1. Anagh Lal Monday, April 14, 2003 1 Chapter 9 – Tree Decomposition Methods Anagh Lal CSCE 990-06 Advanced Constraint Processing

2. Anagh Lal Monday, April 14, 2003 2 Outline Acyclic constraint networks –Hypergraph, dual graph, primal graph, join-tree, hypertree, acyclic constraint network. Solving acyclic constraint networks (DB ) –Algorithm Acyclic-Solving Identifying acyclic constraint networks –Dual-based approach –Primal-based approach Join-tree clustering: general CSP  acyclic CSP

3. Anagh Lal Monday, April 14, 2003 3 Introduction Chapter 4: tree-shaped binary CSPs can be processed efficiently Here, the notion of a constraint tree is extended to non-binary constraints  A new class: acyclic constraint networks Review: –hypergraph –dual of a graph –primal graph

4. Anagh Lal Monday, April 14, 2003 4 Examples on the board Hypergraph Dual graph Reduced dual graph (when tree  join-tree) Primal graph

5. Anagh Lal Monday, April 14, 2003 5 Connectedness property An arc subgraph of a graph contains –all the nodes and –a subset of the edges Two nodes that share variables are connected   a path along which each arc contains the shared variables An arc subgraph of the dual satisfies the connectedness property  all nodes that share a variable are connected  The arc subgraph is a join-graph Connectedness  the running intersection property

6. Anagh Lal Monday, April 14, 2003 6 Has A Constraint Network: R Hypergraph: H Dual of H: H dual Arc subgraph of H dual : G Does G Satisfy Connectedne ss? Yes G is a Join-graph Is G a Tree? G is a Join-tree H is a Hyper-tree R is an Acyclic network Yes

7. Anagh Lal Monday, April 14, 2003 7 Solving acyclic problems An acyclic constraint network can be solved efficiently since its dual graph is a tree of binary constraints. Algorithm Acyclic-Solving –is tree-based (using directional arc-consistency) –solves acyclic problems Domains of variables in the dual problem  constraint tuples of the original problem Acyclic-Solving assumes that any tuples are consistent with domains (Relational arc-consistent)

8. Anagh Lal Monday, April 14, 2003 8 Algorithm Acyclic-Solving Input: –Acyclic constraint network R = (X, D, C) –A join-tree T of R. –Assumes the network is relational arc-consistent Output: –(Global) consistency of network [Yes/No] –A consistent solution to R

9. Anagh Lal Monday, April 14, 2003 9 Acyclic-Solving: 3 steps Step 1: Generate an ordering d = (R 1 …R t ) of the constraints C such that the every constraint/relation appears before its descendants in the join-tree rooted at R 1. Step 2 (tighten relations): –Pick a relation R j from the join-tree in reverse order of d –For each edge such that the edge connects R j and a relation R k that comes earlier in the ordering: Perform a join of the two relations, project the result using the scope of the relation R k Assign the result of above computation to R k (i.e., update R k ) If R k becomes empty: Return with no solution

10. Anagh Lal Monday, April 14, 2003 10 Acyclic-Solving (cont.) Step 3: If we reach this step, then problem is consistent. So, generate a solution. –Pick up the first relation using the ordering d. –Select a tuple from it. –Using this instantiation instantiate values from other relations such that they are consistent with previous assignments –.. And you’re guaranteed to finish!

11. Anagh Lal Monday, April 14, 2003 11 Acyclic-Solving: Correctness Step 2 of the algorithm is essentially applying directional arc-consistency along the join-tree of the dual of a problem. As we saw in Chapter 4, this guarantees a solution for a tree network.

12. Anagh Lal Monday, April 14, 2003 12 Acyclic-Solving: Complexity Complexity of tree algorithms is O(nk 2 ) n: #variables k: domain size Expected complexity is O(rl 2 ) r: #constraints each with most l tuples Using lexicographical sorting  complexity down to O(r.l.log l)

13. Anagh Lal Monday, April 14, 2003 13 Recognizing acyclic networks Two methods 1.Dual-Acyclicity: dual-based recognition –Generates ‘maximal’ spanning tree of dual graph –Tests for connectedness of this spanning tree 2.Primal-Acyclicity: primal-based recognition –Uses the fact that the primal graph for a hypertree must be chordal

