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Constraint Satisfaction Problems Basic Algorithms My Thanks to Roman Bartak

Constraint Satisfaction Problems Basic Algorithms My Thanks to Roman Bartak (for “stealing” some of his slides). Search Algorithms for CSPs. We will study variations of DFS especially for CSPs. These algorithms are based on backtracking search. Simple or Chronological Backtracking (BT)

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Constraint Satisfaction Problems Basic Algorithms My Thanks to Roman Bartak

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  1. Constraint Satisfaction Problems Basic Algorithms My Thanks to Roman Bartak (for “stealing” some of his slides) ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  2. Search Algorithms forCSPs • We will study variations of DFS especially forCSPs. • These algorithms are based on backtracking search Simple or Chronological Backtracking (BT) Backjumping (BJ) and Conflict-Based Backjumping Forward Checking (FC) Maintaining Arc Consistency (MAC) Also two variations of hill climbing • Min-conflicts • Min-conflicts with Random Walk ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  3. Intelligent Backtracking • ΒΤ suffers fromthrashing • it visits again and again the same regions of the search tree becauseit has a very local view of the problem • One way to get rid of the problem is using intelligent backtracking algorithms • BJ, CBJ, DB, Graph-based BJ, Learning • Backjumping (BJ) is different from ΒΤ in the following: • When BJ reaches a dead-end it does not backtrack to the immediately preceding variables. It backtracks to the deepest variable in the search tree which is in conflict with the current variable ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  4. BJ vs. BT We want to color each area in the map with a different color We have three colors red, green, blue ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  5. BJ vs. BT • Let’s consider what ΒΤ does in the map coloring problem • Assume that variables are assigned in the orderQ, NSW, V, T, SA, WA, NT • Assume that we have reached the partial assignment Q = red, NSW = green, V = blue, T = red • When we try to give a value to the next variableSA, we find out that all possible values violate constraints • Dead end! • BT will backtrack to try a new value for variableΤ! • Not a good idea! ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  6. BJ vs. BT • BJ has a smarter approach to backtracking • It tells us to go back to one of the variables which are responsible for the dead-end • The set of these variables is called a conflict set • The conflict set forSA is {Q, NSW, V} • BJ backjumps to the deepest variable in theconflict set of the variable where the dead-end occurred • deepest = the one we visited most recently • CBJ, DB, Graph-based BJ, Learning, Backmarking ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  7. Conflict-based Backjumping (CBJ) • Conflict-based Backjumping is alook-back algorithm that performs intelligent backtracking at dead-ends • In contrast toBJ whichbackjumps only from leaf dead-ends, CBJ can also backjump from dead-ends at inner nodes • for each variablex we have a conflict set • when an assignment(x,a) fails because of a constraint violation with a previous variabley, y is added to the conflict set of x • if there are no values left in the domain of the current variable x,CBJ backjumps to the deepest variable w in the conflict set of x (as BJ) • and the conflict set of x is added to the conflict set of w • then a furtherbackjump can occur fromw ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  8. Forward Checking • Forward Checking (FC) belongs to the family of backtracking algorithms calledlookahead algorithms • The basic idea oflookahead is that when you assign a value to a variablethe problem is reduced throughconstraint propagation • constraint propagation is defined in a different way for each look-ahead algorithm • FC does the following: • When a variablex takes a valuev, for each future variabey which appears in a constraint withx we remove fromDx all the values that are not consistent withv ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  9. Forward Checking • If the domain of some variable becomes empty thenvaluev is rejected and we try the next value ofx • The operation ofFC means that the following holds for each step of the search: • All values of eachfuture variableare compatible with all the values that have been assigned to past variables • FC maintains a restricted form ofarc consistency ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  10. Forward Checking procedure FORWARD_CHECKING (vars,doms,cons) solution FC (vars,Ø,doms,cons) function FC (unlabelled,compound_label,doms,cons) returns a solution or NIL ifunlabelled= Øthen return compound_label else pick a variable x from unlabelled repeat pick a value v from Dx; delete v from Dx doms’ UPDATE(unlabelled-{x},doms,cons,compound_label + {(x,v)}) if no domain in doms’ is empty then result  FC(unlabelled - {x}, compound_label + {(x,v)}, doms’,cons) if result  NIL then return result end until Dx = Ø return NIL end ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  11. Forward Checking function UPDATE (unlab_vars,doms,cons,compound_label) returns an updated set of domains for each variable y in unlab_varsdo for each value v in Dy’ do if (y,v) is incompatible with compound_label with respect to the constraints between y and the variables of compound_label