1. Path Consistency for 1- General CSPs 2- STPs Peter Schlette Wesley Botham CSCE990 Advanced CP, Fall 2009

2. Outline 12/7/2015 2  General CSPs  Review of Path Consistency & PC Algorithms  Path Consistency Algorithms PC-1, PC-2, DPC, PPC, PC-8, PC-2001  STPs  Review of Triangulated Graphs  Path Consistency on STPs Floyd-Warshall, Bellman-Ford,  STP, P 3 C, Prop-STP

3. Path Consistency: Properties 12/7/2015 3  A CSP is path consistent iff it is strongly 3-consistent [van Beek & Dechter, JACM95] [van Beek & Dechter, JACM95]  Domains are filtered by arc consistency  Consistent solutions over 2 variables can be extended to every 3 rd variable  PC algorithms typically iterate over triplets of variables  End variables in triplets need not be distinct  In STP, variables domains are not relevant, thus PC algorithms on STPs enforce only 3-consistency  A given algorithm  Must determine that a CSP is path consistent or not  May or may not filter the constraints as much as possible (e.g., DPC)

4. List of Algorithms Discussed  PC-1 [Mackworth 77,, Dechter Fig 3.10]  PC-2 [Mackworth 77, Dechter Fig 3.11]  DPC [Dechter & Pearl 89, Dechter Fig. 4.9]  PPC [Bliek & Sam-Haroud 99]  PC-8 [Chmeiss & Jégou 98]  PC-2001 [Bessière et al. 05] 12/7/2015 4

5. Path Consistency: Algorithms 12/7/2015 5  They all stop when a relation/domain is empty; omitted for clarity  Queue  Does is it have one?  Edges (e.g., PPC)  Triplets of variables (e.g., PC-2)  Tuples of ‘vv-pair, variable’ (e.g., PC-8 & PC-2001)  Properties  Determines strong 3-consistency?  What is the time and space complexity?  Requires additional data structures to remember supports?  Graph: Complete? Chordal?  What is the practical performance (i.e., phase transition)?

6. PC-1 [Mackworth 77] 12/7/2015 6 1 Repeat until quiescence 2 For i, j, k variables 3 R ij  R ij  R ik  R kj  Has 4 nested loops, iterates over vertices, needs no queue  Updates every edge & every domain ( i = j )  Uses composition and intersection  Determines strong 3-consistency (when i = j )  Time complexity is O(n 5 d 5 )  One sweep costs O(n 3 d 3 )  Number of sweeps O(n 2 d 2 )  Space: no queue, no additional data structure, complete graph

7. PC-2 [Mackworth 77] 12/7/2015 7  Has 1 loop over a queue of triplets of variables  When an edge (domain) is updated, only triplets with an ‘external’ 3 rd node are added to queue  Allows i=j, thus determines strong 3-consistency.  Dechter Fig 3.11 specifies i<j in which case domains are not updated  Theoretically & practically faster than PC-1 (queue)  Time complexity is O(n 3 d 5 )  Space: Queue size is O(n 3 ), no additional data structures, complete graph 1 Q  { ( i,j,k ) | i  j, i  k, j  k } 2 While Q is not empty 3 For ( i,j,k ) from Q 4 R ij  R ij  R ik  R jk 5 If R ij changed, Q  Q U { ( m, i, j ), ( m, j, i ) | m  i, m  j } m m j i k

8.  n=16, a=16, d=30%  PC-1 vs. PC-2 [Botham & Schlette] 12/7/2015 8 PC-2 PC-1

9. PPC [Bliek & Sam-Haroud 99] 12/7/2015 9 Q ← E Until Q is empty do edge ← DEQUEUE(Q) for every triplet i,j,k related to edge R ij ← R ij ∩ (R ik  R kj ) if R ij was changed then EnQueue((i,j), Q)  First triangulates the graph  Keeps Q, a queue of edges  For an edge in Q  Pops edge from Q, retrieves all triplets where edge appears  In each triplet, updates each edge  Each updated edge is added to Q  Does not specify whether or not domains are filtered (some should be for soundness)

10. PPC [Bliek & Sam-Haroud 99] 12/7/2015 10  Triangulated graph is usually sparser than complete graph  For triplet ( i,j,k ) allows i=j, algorithm is sound (not clear in paper)  Enforces  Strong path-consistency  Weaker filtering than PC-2  Time: O(  ed 2 ),  degree of graph  Space: queue O(  e) for storing triplets, no additional data structure, chordal graph  Weakness: if 2 or more edges of a given triplet are in Q  All three edges are updated once for each edge  May do redundant work (fixed in  STP) i vnvn j v2v2 v1v1

