foundations of constraint processing csce421/821, spring 2008:

Foundations of Constraint Processing, Spring 2008

April 16, 2008 Tree-Structured CSPs 1

Foundations of Constraint Processing

CSCE421/821, Spring 2008:

www.cse.unl.edu/~choueiry/S08-421-821/

Berthe Y. Choueiry (Shu-we-ri)

Avery Hall, Room 123B

[email protected]

Tel: +1(402)472-5444

Tree-Based Methods


April 16, 2008

Outline

• Backtrack-Free Search- Principle- Applications

- Backtrack-Bounded Search- Principle- Applications, extensions

Tree-Structured CSPs 2


April 16, 2008

Backtrack-Free Search [Freuder, 1982]

• A CSP can be solved in a backtrack-free manner when it is strong w+1-consistent, where w is the width of the constraint network– Compute w the width of the graph– Reinforce strong w-consistency– May add arcs to the graph, increasing width and

requiring higher-level of strong consistency, etc.

• Approach is of little practical use• Except for trees, width = 1



April 16, 2008

Tree-structured CSPs

• Trees have w = 1

• Enforcing 2-consistency does not alter the width

• Tree-structured CSPs can be solved in polynomial time– Apply Revise(Vi,Vj) for all nodes from leaves

to root– Instantiate variables from root to leaves



April 16, 2008

Exploiting tree structures

• Cycle-Cutset Method – Dechter and Pearl 1997– Dechter Section 10.1.1 pages 273—276

• Independent Set Decomposition– Gompert, FLAIRS 2005

• Graph Reduction (GRED) – Unpublished work by Yaling Zheng 2007



April 16, 2008

Cycle-Cutset Method (1)• Identify a cycle cutset S in the CSP (nodes when

removed yield a tree)• Decompose the CSP into 2 partitions

– The nodes in S– The nodes in T, forming a tree

• Idea– Solve the nodes in S– Try to extend the solution to nodes in T– Iterate



April 16, 2008

Cycle-Cutset Method (2)

• Find a solution to nodes in S (S is smaller than initial problem)

• Repeat until you find a solution – For every solution to S

• Apply DAC from S to T• If no domain is wiped-out, solve T (quick)

• If |S|=c, time is O(dc.(n-c)d2)=O(ndc+2)

• Finding the smallest cutset is NP-hard



April 16, 2008

GRED [Yaling Zheng]

• After each assignment and FC/MAC step– Check the connectivity of the remaining CSP– Identify “dangling trees” using Graham’s graph

reduction operator

• For each dangling tree, – Do DAC from leaf to root – Domain wipe-out indicates unsolvability

• Restrict search to nodes outside the identified dangling trees



April 16, 2008

Outline

• Backtrack-Free Search- Principle- Applications

- Backtrack-Bounded Search- Principle- Applications, extensions



April 16, 2008

j-width [Freuder, 1985]

• Given an ordering– The width of a group of j consecutive nodes is

• the number of nodes

• preceding the j nodes in the group and

• connected to any of them

– The j-width of a node is the minimum, for k=1 to j, of the width of the k consecutive nodes up to and including the node.

– The j-width of an ordering is the maximum j-width of all nodes in the ordering

• The j-width of a graph is the minimum of all j-width of all orderings



April 16, 2008

Separable Graphs (graphs with articulation points)

• Consider a CSP whose graph has articulation nodes

• Assume that the largest biconnected component has size b

• Build a tree whose nodes are the biconnected components, considering that the articulation node belong to the parent

• Build an ordering using a preorder traversal of the tree– The (b-1)-width of the ordering is 1



April 16, 2008

b-Bounded Search

• If for any level h in the search, – We can instantiate variable at level h+1– Considering its possible values, and– Reconsidering at most b-1 previous variables

• In a graph with articulation nodes, – let b be the size of the largest biconnected

component– Ordering the graph ‘along’ its biconnected

components guarantees b-bounded search



April 16, 2008

j-bounded search and j-width• There is an ordering

– That guarantees k-bounded backtrack search – If the graph strongly (i,k)-consistent, where i = j-width

• Problem: enforcing strong (i,k)-consistency may increase the j-width..

• Idea: Consider (1,k)-consistency– (Strong) (1,k)-consistency can be enforced without

altering the structure of the graph – Cost: time exponential in k+1



April 16, 2008

Achieving (1,k)-consistency

• Achieving (strong) k-consistency via the constraint synthesis algorithm– Freuder shows that it also achieves (strong)

(i,j)-consistency for i+j=k, time exponential in k

• Consider the original CSP– Remove the filtered values from the domains,

updating the binary constraints– The resulting network is (1,k)-consistent w/o

altering the graph



April 16, 2008

Graphs with articulation points

• Graphs whose largest biconnected component if b have (b-1)-width = 1

• Enforcing (1,b-1)-consistency • Can be enforced in time exponential in b• While guaranteeing (b-1)-bounded search

Result• A CSP can be solved in time exponential in b where b

is the size of its largest biconnected component



April 16, 2008

Exploiting Separable Graphs

• Tree-Clustering Method– Dechter & Pearl, AIJ 1989– Dechter Section 9.2.1 (Join Tree Clustering)

• Generalization: Tree decompositions– Hinge,Hypertree, etc.



