Chapter 9 – Tree Decomposition Methods- Part II

1. Chapter 9 – Tree Decomposition Methods- Part II Anagh Lal CSCE 990-06 Advanced Constraint Processing 1

2. Outline • Unifying Tree-decomposition schemes • What is a tree-decomposition? • Processing a tree-decomposition, Cluster-Tree Elimination (CTE) • Join tree clustering as tree-decomposition • Adaptive-consistency as tree-decomposition. 2

3. Unifying Tree-Decomposition Schemes • We saw how JTC compiled a general an arbitrary constraint network into an acyclic one. • We also saw an algorithm for solving acyclic networks. • The unifying approach is presented as an algorithm combining the compilation phase and the solution phase of the compiled representation. 3

4. Tree decomposition • Definition: R = (X, D, C) be a CSP problem. A tree decomposition for R is a triple <T,x,> , where T = (V, E) is a tree, x and  are labelling functions which associate each vertex v from V with two sets x(v) Xand (v)  Cthat satisfy the following conditions: • Each constraint from C is part of some set (v) • Each variable in X, the set {vV | the variable is a part of x(v)} induces a connected subtree of T (connectedness property) Let’s see an example… 4

5. More definitions • Tree-width: maximum cardinality from the set of x(v) • Hyper-width: maximum cardinality from the set of (v) • Separator: sep(u, v) = x(u)x(v) 5

6. Tree-decomposition and hypertree embedding • Tree-decomposition defines a hypertree embedding of a hypergraph • Smallest tree-width and hyper-width among all such embeddings are called the tree-width and the hyper-width of the constraint hypergraph, respectively. 6

7. Decomposable subproblem • A subproblem of a constraint network is decomposable relative to the whole network if you can obtain solutions of the subproblem without referring to the remaining network • A subproblem over a subset of Y variables is decomposable relative to the whole network, if its set of solutions is identical to the projection of the network’s solution on Y 7

8. Cluster-Tree Elimination (CTE) • A tree decomposition facilitates solving many reasoning tasks including constraint satisfaction, optimization, and probabilistic reasoning tasks. • Algorithm Cluster Tree Elimination (CTE) is used for processing a tree decomposition. • The algorithm computes a decomposable subproblem for each node in the tree. 8

9. CTE-Algorithm • Input: A tree decomposition <T,x,> for a problem R = <X, D, C>. • Output: An augmented tree whose nodes are clusters containing the original constraints as well as messages received from neighbours. A decomposable problem for each node v. 9

10. Algorithm- Steps for every edge (u,v) in the tree T, do • Compute message m(u,v) ( from u to v), cluster(u) = (u)  {m(i,u) | (i,u)  T, iv} After node u has received messages from all adjacent vertices, except maybe from v Compute and send to v: m(u,v) = sep(u,v)( cluster(u) Ri) end for Return: A tree-decomposition augmented with constraint messages. For every node u  T, return the decomposition subproblem cluster(u) = (u)  {m(i,u) | (i,u)  T, iv} 10

11. Complexity. • N – number of nodes in the tree-decomposition. • w* be the tree-width • sep be its maximum seperator size • r- # constraints • deg – maximum degree in T • Space complexity: O(N.exp(sep)) • Time complexity: O((r+N).deg.exp(w*)) 11

12. Time complexity • For a node u in the tree-decomposition T • # constraints processed = size of cluster = |(u)| + deg – 1 • Time complexity of processing node u (|(u)| + deg -1)exp(|x(u)|) = (|(u)| + deg -1)exp(|w*|) by definition of w*. Summing over all nodes  | (u)| = r So we have time complexity as (r.exp(w*) + N.deg.exp(w*) – N.exp(w*) ) This can be bounded by O(deg.(r+N).exp(w*)) 12

