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Motivation

Motivation. Problem formulation should change to accommodate changes in real world! Some examples: delayed trains, flights, … broken machines, ill workers, … new demands, … interactive solvers, …. Problem area. Dynamic CSP:

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Motivation

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  1. Motivation Problem formulation should change to accommodate changes in real world! • Some examples: • delayed trains, flights, … • broken machines, ill workers, … • new demands, … • interactive solvers, …

  2. Problem area Dynamic CSP: • constraints can be added to and retracted from the constraint model dynamically in an arbitrary order Solving approaches: • robust solutions (still valid after small changes) • new solutions with minimal perturbations (changes) • reusing the original solutions for a new solution • reusing the reasoning process • maintaining maximal arc consistency after constraint addition or retraction

  3. Current situation Dynamic versions of AC algorithms • AC-4  DnAC-4 • fast but large memory consumption • AC-6  DnAC-6 • fast but still large memory consumption • so far fastest algorithm for maintaining dynamic AC • AC-3  AC|DC • low memory consumption but slow • AC-3.1  AC-3.1|DC • substitutes AC-3 in AC|DC by an optimal AC algorithm

  4. Constraint retractionbase approach • restore values that were deleted when propagating through the retracted constraint • propagate domain extensions to other variables (like reverted AC) • algorithms differ in how many values are returned back to domains • remove inconsistencies via “standard” arc consistency algorithms • algorithms differ in used AC algorithm

  5. Our approach • improve practical time efficiency ofAC|DC while keeping low memory consumption • using an optimal (fine-grained) AC algorithm increases memory consumption (AC-3.1|DC) • restore only the most promising values(our experiments showed that AC|DC restores many values that are immediately deleted by AC) How? • use information about the reason and “time” of value removal

  6. Removal information When and why a given value has been removed? • Justification • the first neighboring variable in which the eliminated valuelost all supports (like in DnAC) • Removal time • when the variable was eliminated(a continual counter of all deleted values) Usage:A value V is restored if the neighboring variable • is justification for the value V • has a restored supporter for the value V • has a restored value that has been eliminatedbefore V new

  7. Exampleinitial pruning Constraints are incrementally added to the problem and AC is established after each step B=D (1) B: 2/D6 3/D9 4 D: 1/B1 2/C5 3/C8 4 C=D (3) C: 1/A3 2/A4 3/E7 4 C≠E (4) A<C (2) A: 2 3 4/C2 E: 3 order number when the constraint is added justification for value removalVariableTime

  8. Exampleconstraint retraction Constraint A<C is removed from the problem B=D (1) B: 2/D6 3/D9 4 D: 1/B1 2/C5 3/C8 4 C=D (3) C: 1 2 3/E7 4 C≠E (4) A: 2 3 4 E: 3 These values are returned to domains because they were deleted when propagating via the retracted constraint

  9. Examplepropagation Propagate extended domains B=D B: 2 3/D9 4 D: 1/B1 2 3/C8 4 C=D C: 1 2 3/E7 4 C≠E A: 2 3 4 E: 3 AC is re-established at the end again B=D B: 2 3/D9 4 D: 1/B1 2 3/C8 4 C=D C: 1/D10 2 3/E7 4 C≠E A: 2 3 4 E: 3

  10. The algorithmmodified AC-3 functionpropagate-ac3'(P, data, revise) 1 queue := revise 2 while queue not empty do 3 select and remove a constraint c from queue 4 {u,v} := the variables constrained by c 5 (P,data,revise_u) := filter-arc-ac3'(P, data, c, u, v) 6 (P,data,revise_v) := filter-arc-ac3'(P, data, c, v, u) 7 queue := queue  revise_u  revise_v 8 return (P,data) function filter-arc-ac3'(P, data, c, u, v) 1 modified := false 2 for each d in P.D[u] do 3 if d has no support in P.D[v] w.r.t. c then 4 P.D[v] := P.D[v] - {d} 5 data.justif[u,d].var := v 6 data.justif[u,d].time := data.time 7 data.time := data.time + 1 8 modified := true 9 ifnot modified then 10 return (P,data,Ø) 11 return (P,data,{e in P.C|u is constrained by e and e≠c}) new

  11. The algorithminitialization function retract-constraint-ac|dc2(P, data, c) 1 (P,data,restored_u) := 2 initialize-ac|dc2(P, data, c, u, v) 3 (P,data,restored_v) := 4 initialize-ac|dc2(P, data, c, v, u) 5 P.C := P.C - {c} 6 (P,data,revise) := 7 propagate-ac|dc2(P, data,{restored_u,restored_v}) 8 (P,data) := propagate-ac3'(P, data, revise) 9 return (P,data) function initialize-ac|dc2(P, data, c, u, v) 1 restored_u := Ø 2 time_u :=  3 for each d in (P.D0[u] - P.D[u]) do 4 if data.justif[u,d].var = v then 5 P.D[u] := P.D[u]  {d} 6 data.justif[u,d].var := NIL 7 restored_u := restored_u  {d} 8 time_u := min(time_u, data.justif[u,d].time) 9 return (P,data,(u,time_u,restored_u)) new

  12. The algorithmpropagation function propagate-ac|dc2(P, data, restore) 1 revise := Ø 2 while restore not empty do 3 select and remove (u,time_u,restored_u) from restore 4 for each c in P.C|u is constrained by c do 5 {u,v} := the variables constrained by c 6 restored_v := Ø 7 time_v :=  8 for each d in (P.D0[v] - P.D[v]) do 9 if data.justif[v, d].var = u then 10 if data.justif[v, d].time > time_u then 11 if d has a support in restored_u w.r.t. c then 12 P.D[v] := P.D[v]  {d} 13 data.justif[v,d].var := NIL 14 restored_v := restored_v  {d} 15 time_v := min(time_v, data.justif[v,d].time) 16 restore := restore  {(v,time_v,restored_v)} 17 revise := revise  {e in P.C|u is constrained by e}) 18 return (P,data,revise) new

  13. Experimentsconstraint checks RCSP 100, 50, 0.3, p2 Consistent states Inconsistent states AC|DC • AC|DC-2 DnAC-6 (tightness) constraints added incrementally until a given density reached or inconsistency detected and after that 10% of randomly selected constraints retracted

  14. Experimentsruntime comparison RCSP 100, 50, 0.3, p2 Consistent states Inconsistent states AC|DC • AC|DC-2 DnAC-6 (tightness) constraints added incrementally until a given density reached or inconsistency detected and after that 10% of randomly selected constraints retracted

  15. Experimentsmemory consumption RCSP 100, d, 0.3, p2 NOTE: includes extensional representation of the constraints that is almost identical to memory consumption of AC|DC and AC|DC-2!

  16. A New Algorithm for Maintaining Arc Consistencyafter Constraint Retraction Pavel Surynek and Roman Barták Charles University, Prague pavel.surynek@seznam.cz roman.bartak@mff.cuni.cz

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