11. 1 For k=n downto 1do 2 For i =1 to k R EVISE (i, R ik ) 3 For i,jk i,k connected & j,k connected 4 R ij  R ij  R ik  R jk k-2 k k-4 k-6 DPC [Dechter Fig 4.9, Dechter & Pearl 89]  Given an ordering for the variables  From bottom to top, enforces directional arc-consistency (DAC)  From bottom to top, for every variable, updates the edge between every two of its parents  Properties  Moralizes the graph, determines strong directional path consistency relative to ordering  Time O(min( t.d 3,n 3 d 3 ))  Space: No queue, no additional data structures, chordal graph 12/7/2015 11

12. DPC: Constraint revision [Dechter Fig 3.9] 1 For each ( a,b )  R ij 2 If no c  D k is s.t. ( a,c )  R ik & ( a,b )  R jk 3 Remove ( a,b ) from R ij 12/7/2015 12 k j i  Does not operate on matrices in reality ( ,  )  Iterates over  Tuples in constraints (i.e., (a,b)  R ij ) and  Values in domain (i.e., c  D k )  Not yet tested against PC-1, PC-2, PPC

13. PC-8 [Chmeiss & Jégou 98] PC-8 I NITIALIZE While Q   do P OP ((i,a),k), Q) P ROPAGATE ((i,a),k) UPDATE if WITHOUTSUPPORT((i,a),(j,b),k) REMOVE (a,b) from R ij, (b,a) from R ij Q ← Q U {((i,a,)j),((j,b),i)} 12/7/2015 13 I NITIALIZE Q ←  for i,j,k=1 to n (i<j,k≠i,k≠j) for (a,b)R ij U PDATE ((i,a),(j,b),k) P ROPAGATE for j=1 to n (j≠i,j≠k) for bDj and (a,b)R ij U PDATE ((i,a),(j,b),k)

14. PC-8 Analysis [Chmeiss & Jégou 98]  Queue: a list of (vvp, var) = ((var,val),var)  Determines strong PC-property (if you allow i=j )  Achieves ‘full’ filtering  Time complexity  I NITIALIZATION : O(n 3 d 3 )  P ROPAGATE is called O(n 2 d 2 ) times, each call costs O(nd 2 )  PC-8: I NITIALIZATION + (n 2 d 2 )  P ROPAGATE = O(n 3 d 4 )  Space  Queue O(n 2 d), data structure Status-PC: O(n 2 d)  Graph is complete 12/7/2015 14

15. PC-8 vs. PC-2 [Botham & Schlette] 12/7/2015 15 PC-2 PC-8  n=16, a=16, d=30% 

16. 12/7/2015 16 PC-2001 [Bessière+ 05] PC-2001 1 I NITIALIZE (Q) 2 While Q   3 P OP ((i,a),k) from Q 4 R EVISE P ATH ( (i,a),j ) I NITIALIZE (Q) 1 For i, j, k variables 2 For each ( a, b )  R ij 3 If ( a, b ) has no support c in D k 4 Remove ( a, b ) from R ij 5 Q  Q U {(( i, a ), j ), (( j, b ), i )} 6 Else 7 Last(( i, a ),( j, b ), k )  the first support of ( a, b )  D k

17. 12/7/2015 17 PC-2001 [Bessière+ 05] R EVISE P ATH (( i, a ), k,Q) 1 For each j  k 2 For each b  D j | ( a,b)  R ij 3 support  Last(( i, a ),( j, b ), k ) 4 While support is nil or was deleted 5 support  next value in D k 6 If no supports exist 7 Remove ( a,b ) from R ij 8 Q  Q U {(( i, a ), j ), (( j, b ), i )} 9 Else 10 Last(( i, a ),( j, b ), k )  support  Records supporting values to improve time complexity (at the cost of space overhead)

18. PC-2001 Analysis [Bessière+ 05]  Queue: same as PC-8, list of (vvp,var)  Achieves the same properties as PC-1, PC-2, PC-8  Time: O(n 3 d 3 )  Space  Queue: O(n 2 d)  Data structure: Last structure dominates O(n 3 d 2 )  Graph complete  Compared to PC-8, PC-2001  Is easier to understand and implement  Has lower time complexity  Is faster in general in experiments  Has worse space complexity 12/7/2015 18

19. PC-2001 vs. PC-2, PC-8 [Botham & Schlette] 12/7/2015 19 PC-8 PC-2001 PC-2  n=16, a=16, d=30% 

20. 12/7/2015 20 QueueDataStrTimeGraphPC-p?FilteringEmpirically PC-1None O(n 5 d 5 )CompleteYesFull PC-2 Triplets O(n 3 ) NoneO(n 3 d 5 ) Complete YesFullBetter than PC-1 in all cases PPC Edges O(  e) None O(  ed 2 ) ChordalYesPartial Advantageous on sparse graphs DPCNone O(min(td 3, n 3 d 3 ) ChordalNo‘weak’ partial Not evaluated yet, but likely best PC-8 {(vvp,var) * } O(n 2 d) Status O(n 2 d) O(n 3 d 4 )CompleteYesFull Significantly better than PC- 2 around phase transition, ambiguous otherwise PC-2001 {(vvp,var) * } O(n 2 d) Last O(n 3 d 3 ) CompleteYesFull Better than PC-8 in nearly all cases Summary of PC Algorithms for General CSPs  Comparisons: CPU time, #CC for preprocessing

