1. Wednesday, January 29, 2003CSCE 990-06 Spring 2003 B.Y. Choueiry Directional Consistency Chapter 4

2. Wednesday, January 29, 2003CSCE 990-06 Spring 2003 B.Y. Choueiry Section 4.1 Problem ‘simplification’ as a justification for humans’ ability to handle difficult problems –Abstraction, reformulation, approximation –Approximate solutions or heuristic guidance Easy problems  solvable in polynomial time Search: easy problems  backtrack-free Bactrack-free: partial solutions can always be extended consistently to one more variable Tractability: –Graph topology  focus of this chapter –Constraint semantic

3. Wednesday, January 29, 2003CSCE 990-06 Spring 2003 B.Y. Choueiry Section 4.2 Tractability by restricting graph topology Graph parameter: (induced) width

4. Wednesday, January 29, 2003CSCE 990-06 Spring 2003 B.Y. Choueiry Section 4.2.1: Width Given an undirected graph Given an ordering of the nodes Node width in the ordering is the number of parents of the node in the ordering Width of ordering is the max of width of nodes Graph width is the minimal width across all possible orderings

5. Wednesday, January 29, 2003CSCE 990-06 Spring 2003 B.Y. Choueiry Section 4.2.1: Induced width Induced graph of an ordered graph (elimination, moralization) Induced width of ordered graph is the width of the moralized graph Induced width of the graph is the minimal induced width over all orderings

6. Wednesday, January 29, 2003CSCE 990-06 Spring 2003 B.Y. Choueiry Section 4.2.1: Width of trees Trees have no cycles G is a tree  (induced) width of G is 1 Width of a tree: polynomial Induced width of a tree: NP-complete Deciding whether there is an ordering with induced width k is O(n k ) Greedy approximation: eliminate node of min degree, connect neighbors

7. Wednesday, January 29, 2003CSCE 990-06 Spring 2003 B.Y. Choueiry Section 4.2.1: Chordal graphs Chordal  triangulated, subclass of Perfect Graphs Maximal cliques is easy on chordal graphs –Use max-cardinality ordering Max-cardinality ordering identifies chordal graphs –Iff each vertex and its parents form a clique –Ordered graph  induced graph –Width  induced width Proposition 4.2.5: induced graphs are chordal

8. Wednesday, January 29, 2003CSCE 990-06 Spring 2003 B.Y. Choueiry Section 4.2.1: k-trees K-trees are a subclass of chordal graphs –Max cliques are (exactly) of size (k+1) A graph G can be embedded in a k-tree  induced width of G w* is  k

9. Wednesday, January 29, 2003CSCE 990-06 Spring 2003 B.Y. Choueiry Section 4.3 Goal: amount of inference to guarantee backtrack- free search Amount of inference –Constraint propagation level (e.g., full arc, full path, full i-consistency) –Limit inference to a given ordering of the variables Example: arc-consistency is required only in the direction to be exploited by search [sic] FAC/FPC versus DAC/DPC

10. Wednesday, January 29, 2003CSCE 990-06 Spring 2003 B.Y. Choueiry Section 4.3: directional AC/PC Section 4.3: directional consistency –Directional arc, path, i-consistency –Section 4.5: adaptive consistency –Chapter 8: relational consistency DAC/DPC: –Simplest forms: binary constraints –Generalized to arbitrary constraints: generalized arc-consistency and relational consistency

11. Wednesday, January 29, 2003CSCE 990-06 Spring 2003 B.Y. Choueiry Section 4.3.1: Directional AC In DAC, each constraint is processed exactly once, O(ek 2 ) k is domain size –Computational advantage of DAC vs FAC FAC  DAC, DAC is weaker than FAC –a fortiriori, DAC is not sufficient in general and higher levels of consistency are neccessary –but is sometimes sufficient to yield a backtrack- free search (see example page 101).

12. Wednesday, January 29, 2003CSCE 990-06 Spring 2003 B.Y. Choueiry Section 4.3.2: Directional PC DPC vs FPC: see Example 4.3.6 DPC in Fig 8, relative to ordering d: –realizes DAC (updates domains) –realizes DPC (updates binary constraints) –manages addition of new arcs

13. Wednesday, January 29, 2003CSCE 990-06 Spring 2003 B.Y. Choueiry Section 4.3.3: Directional iC To generalize to i-consistency we need to look a larger scopes (than 2 or 3)