Csp Module

7/27/2019 Csp Module

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Constraint Satisfaction Problems

[Read chapter 5 of Russell & Norvig]


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Homework #2: Formulate the

Following Problem as a CSPGiven a set of n data points and a distance

function d(xi, xj) between all pairs ofpoints. Partition the data set into a twosubsets so that the maximum diameter ofthe two subsets is minimized.


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Constraint satisfaction problems

(CSPs) Simple example of a formal representation language

Later we'll define much more complex languages in PLand FOL. Trade-off b/w complexity and solvability

Standard search problem: state is a "black box any datastructure that supports successor function and goal test

state is defined by variablesXi with values from domainDi

goal test is a set of constraints specifying allowable combinationsof values for subsets of variables

Focus on useful general-purpose algorithms with morepower than standard search algorithms (BFS, DFS) anddon't require user generated heuristics like A*.


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Example: Map-Coloring

VariablesWA, NT, Q, NSW, V, SA, T

DomainsDi = {red,green,blue} Constraints: adjacent regions must have different colors e.g., WA NT


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Example: Map-Coloring

Solutions are complete and consistent assignments

e.g., WA = red, NT = green, Q = red, NSW =green,V = red,SA = blue,T = green


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Constraint graph

Binary CSP: each constraint relates two variables

Constraint graph: nodes are variables, arcs are constraints


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Varieties of CSPs

Discrete variables finite domains:

n variables, domain size d O(dn) complete assignments e.g., Boolean CSPs, incl. Boolean satisfiability (NP-complete)e

infinite domains: integers, strings, etc.

e.g., job scheduling, variables are start/end days for each job

need a constraint language, e.g., StartJob1 + 5 StartJob3

Continuous variables e.g., start/end times for Hubble Space Telescope observations

linear constraints solvable in polynomial time by LP


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Varieties of constraints

Unary constraints involve a single variable,

e.g., SA green

Binary constraints involve pairs of variables,

e.g., SA WA

Higher-order constraints involve 3 or more

variables,

Eventually we'll be able to specify constraints as a

sentence in a logic.


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Real-world CSPs

Assignment problems e.g., who teaches what class

Timetabling problems e.g., which class is offered when and where?

Transportation scheduling Factory scheduling

Notice that many real-world problems involvereal-valued variables

Related techniques involving constraints?

Why can't we use it for our homeworkproblem?


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Standard search formulation

(incremental)Let's start with the straightforward approach, then fix it

States are defined by the values assigned so far

Initial state: the empty assignment { } Successor function: assign a value to an unassigned variable that does

not conflict with current assignment fail if no legal assignments

Goal test: the current assignment is complete

This is the same for all CSPs Every solution appears at depth n with n variables use depth-first search

Path is irrelevant, so can also use complete-state formulation b = (n - l )d at depth l, hence n! dn leaves


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Backtracking search

Variable assignments are commutative, i.e.,[ WA = red then NT = green ] same as [ NT = green then WA = red ]

=> Only need to consider assignments to a single variable ateach node

Depth-first search for CSPs with single-variable assignments iscalled backtracking search

Can solve n-queens for n 25


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Behavior of this algorithm for n=4n-queens problem


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14


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Backtracking example


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What are the possible improvements to basic BT search?

What guiding principles/aims can we use to help answer this question?


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Improving backtracking efficiency

General-purpose methods can give huge

gains in speed:

Which variable should be assigned next?

In what order should its values be tried?

Can we detect inevitable failure early?


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When Building The Tree How

Should We Order Variables?

What should we assign next, SA, QLD,

NSW, VIC, TAS? Why?


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Most constrained variable

Most constrained variable:

choose the variable with the fewest legal values

a.k.a. minimum remaining values (MRV)

heuristic


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Will the MRV Heuristic Help Us

Choose the Root of the Tree? If not, whats a reasonable method to choose

and why?


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Most constraining variable

Tie-breaker among most constrained

variables

Most constraining variable:

choose the variable with the most constraints on

remaining variables


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Okay. We How About Choosing

What Values to Explore First? If we are expanding QLD which value should we

instantiate first?


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Least constraining value

Given a variable, choose the leastconstraining value:

the one that rules out the fewest values in the

remaining variables

Combining these heuristics makes 1000

queens feasible.

How is this philosophically different to our

heuristic search with A*


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What was nice about A*'s behavior?


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So how can we do pruning in this context?


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Forward checking

Idea:

Keep track of remaining legal values for unassigned variables

Terminate search when any variable has no legal values


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Forward checking

Idea:




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Forward checking

Idea:




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Forward checking

Idea:



Using these heuristics in combination. What node ordering

heuristic is a natural partner to forward checking?