21. Outline 12/7/2015 21  General CSPs  Review of Path Consistency & PC Algorithms  Path Consistency Algorithms PC-1, PC-2, DPC, PPC, PC-8, PC-2001  STPs  Review of Triangulated Graphs  Path Consistency on STPs Floyd-Warshall, Bellman-Ford,  STP, P 3 C, Prop-STP

22. Triangulated Graphs: Motivation  [Bliek & Sam-Haroud 99] showed that PPC  Operates on triangulated graphs  Determines the property of strong path consistency  When constraints are convex, PPC also yields minimal CSP  [Xu & Choueiry 03] studied STP  In STP constraints are convex  Proposed  STP, which adapts PPC to STPs w/o updating domains (‘weak’ path consistency)  May do fewer updates than PPC: queue of edges versus queue of triangles  [Planken et al. 08] studied STP  Showed that  STP is O(t 2 ), t is the number of triangles  Proposed P 3 C, for STP, that is O(t) 12/7/2015 22

23. Vertex elimination  Vertex elimination operation: When removing a vertex, connect all neighbors if they are not already connected  Fill edges: are the edges added when eliminating a vertex  Simplicial vertex  Vertex whose neighbors are all connected (form a clique)  Eliminating a simplicial vertex does not add any edges  Perfect elimination ordering:  There is always a simplicial vertex to be eliminated.  All nodes can be eliminated w/o adding any fill edges 12/7/2015 23

24.  A graph is triangulated iff it has a perfect elimination order  The width of the triangulated graph is equal to the size of its largest clique -1. Why?  Finding the width is tractable, thus max. clique on triangulated graph is tractable (usually, NP-hard)  Using the reverse of the perfect elimination ordering of a triangulated graph yields  A moralized graph  The induced width of this ordering is equal to the width of the triangulated graph, why?  Moralizing an arbitrary ordering of a graph yields a triangulated graph. Why? Triangulated Graphs 12/7/2015 24

25. Outline 12/7/2015 25  General CSPs  Review of Path Consistency & PC Algorithms  Path Consistency Algorithms PC-1, PC-2, DPC, PPC, PC-8, PC-2001  STPs  Review of Triangulated Graphs  Path Consistency on STPs Floyd-Warshall, Bellman-Ford,  STP, Prop-STP, P 3 C (Prop-STP is not discussed for lack of time)

26. Floyd-Warshall for STP [CLR]  Basic STP solver, three nested loops  Initialization: builds the distance graph  R ij = [a,b] gives e ij  b and e ji  -a  When edge does not exist, add infinite distance (complete graph)  Time  (n 3 ), Space: No queue but O(n 2 ) new edges 12/7/2015 26 F LOYD W ARSHALL For k 1 to n For i  1 to n For j  1 to n w(e ij ) M IN (w(e ij ),w(e ik )+w(e kj ))

27. d[s]  0 for each vertex i other than source ( i  s) d[ i ]   Repeat n-1 times for each edge e ij if d[i] + w( e ij ) < d[ j ] then d[j]  d[ i ] + w( e ij ) for each edge e ij if d[ i] + w( e ij ) < d[ j] then return inconsistent 12/7/2015 27 Bellman-Ford for STP [CLR]  Time: O(en), Space: No queue but O(n) new edges  Detects path consistency  Edges are not guaranteed minimal

28.  PPC  Operates on triangulated graphs  ‘Fully’ filters convex constraints  ∆STP adapts & refines PPC to STP  Keeps a queue of triangles (vs. a queue of edges)  Pops a triangle from queue & updates all 3 edges  Implicitly separate graph in biconnected components  Enqueues triangles adjacent to only the updated edge  Best performance when queue is FIFO 12/7/2015 28 ∆STP [Xu & Choueiry 03]

29. 12/7/2015 29 Performance on STPs: F-W, PPC,  STP [Xu & Choueiry 03]

30. 12/7/2015 30 Constraint checks for selected STP solvers Performance on STPs: BF, DPC,  STP [Shi+ 05]

