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

2. 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)
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 from thrashing
it visits again and again the same regions of the search tree because it 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 order Q, 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 variable SA, we find out that all possible values violate constraints
Dead end!
BT will backtrack to try a new value for variable Τ!
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 for SA is {Q, NSW, V}
BJ backjumps to the deepest variable in the conflict 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 a look-back algorithm that performs intelligent backtracking at dead-ends
In contrast to BJ which backjumps only from leaf dead-ends, CBJ can also backjump from dead-ends at inner nodes
for each variable x we have a conflict set
when an assignment (x,a) fails because of a constraint violation with a previous variable y, 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 further backjump can occur from w
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

8. Forward Checking
Forward Checking (FC) belongs to the family of backtracking algorithms called lookahead algorithms
The basic idea of lookahead is that when you assign a value to a variable the problem is reduced through constraint propagation
constraint propagation is defined in a different way for each look-ahead algorithm
FC does the following:
When a variable x takes a value v, for each future variabe y which appears in a constraint with x we remove from Dx all the values that are not consistent with v
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

9. Forward Checking
If the domain of some variable becomes empty then value v is rejected and we try the next value of x
The operation of FC means that the following holds for each step of the search:
All values of each future variable are compatible with all the values that have been assigned to past variables
FC maintains a restricted form of arc 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
if unlabelled = Ø then 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
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_vars do
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
return doms’
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

12. FC in operation WA NT Q NSW V SA T initial domains R G B R G B B G R B
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 variable X is node 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: Then we say that a supports b
A variable X is arc consistent iff for each other variable Y the following holds: For each value a of Χ there is at least one value b of Υ such that a and b are compatible
Then we say that a supports b
An algorithm that applies arc consistency deletes values from the domain of 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 X Y 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. Arc Consistency propagation: Crossword Puzzle example1 2 3 4 5
….No more changes! X1 X2 X4
astar
live
live
happy
load
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 applies arc consistency is called MAC (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
if unlabelled = Ø then 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
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

21. Algorithms for Arc 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 simple AC algorithm
AC-3 and AC-2001 use:
a queue (or stack) where the variables that are checked for arc consistency are inserted
a routine Revise 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):
procedure AC-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;
procedure REVISE (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 of d
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 achieves the 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
procedure REVISE-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 total number of revisions depends on the number of arcs but also on the size of domains (while cycle)
Is it possible to weaken AC in such a way that every arc is revised just once?
Definition: A CSP is directional arc consistent using a given order of variables iff every arc (i,j) such that i<j is arc consistent
Again, every arc has to be revised, but revision in one direction 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 of Consistency
Beyond arc consistency there are numerous other levels of consistency
path consistency
singleton arc consistency
neighborhood inverse consistency …
These are stronger than arc 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 checking by remembering conflicting level for 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 steps backtracking - 5 steps propagation - 2 steps
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

33. 4-queen problem Q1, Q2, Q3, Q4  Q1Q2, Q1Q3, Q1Q4,
Place 4 queens so that no two queens are in attack.
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
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

34. 4-queen problem first solution1 2 3 4 Q1 Q2 Q3 Q4
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
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

35. 4-queen problem, BT algorithm1 2 3 4 Q1 Q2 Q3 Q4 1 2 3 4 Q1 Q2 Q3 Q4 1 2 3 4 Q1 Q2 Q3 Q4 1 2 3 4 Q1 Q2 Q3 Q4 1 2 3 4 Q1 Q2 Q3 Q4
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

36. 4-queen problem, FC algorithm1 2 3 4 Q1 Q2 Q3 Q4 1 2 3 4 Q1 Q2 Q3 Q4 1 2 3 4 Q1 Q2 Q3 Q4 1 2 3 4 Q1 Q2 Q3 Q4 1 2 3 4 Q1 Q2 Q3 Q4 1 2 3 4 Q1 Q2 Q3 Q4 1 2 3 4 Q1 Q2 Q3 Q4
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

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

38. Hybrid Algorithms
We can combine the operations of various backtracking algorithms to design hybrid algorithms
For example we can combine the lookahead function of forward checking and the lookback function of BJ
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 for CSPs ?
Time / Space Complexity
not very useful. They all have exponential time complexity!
cpu times
number of nodes they visit in the search tree
amount of consistency 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 for CSPs
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 for CSPs
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

51. Stochastic and Local Search Methods
local search - chooses best neighbouring configuration
hill climbing neighbourhood = value of one variable changed
min-conflicts neighbourhood = value of selected conflicting variable changed
can we avoid local optima?
restarts
if at a local optimum, start procedure from scratch
random-walk sometimes picks neighbouring configuration randomly
tabu search few last configurations are forbidden for next step
local search does not guarantee completeness
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

52. The Min-Conflicts Algorithm
Start with a random assignment of values to variables
or a seemingly good one according to some heuristic
some constraints will be violated
Try to repair it
change the value assignment that resolves the greatest numbers of constraints
local optima can be escaped using random restarts
Otherwise:
Simulated annealing
Tabu search
Random walk
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

53. Min-Conflicts (version 1)
procedure Min_Conflicts(P, maxTries, maxChanges)
for i :=1 to maxTries do
A := initial complete assignment of the variables in P
for j:=1 to maxChanges do
if A satisfies P then return (A)
else
x := randomly chosen variable whose assignment is in conflict
(x,a) := alternative assignment of x which satisfies the maximum number of constraints under the current assignment A
if by making assignment (x,a) you get a cost ≤ current cost then
make the assignment
endif
endfor
return (“No solution found”)
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

54. Min-Conflicts (version 2)
procedure Min_Conflicts(P, maxTries, maxChanges)
for i :=1 to maxTries do
A := initial complete assignment of the variables in P
for j:=1 to maxChanges do
if A satisfies P then return (A)
else
(x,a) := the alternative assignment of a variable x which minimizes the number of constraint violations under the current assignment A
if by making assignment (x,a) you get a cost ≤ current cost then
make the assignment
else break
endif
endfor
return (“No solution found”)
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

55. Min-Conflicts with Random Walk
How can we leave the local optimum without a restart
(i.e. via a local step)?
By adding some “noise” to the algorithm!
Random walk
a state from the neighbourhood is selected randomly (e.g., the value is chosen randomly)
such technique can hardly find a solution
so it needs some guide
Random walk can be combined with the heuristic guiding the search via probability distribution:
p - probability of using the random walk
(1-p) - probability of using the heuristic guide
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

56. Min-Conflicts with Random Walk (version 1)
procedure Min_Conflicts(P, maxChanges,p)
A := initial complete assignment of the variables in P
for j:=1 to maxChanges do
if A satisfies P then return (A)
else
if probability p verified
x := randomly chosen variable whose assignment is in conflict
(x,a) := randomly chosen alternative assignment of x
(x,a) := the alternative assignment of a variable x which minimizes the number of constraint violations under the current assignment A
make the assignment (x,a)
endif
endfor
return (“No solution found”)
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1

57. Min-Conflicts with Random Walk (version 2)
procedure Min_Conflicts(P, maxChanges,p)
A := initial complete assignment of the variables in P
for j:=1 to maxChanges do
if A satisfies P then return (A)
else
x := randomly chosen variable whose assignment is in conflict
if probability p verified
(x,a) := randomly chosen alternative assignment of x
(x,a) := the alternative assignment of x which satisfies the maximum number of constraints under the current assignment A
make the assignment (x,a)
endif
endfor
return (“No solution found”)
ΑΝΑΠΑΡΑΣΤΑΣΗ ΓΝΩΣΗΣ - Lecture 1