Foundations of Constraint Processing

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Tree-Based Methods

• 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
• choueiry_at_cse.unl.edu
• Tel [phone]

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Outline

• Backtrack-Free Search
• Principle
• Applications
• Backtrack-Bounded Search
• Principle
• Applications, extensions

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Backtrack-Free Search Freuder, 1982

• A CSP can be solved in a backtrack-free manner
  when it is strong w1-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

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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

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Exploiting tree structures

• Cycle-Cutset Method
• Dechter and Pearl 1997
• Dechter Section 10.1.1 pages 273276
• Independent Set Decomposition
• Gompert, FLAIRS 2005
• Graph Reduction (GRED)
• Unpublished work by Yaling Zheng 2007

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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

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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 Sc, time is O(dc.(n-c)d2)O(ndc2)
• Finding the smallest cutset is NP-hard ?

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GRED Yaling Zheng

• After each assignment and FC/MAC step
• Check the connectivity of the remaining CSP
• Identify dangling trees using Grahams 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

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Outline

• Backtrack-Free Search
• Principle
• Applications
• Backtrack-Bounded Search
• Principle
• Applications, extensions

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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 k1 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

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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

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b-Bounded Search

• If for any level h in the search,
• We can instantiate variable at level h1
• 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

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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 k1

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Achieving (1,k)-consistency

• Achieving (strong) k-consistency via the
  constraint synthesis algorithm
• Freuder shows that it also achieves (strong)
  (i,j)-consistency for ijk, 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

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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

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Exploiting Separable Graphs

• Tree-Clustering Method
• Dechter Pearl, AIJ 1989
• Dechter Section 9.2.1 (Join Tree Clustering)
• Generalization Tree decompositions
• Hinge,Hypertree, etc.

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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

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Tree-Clustering Method Complexity

• n number of variables
• Triangulation O(n2)
• MinFill and MaxCardinality O(ne)
• 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)

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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.

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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
• Each constraint appears in at least on node in
  the tree, and all its variables are in that node
• Nodes where any variable appears induce a single
  connected subtree (connectedness property)

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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

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Structural decomposition methods
HYPERTREE Gottlob et al., 2002
HINGETCLUSTER Gyssens et al., 1994
HYPERCUTSET Gottlob et al., 2000
CaT
HINGE
TRAVERSE
TCLUSTER Dechter Pearl, 1989
HINGE Gyssens et al., 1994
CUT
BICOMP Freuder, 1985
CUTSET Dechter, 1987
The techniques in blue were proposed by Yaling
Zheng in 2005.
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Constraint hypergraph
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• 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

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HINGE Gyssens et al., 94
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In HINGE, the cut size is limited to 1
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hw 12
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HINGE maximum cut size is a parameter
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HINGE with maximum cut size of 2
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hw 5
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Bibliography

• Freuder, 1982 A Sufficient Condition for
  Backtrack Free Search, JACM 29 (1), pages 2432.
• 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 224230.
• Dechter and Pearl, 1989 R. Dechter and J.
  Pearl. Tree Clustering for Constraint Networks.
  Artificial Intelligence, 38353366.