31. 12/7/2015 31  Designers of ∆STP became aware of relevance of simplicial ordering in ∆STP in 2005 (ref. Nic Wilson)  Designers of Prop-STP exploited the idea  The authors of P 3 C formalize the flaw of ∆STP  Identified a pathological case where ∆STP does unnecessary work (not useful filtering)  Characterized it as set of problems where ∆STP runs in Ω (t 2 ), where t is the number of triangles  P 3 C addresses flaw by using a simplicial ordering  Proves that propagation can be achieved in  (t) P 3 C: The Idea [Planken+ 08]

32. P 3 C: Pathological Case [Planken+ 08]  {c i →i+1 | 0 ≤ i ≤ t+1} with zero weight  {c i→j | (1 ≤ i ≤ j−2 < t) ∧ i+j−t ∈ {1,2}} with weight j−i−1  {c j→i | (1 ≤ i ≤ j−2 < t) ∧ i+j−t ∈ {1,2}} with weight t−(j−i−1) 12/7/2015 32

33. Path. Case: Empirically [Planken+ 08] 12/7/2015 33 cubic quadratic linear

34. P 3 C: The Algorithm [Planken+ 08]  Given: A triangulated graph & a perfect elimination order 12/7/2015 34 k j i k j i  The algorithm has two steps  Bottom up: For every node Considers every pair of parents Updates the edge between parents (ref. DPC)  Top down: For every node Considers every pair of parents Updates edges adjacent to node 1 2

35. P 3 C: Bottom up [Planken+ 08]  DPC: For each node update the edges between all parents given the edges with the node 12/7/2015 35

36. P 3 C: Top Down [Planken+ 08]  Update the edges between every node and its parent 12/7/2015 36

37. P 3 C: Sound & complete [Planken+ 08]  Claim: on iteration k of P 3 C’s second half, all edges in the subgraph consisting of { V i | i ≤ k } are minimal  Base case: k = 2  We know that c 1,2 (in orange) must be minimal if it exists, due to DPC; the subgraph will always be minimal for k = 2 12/7/2015 37 k=2 1 2 1 k=3

38.  Induction hypothesis: G k-1 ={ V i ≤ k -1 } forms a minimal subgraph  Induction step: G k-1 minimal G k minimal  In the k th iteration, w ik =min( w ik, w ij + w jk )  Any part of a theoretical shorter path that extends out of the G k-1 can be replaced by its two endpoints within G k-1, due to the filtering from DPC  The only way w ik could be non-minimal is if there were a shorter path w ij + w jk, but this path was checked in the k th iteration  We have a base step and an inductive step, so the proof is complete! 12/7/2015 38 P 3 C: Sound & complete [ Planken+ 08] k i j GkGk k-1 G k-1 1

39. Time Complexity [Planken+ 08]  P 3 C runs in Θ (t), t is the number of triangles  O(t) ⊆ O(n  w* 2 ), w* is the min. induced width  O(n  w* 2 ) ⊆ O(n δ 2 ), where δ is max. degree  O(n δ 2 ) ⊆ O(n 3 )  DPC visits each triangle exactly once  The second half of P 3 C visits each triangle exactly once  Thus we have a constant number of visits to each triangle and a linear time complexity in the number of triangles 12/7/2015 39

40. Jobshop (enforced consistency) [Planken+ 08] 12/7/2015 40 In spite of theoretical bounds, the two algorithms are quite close

41. Summary of STP Algorithms GraphPC-p?Minimality?TimeQueueExperiments FWCompleteYes Θ(n 3 ) NoneWorst BFPartialYesNoO(en)NoneBetter than FW PPCChordalYes O(  ed 2 )O(  e) DPCChordalYesNo  (t) None Low density: less good than  STP, high density same as  STP  STP ChordalYes O(min(t 2,  ed 2 )) O(  e) Faster than PPC, better than DPC on low densities P3CP3CChordalYes  (t) NoneBest reported, 12/7/2015 41

42. Future Work [Planken+ 08]  Extend P 3 C to general CSPs  Investigate efficiency of triangulation algorithms vs. P 3 C  Incremental P 3 C solver for STPs 12/7/2015 42

43. References  CSP  PC-1, PC-2: see [Mackworth, AIJ 1977]  PPC: see [Bliek & Sam-Haroud, IJCAI 1999]  DPC: see [Dechter 4.2.2, Dechter & Pearl, AIJ 1987]  PC-8: see [Chmeiss & Jégou, IJAITools 1998]  PC-2001: see [Bessière et. Al, AIJ 2005]  STP  Floyd-Warshall, Bellman-Ford: see CLR textbook  DPC: see [Dechter et al., AIJ 1991]  ∆STP: see [Xu and Choueiry, TIME 2003]  Prop-STP: see [Bui, Tyson, and Yorke-Smith, AAAI 07, Workshop]  P 3 C: see [Planken et al., ICAPS 2008] 12/7/2015 43