Advanced Constraint Processing, Fall 2009 An Efficient Consistency Algorithm for the Temporal...

Advanced Constraint Processing, Fall 2009

An Efficient Consistency Algorithm for the Temporal

Constraint Satisfaction Problem

Berthe Y. Choueiry & Lin Xu, AI Comm. 2004

Presented by Shant KarakashianAdvanced Constraint Processing

CSCE 990 02, Fall 2009

19/8/2009


Outline

• Introduction• Solving the TCSP• Motivation: Constraint Propagation & Fragmentation• ∆AC Algorithm• Identifying the Triangles • ∆AC Initialization• Advantages of ∆AC• Experiment Results• Conclusion

29/8/2009


Introduction• Temporal relations in a problem can be represented

& reasoned about by modeling the problem as a Temporal Constraint Satisfaction Problem (TCSP)

• TCSP can be solved by assigning a value to each time point, such that all the constraints are simultaneously satisfied

• A TCSP can have exponential number of STPs, where each STP can be solved efficiently

39/8/2009


Outline

• Introduction• Solving the TCSP• Motivation: Constraint Propagation & Fragmentation• ∆AC Algorithm• Identifying the Triangles • ∆AC Initialization• Advantages of ∆AC• Experiment Results• Conclusion

49/8/2009


Solving TCSP [Dechter et al. 91]

• Dechter described a backtrack search procedure for determining the consistency of the TCSP.

• TCSP is modeled as a meta-CSP – With n vertices, e edges, and k intervals per edge– The edges in TCSP are the variables in meta-CSP– The labels of an edge in TCSP form the domain of the corresponding variable – The only constraint is a global constraint, stating that the edge-interval pairs

form a consistent STP

59/8/2009


Outline

• Introduction• Solving TCSP• Motivation: Constraint Propagation & Fragmentation• ∆AC Algorithm• Identifying the Triangles • ∆AC Initialization• Advantages of ∆AC• Experiment Results• Conclusion

69/8/2009


Constraint Propagation by Domain Filtering• Constraint propagation effectively enhances the performance of search

• The single constraint for the meta-CSP has size O(ke), which makes it unrealistic to apply GAC

• ∆AC remedies this difficulty by imposing a new ternary constraint over every existing triangle in the TCSP graph

8/25/2009 Topic 7

Arc consistency– Single n-ary constraint– GAC is NP-hard

AC– Works on existing triangles– Poly # of poly constraints


Fragmentation• Algorithms such as NPC-1 and NPC-2 modify intervals, which causes

fragmentation• AC removes values that are not supported by the ternary constraint

For every interval in the domain of an edge, there must existintervals in the domains of the 2 other edges such that the 3intervals verify the triangle inequality rule

8/25/2009 Topic 8

[1,3] in e3 has no support in e1 and e2

AC removes [1,3] from domain of e3

• ∆AC either keeps an interval as is or removes it, hence avoids fragmentation


Outline

• Introduction• Solving TCSP• Motivation: Constraint Propagation & Fragmentation• ∆AC Algorithm• Identifying the Triangles • ∆AC Initialization• Advantages of ∆AC• Experiment Results• Conclusion

99/8/2009


∆AC Algorithm (1)

• ∆AC keeps support tables (à la AC-4 / AC-2001)– Calls InitializeSupport

• Initializes the queue of (edge,interval) pairs• Builds 2 support tables: Supports, SupportedBy

– Loops through the queue, checking if a (edge-interval) pair has a support in all triangles where the edge appears

8/25/2009 Topic 10

A= {a1, a2,.., ak}


∆AC Algorithm (2)Given the triangle: XYZREVISE A={a1,a,2,.., ak}, finding a support for each interval

For ai in A

If FINDSUPPORT(A, ai) = false

Then DELETE ai from A

ENQUEUE the intervals supported by ai

FINDSUPPORT (A, ai)

For each interval bj in B, ck in C:

If (bj •ck) ai is not empty

Then ADD a to list of intervals supported by b j, ck

Return trueReturn false

11

Y

X Z

C ={c1,c2,..,ck }

B ={b1,b2,..,bk}

A= {a1, a2,.., ak}

• 9/8/2009

[1,3] in e3 has no support in e1 and e2


Outline

• Introduction• Solving TCSP• Motivation• Filtering intervals with ∆AC• Identifying the Triangles • ∆AC Initialization• Advantages of ∆AC• Experiment Results• Conclusion

129/8/2009


Identifying the Triangles

13

X Y

Z W

A={a1, a2}

C={c1, c2}

E={e1, e2}

D={d1, d2}

F = {[-∞,+∞)}

The interval i11 is an absorbing element: (-∞,+∞)• (-∞,+∞) composed with any interval is: (-∞,+∞)• (-∞,+∞) intersected with any interval x is x, not null

B={b1, b2}

9/8/2009


Outline


149/8/2009


∆AC: Initialization

15

(A, a1)(A ,a2)(B,b1)(B,b2)(C,c1)(C,c2)(D,d1)

(D,d2)

Q

X Y

Z W

A={a1, a2}

C={c1, c2}

B={b1, b2}

D={d1, d2}

d2 supports b1

c1 supports b1

B={b1}

C={c2}

(B,b1)

9/8/2009

b1 supported by d2

b1 supported by c1


Outline


169/8/2009


Advantages of AC

• Powerful, especially for dense TCSPs

• Sound and cheap O(n |E| k3)

• It may be optimal

– Uses polynomial-size data-structures:

Supports, Supported-by as in AC-4

17

[borrowed from lecture slides]

9/8/2009


Outline


189/8/2009


Reduction of meta-CSP’s size

19


9/8/2009


Effect of AC on #nodes visited

20


9/8/2009


Cumulative Improvement

21

Before, after AP, after NewCyc,… … and now (AC, STP, NewCyc, EdgeOrd)

Max on y-axis 5.000.000 Max on y-axis 18.000, 2 orders of magnitude improvement


9/8/2009


Outline


229/8/2009


Conclusion

• The paper proposes ∆AC that:– Defines ternary constraints on a TCSP– Propagates the constraints to filter the intervals in the

TCSP that are the domains of the variables in the meta-CSP

• ∆AC is used as a preprocessing step– Should be also used for look-ahead

• Experiments show that the algorithm is able to filter the domains tremendously specially as the density of the graph increases

239/8/2009

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