13. Space complexity The computation of message is done as follows: m(u,v) = sep(u,v)(JoinRicluster(u) Ri) Computing joins, storing them and then projecting will lead to a space complexity exponential in |x(u)| or |w*|. But projecting after every join step will lead to a space complexity exponential in the separator size (sep). Thus for all nodes N, the space complexity is O(N.exp(sep)) 13

14. JTC as tree-decomposition • Example on board. • Space complexity • JTC  exponential in w* • CTE  exponential in separator size, sep • Time Complexity • JTC  O(r.exp(w*)), If N < r • CTE  O( r.deg.exp(w*)) • There are tree-decompositions that will not be created by JTC. Consider the example 9.2.13 … 14

15. Adaptive-consistency (AC) as tree-decomposition • AC can be viewed as a message passing algorithm along a “bucket-tree”, which is a special case of tree-decomposition. • Bucket-tree structure, first approach to description: • Consider a problem R = (X,D,C) and ordering d • Each bucket Bxicontains those constraints in C whose latest variable in d is xi • A bucket-tree of R and an ordering d, has buckets as its nodes, and bucket Bx is connected to bucket Byif the constraint generated by adaptive-consistency in bucket Bx is placed in By • In a bucket-tree every node Bx has one parent node By 15

16. Bucket-tree structure • Bucket-tree structure, graph based description: • Let Gd bethe induced graph along ordering d of problem R having primal graph G. • Each variable x, and all of its earlier neighbours in the induced-graph reside in bucket Bx • Each node Bx points to By(By is the parent of Bx) if y is the latest earlier neighbour of x in Gd 16

17. Bucket-tree is a tree-decomposition • Proof: Given problem R, its bucket-tree and two mappings: • x(Bx) contains x and its earlier neighbours in the induced graph along ordering d. • (Bx) contains all constraints whose highest-ordered argument is x • To prove • Each constraint from C is part of some set (v) and • Each variable in X, the set {vV | the variable is a part of x(v)} induces a connected subtree of T (connectedness property) 17

18. Proof • By construction of the bucket-tree, the first requirement (for the labelling x) holds • Proof of connectedness: Proof by contradiction. 18

19. Adaptive Tree Consistency • Since the bucket-tree is a tree-decomposition, it can be processed by CTE. • CTE adds a bottom-up message passing to adaptive consistency yielding Adaptive Tree Consistency (ATC). • Top-down phase: Each bucket receives messages from its children and sends to its parent  AC. • Bottom-up phase: Each bucket receives a constraint from its parent and sends constraints to its children. 19

20. Algorithm • Input: Problem R =(X,D,C), ordering d • Output: Augmented buckets containing and all the  constraints received from neighbours in the bucket tree. • Steps: • Step 0: Pre-processing • Step 1: Top-down phase • Step 2: Bottom-up phase 20

21. Steps-0,1 • Pre-processing: • Generate a bucket-tree using induced graph Gd • Top-down phase (AC): • For i= n to , process bucket Bxi: • Let 1,…, j be all the constraints in Bxi, including the original constraints of R. • The constraint yxi sent fromxito its parent y yxi (sep (xi,y)) =  sep(xi,y) (Join)ji=1 i 21

22. Step-2 • Bottom-up phase: • For i= n to , process bucket Bxi: • Let 1,…, j be all the constraints in Bxi, including the original constraints of R • The constraint zjxi sent fromeach child zj zjxi (sep (xi, zj)) =  sep(xi,zj) (Join)ji=1 i 22

23. Interesting note • Since a bucket-tree is a tree decomposition, and since it can be shown theat CTE applied to a bucket-tree is equivalent to ATC, then ATC : • Generates back-track free representation along certain orderings (width = 1) and, • Augments this representation with the generation of minimal subproblems. 23

24. Complexity of ATC • Notation: • w* - induced width along ordering d • deg – maximum degree in the bucket-tree • r- # constraints • Time complexity • O(r.deg.exp(w*)) • Space complexity • O(n.exp(w*)) 24

25. Discussion • Questions • Comments 25