tasks: scheduling, sequencing, planning
Constraint Programming
CP = solve problems by specifying
constraints on acceptable solutions
Why CP? constraints often a natural part of problems once problem is modeled using constraints, wide
selection of solution techniques available
Constraint-based problem solving
Model problem specify in terms of constraints on acceptable solutions define variables (denotations) and domains define constraints in some language
Solve model define search space / choose algorithm
– incremental assignment / backtracking search
– complete assignments / stochastic search
design/choose heuristics
Verify and analyze solution
Model problem specify in terms of constraints on acceptable solutions define variables (denotations) and domains define constraints in some language
Solve model define search space / choose algorithm
– incremental assignment / backtracking search
– complete assignments / stochastic search
design/choose heuristics
Verify and analyze solution
Constraint-based problem solving
ConstraintSatisfactionProblem
Constraint satisfaction problem
A CSP is defined by a set of variables a domain of values for each variable a set of constraints between variables
A solution is an assignment of a value to each variable that
satisfies the constraints
Model problem specify in terms of constraints on acceptable solutions define variables (denotations) and domains define constraints in some language
Solve model define search space / choose algorithm
– incremental assignment / backtracking search
– complete assignments / stochastic search
design/choose heuristics
Verify and analyze solution
Constraint-based problem solving
Example: Assembly line sequencing
What order should the cars be manufactured?
Constraints: even distributions changes in colors run length constraints
Example: Scheduling
Algy Bertie Charlie Digby1st FT 60 Guardian 75 Express 5 Sun 902nd Guardian 30 Express 3 Guardian 15 FT 13rd Express 2 FT 25 FT 10 Guardian 14th Sun 5 Sun 10 Sun 30 Express 1
What is the earliest that they can all set of for school?
Four students, Algy, Bertie, Charlie, and Digby share a flat.Four newspapers are delivered.Each student reads the newspapers in a particular order and for a specified amount of time.Algy arises at 8:30, Bertie and Charlie at 8:45, Digby at 9:30.
Schedule
Sun
8 am 9 10 11
Express
Guardian
FT
Example: Graph coloring
Given k colors, does there exist a coloring of the nodes such that adjacent nodes are assigned different colors
Example: 3-coloring
Variables: v1, v2 , v3 , v4 , v5
Domains: {1, 2, 3}
Constraints: vi vj if vi and vj
are adjacent
v2
v3
v1
v5
v4
Example: 3-coloring
One solution:
v1 1
v2 2
v3 2
v4 1
v5 3
v2
v3
v1
v5
v4
Example: Boolean satisfiability
Given a Boolean formula, does there exist a satisfying assignment (an assignment of true or false to each variable such that the formula evaluates to true)
Example: 3-SAT
Variables: x1, x2 , x3 , x4 , x5
Domains: {True, False}
Constraints: (x1 x2 x4), (x2 x4 x5), (x3 x4 x5)
(x1 x2 x4) (x2 x4 x5) (x3 x4 x5)
Example: 3-SAT
One solution:x1 False
x2 False
x3 False
x4 True
x5 False
(x1 x2 x4) (x2 x4 x5) (x3 x4 x5)
Example: n-queens
Place n-queens on an n n board so that no pair of queens attacks each other
Example: 4-queens
4
3
2
1
x1 x2 x3 x4Variables: x1, x2 , x3 , x4
Domains: {1, 2, 3, 4}
Constraints: xi xj and | xi - xj | | i - j |
Example: 4-queens
One solution:
x1 2
x2 4
x3 1
x4 3 4
3
2
1
x1 x2 x3 x4
Q
Q
Q
Q
Search tree for 4-queens
x1
x2
x3
x4
1 2 3 4
(1,1,1,1) (4,4,4,4)(2,4,1,3) (3,1,4,2)
Specification of forward checking
Invariant:1 i c,c j n,
(xj, xi) is arc-consistent
x1 xc-1 xc xc+1 xn
currentpast future
Forward checking
{2} {5}Q
Q
Q
1
1
11
1
1
1
1
1
2
2
2
22
2
3
3 3
33
{3}
{1,4,6}
{1,3,4}
{3,4,6}
x1 x5
x6
x3
x4
x2
4321 5 6 x1
x5
x6
x3
x4
x2
1
Enforcing arc-consistency
{a, b, c} {a, b, c}<
xi xj
Enforcing path-consistency
<
{a, b, c}
xi
{a, b, c}
xj
{a, b, c}
xk
<
<
Search graph for 4-queens
(1,1,1,1)(1,1,4,2)
(2,4,1,3)
(3,1,4,2)
(1,1,1,2)
(1,1,1,3)
(1,1,1,4)(1,4,1,3)
(4,1,4,2)
0
0
4
6
1
3
41
1
Stochastic search
Q
Q
Q x1
Q
Initial assignment
x2
x3
x4
1 12
Pick queen in conflict
Q
Q
Q x1
Q
x2
x3
x4
Move to min. conflicts
Q
Q
Q x1
Q
x2
x3
x4
0 2
Pick queen in conflict
Q
Q
Q x1
Q
x2
x3
x4
2
Tractability
NP
NP-Complete
P
(SAT, TSP, ILP, CSP, …)
Reducibility
NP-Complete
3-SAT
ILP
CSP binary CSP
(0,1)-ILP
CSP, binary CSP, SAT, 3-SAT, ILP, ...
Model and solve in one of these languages Model in one language, translate into
another to solve
Options
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