April 16, 2008

Tree-Clustering Method

• Sketch of algorithm – Triangulate the graph (MinFill, Fig 4.4, page 89)

– Find maximal cliques (Max-Cardinality, Fig 4.5, page 90)

– Create a tree structure over the cliques– Repeat

• Solve a clique (all solutions), at each node in tree• Apply DAC from leaves to root• Generate solutions in a BT-free manner



April 16, 2008

Tree-Clustering Method: Complexity

• n: number of variables• Triangulation: O(n2)

– MinFill and MaxCardinality: O(n+e)

• Finding cliques: linear in n• Solving clusters: O(kr), k is domain size, r is size of

largest clique• Generating a solution O(n t log t)

– t is #tuples in each cluster, (sorted) domain of a ‘super’ variable – in best case, we have n cliques

• Complexity bounded by size of largest clique:– O(n2)+O(kr)+O(n t log t)=O(n kr log kr)=O(n r kr)



April 16, 2008

Tree Decompositions• Tree must obey specific properties• Inspired from Database Theory (acyclic queries)• Studied for non-binary CSPs• Complexity of solving the CSP is bounded by a ‘width’

parameter – Max number of nodes in cluster– Max number of constraints in cluster

• Used to characterize tractable classes of CSPs• Little use beyond theoretical characterization, except

BTD/BTD+ by Jégou et al.



April 16, 2008

Tree decomposition Def 9.4, p. 257

• A tree decomposition (T, ,) of a CSP P=(X,D,C) where

• T=(V,E), = chi, =psi: labeling functions (v) X, (v) C

1. Each constraint appears in at least on node in the tree, and all its variables are in that node

2. Nodes where any variable appears induce a single connected subtree (connectedness property)



April 16, 2008

Parameters of a tree decomposition

• Treewidth (tw)– Maximum number of variables in any node in

tree - 1

• Hyperwidth (hw)– Maximum number of constraints in any node

in tree

• Separator of two nodes– Number of common variables



April 16, 2008 Tree-Structured CSPs 22

HINGETCLUSTER

Gyssens et al., 1994HYPERCUTSETGottlob et al., 2000

TCLUSTERDechter & Pearl,

1989

BICOMPFreuder, 1985

HYPERTREEGottlob et al., 2002

HINGEGyssens et al.,

1994

CaT

CUT

HINGE+ TRAVERSE

CUTSETDechter, 1987

Structural decomposition methods

The techniques in blue were proposed by Yaling Zheng in 2005.


April 16, 2008

Constraint hypergraph


S1 S2

S4

S3 S5 S7S6

S15

S8

S12 S13 S14S11

S10

S9

S16S17

• A vertex represents a variable• A hyperedge represents a constraint (delimits

its scope)• Cut is a set of hyperedges whose removal

disconnects the graph• Cut size is the number of hyperedges in the cut


April 16, 2008

HINGE [Gyssens et al., 94]


S2

S4

S3 S5 S7S6 S8

S12 S13 S14S11

S9

hw = 12

S1 S2

S4

S3 S5 S7S6

S15

S8

S12 S13 S14S11

S10

S9

S16S17

S10S9

S16S9

S15S9

S17S11

S1 S2

In HINGE, the cut size is limited to 1


April 16, 2008

HINGE+: maximum cut size is a parameter


S2

S4

S3

S5

S7

S6

S12

S13

S11S9

S16

S17

S9

S11

S5

S4S6

S12

S8

S14

hw = 5

S1 S2

S4

S3 S5 S7S6

S15

S8

S12 S13 S14S11

S10

S9

S16S17

S8

S14

S7

S13

S10S9

S15S9

S1 S2

HINGE+ with maximum cut size of 2:


April 16, 2008

Bibliography• [Freuder, 1982] A Sufficient Condition for Backtrack Free Search,

JACM 29 (1), pages 24—32.• [Freuder, 1985] A Sufficient Condition for Backtrack Bounded

Search, JACM.• [Dechter & Pearl, 1987] The Cycle-Cutset Method for improving

Search Performance in AI Applications. In Third IEEE Conference on AI Applications, pages 224–230.

• [Dechter and Pearl, 1989] R. Dechter and J. Pearl. Tree Clustering for Constraint Networks. Artificial Intelligence, 38:353–366.


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