then Dy’  Dy’ – {v} end end returndoms’ ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  12. FC in operation V NSW T Q SA WA NT initial domains after WA=R after Q=G after V=B ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  13. Consistency Techniques • removing inconsistent values from variables’ domains • graph representation of the CSP • binary and unary constraints only (relatively easy) • nodes = variables • edges = constraints • node consistency (NC) • arc consistency (AC) • path consistency (PC) • (strong) k-consistency A>5 A A<C AB C B B=C ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  14. Node Consistency • A variableX isnode consistent iff each value a of X satisfies all the unary constraints on X • Node consistency can be applied as a preprocessing step before starting search to remove all the node inconsistent values A>5 If D(A)={0,…,9} node consistency will remove values 0,…,5 A A<C AB C B B=C ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  15. Arc Consistency • Definition: • A variableX isarc consistent iff for each other variableYthe following holds: For each valuea ofΧthere is at least one valueb ofΥsuch thata andb are compatible • Then we say thata supportsb • An algorithm that appliesarc consistency deletes values from the domainof a variable when they are not supported by any value in the domain of another variable ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  16. Arc Consistency (AC) • the most widely used consistency technique (good simplification/performance ratio) • deals with individual binary constraints • repeated revisions of arcs • Directional (one pass) AC a b c a b c a b c Y X Z ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  17. AC - Example • Problem:X::{1,2}, Y::{1,2}, Z::{1,2}X = Y, X  Z, Y > Z X X 1 2 1 2 1 2 1 2 Y Y 1 2 1 2 Z Z ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  18. 1 2 3 4 5 Arc Consistency propagation: Crossword Puzzle example ….No more changes! X1 X2 X4 astar live live load happy load peal peal hello peel peel hoses save save talk talk ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  19. Arc Consistency • We apply arc consistency: • As a (preprocessing) step before we start search • in that way we can reduce the size of the search tree • and in some cases discover inconsistent problems • While searching after an assignment of a value to a variable • constraint propagation  fast discovery of dead ends • The search algorithm which appliesarc consistency is calledMAC (maintaining arc consistency) ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  20. MAC procedure Maintaining Arc Consistency (vars,doms,cons) solution MAC (vars,Ø,doms,cons) function MAC (unlabelled,compound_label,doms,cons) returns a solution or NIL ifunlabelled= Øthen return compound_label else pick a variable x from unlabelled repeat pick a value v from Dx; delete v from Dx doms’ AC(unlabelled-{x},doms,cons,compound_label + {(x,v)}) if no domain in doms’ is empty then result  MAC(unlabelled - {x}, compound_label + {(x,v)}, doms’,cons) if result  NIL then return result end until Dx = Ø return NIL end ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  21. Algorithms forArc Consistency • Arc consistency can be enforced with Ο(ed2) optimal worst case time complexity • AC-4, AC-6, AC-7, AC-2001 • AC-3: non-optimal,but simpleAC algorithm • AC-3 andAC-2001 use: • a queue (or stack) where the variables that are checked for arc consistency are inserted • a routineRevise which deletes values that are not supported • AC-4, AC-6, AC-7 use more complex data structures • support lists ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  22. Achieving Arc Consistency • From Mackworth (1977a): procedureAC-3(G) Let Q be the set of (directed) arcs of G (not self-cyclic) while Q not empty do select and remove any arc (x,y) from Q; REVISE(x,y) if REVISE(x,y) changed the domain of x then add to Q the set of all arcs of G (z,x) that go into x; procedureREVISE(x,y) for each value a in domain of x do if there is no value b in the domain of y such that (a,b) is consistent then delete a from the domain of x ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  23. Achieving Arc Consistency • Runtime of AC-3: O(ed3) for graph e binary constraints, and maximum domain size ofd • For one constraint, function Revise costs O(d2) and it can be called d times • there are e constraints, so the complexity is O(ed3) • AC-2001/3.1 achievesthe optimal Ο(ed2) complexity by using a set of pointers Lastx,a,y • For each value a of a variable, Lastx,a,y points to the most recently discovered value in the domain of y that supports a procedureREVISE-2001/3.1(x,y) for each value a in domain of x do if there is no value b in the domain of y such that b> Lastx,a,y and (a,b) is consistent then delete a from the domain of x else Lastx,a,y = first such value ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  24. Algorithms for Arc Consistency • In some cases we can exploit the semantics of certain binary constraints to achieve an even better complexity • functional, anti-functional, monotonic, piecewise functional, etc. • algorithm AC-5 • What the complexity of AC processing for a constraint of the following types? • x = y • x ≠ y • x < y • x > y ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  25. Directional Arc Consistency (DAC) • Observation:AC has to repeat arc revisions; the totalnumber of revisions depends on the number of arcs butalso on the size of domains (while cycle) • Is