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Forward Checking is an Example ofConstraint Propagation.


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Constraint propagation Forward checking propagates information from assigned to

unassigned variables, but doesn't provide early detection for

all failures:

NT and SA cannot both be blue!

Constraint propagation repeatedly enforces constraints

locally


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In your Class Groups of Three

Use the allocated technique: minimum remaining values, most constrainedvariable, least constraining value or forward checking.

Which is the odd one out and why?

Show the generated search tree.


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Arc consistency

Simplest form of propagation makes each arc consistent

XYis consistent iff

for every valuex ofXthere is some allowedy


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Arc consistency Simplest form of propagation makes each arc consistent

XYis consistent iff



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XYis consistent iff


IfXloses a value, neighbors ofXneed to be rechecked


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XYis consistent iff


IfXloses a value, neighbors ofXneed to be rechecked

Arc consistency detects failure earlier than forward checking

Can be run as a preprocessor or after each assignment


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So what if we find a nodes xi and xj are not arcconsistent what should we do?


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Arc consistency algorithm AC-3

Time complexity: O(n2d3)

Checking consistency of an arc is O(d2)


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I l CSP f l i


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In your class groups use a CSP formulation tosolve this problem.

Given a set of n data points and a distancefunction d(xi, xj) between all pairs ofpoints. Partition the data set into a twosubsets so that the maximum diameter ofthe two subsets is minimized.

Hint: Construct a simple example to verifyyour algorithm.

Is the problem solvable in polynomial timewith respect to the number of instances?

Core Decision Problem: Boottleneck Diameter Problem (BDP):


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Core Decision Problem: Boottleneck Diameter Problem (BDP):

Given: Point set P = \{p_1, p_2, ..., p_n\}, the distance d(p_i,p_j) for each pair of points, a distance \alpha.

Question: Is it possible to partition P into two sets P_1 andP_2 such that dia(P_1)


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Putting It All Together: Sudoku

85

14783

35179

325

1926

849

42853

95162

39


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Sudoku

http://www.websudoku.com/

Each Sudoku has a unique solution that can bereached logically without guessing. Enter digitsfrom 1 to 9 into the blank spaces. Every row must

contain one of each digit. So must every column,as must every 3x3 square.
http://www.websudoku.com/http://en.wikipedia.org/wiki/Sudokuhttp://en.wikipedia.org/wiki/Sudokuhttp://www.websudoku.com/


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Problem Formulation As Search

What is the state space? What is the initial state? What is the successor function?

What is the goal test? What is the path cost? Admissible heuristic?

Why does using these search techniques notmake sense for this problem?


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Points

Large search space

commutativity

what about using bfs or dfs?fixed depth


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CSP as a Search ProblemCSP as a Search Problem

Initial state: empty assignment

Successor function: a value is assigned to any unassignedvariable, which does not conflict with the currently assigned

variables Goal test: the assignment is complete

Path cost: irrelevant

Branching factor b at the top level is nd. b=(n-l)d at depth l, hence n!dn leaves (only dn complete

assignments).

We can do better.


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What else is needed?What else is needed?

Not just a successor function and goal test But also a means to propagate the

constraints imposed by one move on theothers and an early failure test

Explicit representation of constraints and

constraint manipulation algorithms


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Example: Sudoku CSPExample: Sudoku CSP

variables:

domains:

constraints:

Goal: Assign a value to every variable such that allconstraints are satisfied


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Example: Sudoku CSPExample: Sudoku CSP

variables: X11, , X99

domains: {1,,9}

constraints: row constraint: X11 X12, , X11 X19

col constraint: X11 X12, , X11 X19

block constraint: X11 X12, , X11 X33

How do we encode existing board values as constraints?

Goal: Assign a value to every variable such that allconstraints are satisfied

G N D A h


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Great. Now Do AnotherFormulation using Propositional

Logic s_{xyz} is the proposition that tile locationx,y value is z

Why is it so important to state our problem inpropositional or first order calculus?

G N D A h


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Great. Now Do AnotherFormulation using Propositional

Logic


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Other techniques for CSPs

k-consistency

Tradeoff between propagation and branching

Symmetry breaking


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Structured CSPs

Tree-structured CSPs Removes


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Tree structured CSPs Removes

The Need For Back-tracking


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Algorithm for tree-structured CSPs

Ne rl tree str ct red CSPs


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Nearly tree-structured CSPs

(Finding the minimum cutset is NP-complete.)

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