14. Anagh Lal Monday, April 14, 2003 14 Dual-Acyclicity Input: Hypergraph H of constraint network R(X, D, C) Output: A join-tree T(S, E) if R is acyclic Algorithm: – Generate a maximal spanning tree T of the dual of R (weight = number of shared variables) –For every two nodes u and v in T, do If the unique path connecting them does not satisfy the connectedness property, then Exit (R is not acyclic) – R is acyclic, and T is the join-tree

15. Anagh Lal Monday, April 14, 2003 15 Dual-Acyclicity: complexity O(e 3 ), where e is the number of constraints

16. Anagh Lal Monday, April 14, 2003 16 Refresher of Chapter 4 Maximal cardinality ordering Chordal graph A graph is chordal if, in a maximum cardinality ordering, each node’s parents are connected Finding max-cliques in chordal graphs can be done in linear time

17. Anagh Lal Monday, April 14, 2003 17 Primal-Acyclicity: principle Uses 2 tests –the primal graph G of the network is chordal and –the max-cliques of the chordal primal graph G satisfies the conformatlity property Conformality –A chordal primal graph is conformal relative to a constraint hypergraph  there is a 1-to-1 mapping between maximal cliques and scopes of constraints

18. Anagh Lal Monday, April 14, 2003 18 Primal-Acyclicity Input: A network R = (X, D, C) and its primal graph G Output: A join-tree iff the problem is acyclic Algorithm: 3 steps –Test chordality of the primal graph G –Test conformality of the primal graph G –If acyclicity, then create the join-tree for R

19. Anagh Lal Monday, April 14, 2003 19 Primal-acyclicity: the steps 1.Test chordality: Using a max-cardinality ordering d of G and starting from the last node check if the parents of all the nodes are connected. If for any node they are not then, exit: problem is not acyclic 2.Test conformality: Let C 1,…,C t be the cliques indexed by their highest variable in the ordering. Check if the cliques correspond to scopes of constraints, if so R is acyclic 3.Create a join-tree: From t to 1, connect every clique to an earlier clique with which it shares a maximal number of variables Return the tree of cliques

20. Anagh Lal Monday, April 14, 2003 20 Join-tree clustering Since acyclic constraint networks can be solved efficiently, try to convert an arbitrary constraint network into an acyclic one This can be achieved by grouping subsets of constraints into clusters, or sub-problems, whose scopes constitute a hypertree, thus transforming a constraint hypergraph into a constraint hypertree By replacing each sub-problem with its solution we obtain an acyclic constraint problem, which can be solved using the acyclic solving algorithm. This compilation process is called join-tree clustering

21. Anagh Lal Monday, April 14, 2003 21 Join-tree clustering example

22. Anagh Lal Monday, April 14, 2003 22 Hypertree embedding A hypertree embedding of a hypergraph R = (X,H) is a hypertree T = (X,S)such that, for every h element of H there is a h1 element of S such that h is a subset of h1.

23. Anagh Lal Monday, April 14, 2003 23 Decomposing a hypergraph into a hypertree Consider the primal-acyclicity algorithm to detect acyclic networks. Instead of terminating when we find that the graph is acyclic we could repair the primal graph and proceed to convert the hypergraph to a hypertree. Enforce chordality by connecting the parents Enforce conformality by generating unique constraint for every maximal clique. Algorithm Join Tree Clustering (JTC) does this.

24. Anagh Lal Monday, April 14, 2003 24 Joint Tree Clustering Input: Constraint problem R = (X, D, C) and its primal graph G = (X, E) Output: An equivalent acyclic problem and its join-tree: T=(X, D, C ’ )

25. Anagh Lal Monday, April 14, 2003 25 JTC algorithm (cont.) Select a variable ordering d (max-cardinality recommended) Triangulation: Connect the parents. Generate an induced graph along d. Create a join-tree for the induced graph –Identify all max cliques in the chordal graph. (C 1,…,C t ), from the last variable to the first. –Connect each C i to one single C j (j < i) with whom it shares the largest subset of variables.

26. Anagh Lal Monday, April 14, 2003 26 JTC algorithm (cont.) Place each constraint from R in one clique containing its scope, and let P i be the constraint subproblem associated with C i Solve each P i and let R’ i be its set of solutions Return C’ = {R’ 1,…, R’ t }, and their join- tree, T

27. Anagh Lal Monday, April 14, 2003 27 Complexity Time and space complexity: –O(r.k (w*(d)+1) )

28. Anagh Lal Monday, April 14, 2003 28 Discussion Questions? Comments?