it possible to weaken AC in such a way that every arc isrevised just once? • Definition: A CSP is directional arc consistentusing a givenorder of variables iff every arc (i,j) such that i<j is arcconsistent • Again, every arc has to be revised, but revision in onedirection is enough now ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  26. Arc Consistency as a Solution Method • Question: • Are there cases where we can guarantee that solubility (or insolubility) will be determined by applying arc consistency? • Answer (Freuder 1982): • When the constraint graph of the problem is a tree • In this case, a solution can be found (if one exists) in a backtrack-free manner by first applying directional arc consistency • A case of polynomially solvable CSPs • Many other such cases exist depending on the structure of the constraint graph and the nature of the constraints ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  27. Is AC enough? • empty domain => no solution • cardinality of all domains is 1 => solution • Problem:X::{1,2}, Y::{1,2}, Z::{1,2}X  Y, X  Z, Y  Z X 1 2 1 2 Y Z 1 2 ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  28. Stronger Levels ofConsistency • Beyondarc consistency there are numerous other levels ofconsistency • path consistency • singleton arc consistency • neighborhood inverse consistency • … • These are stronger thanarc consistency (i.e. they delete more inconsistent values when they are applied) • But they are more expensive (higher time complexity) • We will review some of them in the next lecture ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  29. Constraint Propagation • systematic search only => no efficient • consistency only => no complete • combination of search (backtracking) with consistency techniques • methods: • look back (restoring from conflicts) • look ahead (preventing conflicts) look back look ahead Labelling order ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  30. Look Back Methods • intelligent backtracking • consistency checks among instantiated variables • backjumping • backtracks to the conflicting variable • backchecking and backmarking • avoids redundant constraint checkingby remembering conflicting levelfor each value jump here a conflict b b b still conflict ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  31. Look Ahead Methods • preventing future conflicts via consistency checks among not yet instantiated variables • forward checking (FC) • AC to direct neighbourhood • partial look ahead (PLA) • DAC • (full) look ahead (LA) • Arc Consistency • Path Consistency instantiated variable labelling order ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  32. Look Ahead - Example • Problem:X::{1,2}, Y::{1,2}, Z::{1,2}X = Y, X  Z, Y > Z generate & test - 7 stepsbacktracking - 5 stepspropagation - 2 steps ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  33. Q1 Q2 Q3 Q4 1 2 3 4 Qi: line number of queen in column i, for 1i4 Q1, Q2, Q3, Q4 Q1Q2, Q1Q3, Q1Q4, Q2Q3, Q2Q4, Q3Q4, Q1Q2-1, Q1Q2+1, Q1Q3-2, Q1Q3+2, Q1Q4-3, Q1Q4+3, Q2Q3-1, Q2Q3+1, Q2Q4-2, Q2Q4+2, Q3Q4-1, Q3Q4+1 4-queen problem Place 4 queens so that no two queens are in attack. ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  34. Q1 Q2 Q3 Q4 1 2 3 4 There is a total of 256 valuations GT algorithm will generate 64 valuations with Q1=1; + 48 valuations with Q1=2, 1Q23; + 3 valuations with Q1=2, Q2=4, Q3=1; = 115 valuations to find first solution 4-queen problem first solution ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  35. Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 1 1 2 2 3 3 4 4 Q1 Q2 Q3 Q4 1 2 3 4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 1 1 2 2 3 3 4 4 4-queen problem, BT algorithm ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  36. Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 1 1 2 2 3 3 4 4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 1 1 2 2 3 3 4 4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 1 1 1 2 2 2 3 3 3 4 4 4 4-queen problem, FC algorithm ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  37. Q1 Q2 Q3 Q4 1 2 x 3 x 4 x Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 1 1 1 2 2 2 3 3 3 4 4 4 4-queen problem, MAC algorithm Value 3 of Q2 is unsupported in Q3, Value 4 of Q3 is unsupported in Q2, Value 2 of Q3 is unsupported in Q4, ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  38. Hybrid Algorithms • We can combine the operations of various backtracking algorithms to designhybrid algorithms • For examplewe can combine thelookahead function offorward checking and thelookback function ofBJ • FC-BJ • FC-CBJ • MAC-BJ • MAC-CBJ • … ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  39. FC-CBJ • Forward Checking with Conflict-based Backjumping • FC-CBJ combines the look-ahead of FC and the intelligent backjumping of CBJ • each variable is associated with a conflict set • when the forward checking of an assignment (x,a) results in a value deletion from the domain of a variable y, x is added to the conflict set of y • if after the forward checking of an assignment (x,a) the domain of a variable y is wiped out, the variables in the conflict set of y are added to the conflict set of x • why is this done? • if there are no more values left in the domain of the current variable x, FC-CBJ backjumps to the deepest variable w in the conflict set of x • the conflict set of x is added to the conflict set of w ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  40. Evaluation of Backtracking Algorithms • How can we compare backtracking algorithms forCSPs ? • Time / Space Complexity • not very useful. They all have exponential time complexity! • cpu times • number of nodes they visit in the search tree • amount ofconsistency checks they perform • amount of backtracks they perform ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  41. Evaluation of Backtracking Algorithms • Some theoretical results: • Search tree nodes visited FC-CBJ  FC-BJ  FC  BJ  BT CBJ  BJ • Number of consistency checks CBJ  BJ ΒΤ FC-CBJ  FC-BJ  FC • CPU times ? We always need experiments!!! ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  42. Heuristic Methods forCSPs • Search algorithms must take decisions: • Which will be the next variable to assign ? • Which value should I give it ? • Which constraint should I check ? • The decisions that the algorithm takes at each step have a drastic effect on the search space (and the efficiency of the algorithm) • Especially decision (1) • Heuristics help the algorithms take correct decisions • fail first principle ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  43. Heuristic Methods forCSPs • Variable ordering heuristics • static heuristics • MaxDegree, Bandwidth, … • dynamic heuristics • MRV, Brelaz, dom/deg, dom/wdeg… • Value ordering heuristics • Geelen’s promise, least-constraining… • Heuristics for constraint ordering • based on the cost of propagation ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  44. Variable Ordering Heuristics • Minimum Width • The width of a variable x is the number of variables that are before x, according to a given ordering, and are constrained with x • The width of an ordering is the maximum width of all the variables under that ordering • The width of a constraint graph is the minimum width of all possible orderings • Variables are ordered in descending width • useful when the degree of the nodes varies significantly • Problem: How many are the possible orderings? ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  45. Variable Ordering Heuristics • Maximum Degree • Variables are ordered in decreasing order of their degree in the constraint graph • degree is the number of adjacent variables in the graph • Heuristic to find a minimum width ordering • Maximum Cardinality • Selects the first variable arbitrarily • Then, at each stage, selects the variable that is adjacent to the largest set of already selected variables. ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  46. Variable Ordering Heuristics • Minimum Bandwidth • The bandwidth of a variable x, according to a given ordering, is the maximum distance between x and any other variable which is adjacent to x • The bandwidth of an ordering is the maximum bandwidth of all the variables under that ordering • The bandwidth of a constraint graph is the minimum bandwidth of all possible orderings • Idea: The closer the variables involved in a constraint are placed to each other the less backtracking will be required • Problem: Computing the minimum bandwidth is NP-complete ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  47. Dynamic Variable Ordering Heuristics • Minimum Remaining Values (MRV) or Smallest Domain (SD) • At each stage of search select the variable with the smallest domain size • How do we break ties? • Select a variable randomly • Select the variable with the highest degree in the original graph • Select the variable with the highest future degree (i.e. the one involved in the maximum number of constraints with future variables). This is called the Brelaz heuristic • Many variations have been proposed • dom/deg, dom/fdeg ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  48. State-of-the-art Dynamic Variable Ordering Heuristics • Weighted degree heuristics • each constraint is associated with a weight initially set to 1 • each time a constraint c removes the last value from a domain (i.e. causes a domain wipeout - DWO) its weight is incremented by 1 • the weighted degree of a variable x is the sum of the weights of the constraints that include x • wdeg heuristic • selects the variable with maximum weighted degree • dom/wdeg heuristic • selects the variable with minimum ration of domain size to weighted degree • What is the rationale behind these heuristics? • they use information gathered throughout search – not just from the current search state like dom/fdeg ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  49. Value Ordering Heuristics • Min-Conflicts • Associate with each value a the total number of values in future variables that are incompatible with a • Select the value with lowest sum • Alternative: Divide the number of incompatible values of future variable x with the domain size of x • Geelen’s Promise • For each value a count the total number of values in each future variable that are compatible with a • Take the product of the counts. This is called the promise of value a • Select the value with the maximum promise ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

  50. Constraint Ordering Heuristics • Is this issue important? • not very much when maintaining arc consistency • but there exist heuristics for ordering the constraints in the propagation queue. Can you think of such a heuristic? • but very important in modern advanced solvers that use propagators for the various (global) constraints • the idea here is to propagate the less expensive constraints first ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

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