1. Solve Your Problem Faster - by changing the model
Barbara Smith

2. Background Assumptions
A constraint programming tool providing:
a systematic search algorithm
combined with constraint propagation
a set of pre-defined constraints
A problem that can be represented as a finite domain constraint satisfaction or optimization problem
Caveat: the experiences described are based on ILOG Solver
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What do I mean by Faster?
e.g.`Peaceable Armies of Queens’ problem
place m black & m white queens on a chessboard so that the black queens don’t attack the white queens, and maximize m
Optimal solution for an 11 x board:
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Results for 8 x 8 board
Backtracks to
find optimal
prove optimality
Time
(sec.)
Basic model
5270
12,002,608
2100
Eliminate symmetry
5270
938,652
240
New model
4676
478,012 72
Variable order-ing heuristic
2973
44,276 12
Just changing the model makes a huge difference to the time to solve the problem
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What are the options?
Given a CSP representation of the problem:
if there is symmetry in the CSP, eliminate it
find a related representation to use instead/as well
add variables – to express different aspects of the problem
add constraints
to relate new variables to old
to prune dead-ends earlier
change the search strategy
(other possibilities won’t be considered)
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Symmetry in the CSP
A symmetry transforms any (full or partial) assignment into another so that consistency/inconsistency is preserved
Symmetry causes wasted search effort: after exploring choices that don’t lead to a solution, symmetrically equivalent choices will be explored
Especially difficult if there is no solution, or if all solutions are wanted, or in optimization problems
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7. Symmetry Breaking During Search (SBDS)
See Gent & Smith, ECAI’2000
A = set of assignments
made so far
var = val
var != val
+ g(var!= val) for any unbroken
symmetry g, i.e. if g(A) is
(or will be) true
On backtracking to a choice point, add symmetry breaking constraints to the 2nd branch explored
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8. Symmetries of ‘armies of queens’
Variable s[i,j] represents the square i, j
The values are b, w or e (black, white or empty)
7 chess board symmetries: x (horizontal reflection), y, d1, d2, r90, r180, r bw (swap black & white queens) + 7 combinations
x(s[i,j]=v)  s[n-i+1,j]=v
bw(s[i,j] = v)  s[i,j] = v’ , (v’ = b if v = w, w if v = b, e if v = e )
bw_x(s[i,j]=v)  s[n-i+1,j]=v’ , etc.
15 symmetry functions are needed in all
each takes a constraint as input, e.g. s[i,j]=v, and returns the symmetric constraint, e.g. s[i,j] = v’
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9. Armies of Queens – 8 x 8 Board
s[1,1]=w
s[1,1] != w
d1: s[1,1]!=w
bw_x: s[8,1]!=b, etc.
x: s[8,1]!=w
bw: s[1,1]!=b
r90: s[1,8]!=w
s[1,2]=w
s[1,2] != w
All symmetries swapping colours now broken on this branch
x: if s[8,1]=w then
s[8,2]!=w , etc.
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Results for 8 x 8 board
Fails to find
optimal
Fails to prove
optimality
Time (sec.)
Basic model
5270
12,002,608
2100
Eliminate symmetry
5270
938,652
240
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11. Optimizing SONET Rings
See Sherali & Smith*, ‘Improving Discrete Model Representations via Symmetry Considerations’ (*no relation)
Transmission over optical fibre networks
Known traffic demands between pairs of client nodes
A node is installed on a SONET ring using an ADM (add-drop multiplexer)
If there is traffic demand between 2 nodes, there must be a ring that they are both on
Rings have capacity limits (number of ADMs, i.e. nodes, & traffic)
Satisfy demands using the minimum number of ADMs
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12. Example & Optimal Solution
Up to 7 rings, maximum capacity 5 ADMs
Ignore traffic capacity (for now)
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A CSP Model
Variables: xij = 1 if node i is on ring j
At most 5 nodes on any ring:
If there is a demand between nodes k and l :
Minimize
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14. Symmetry in the SONET CSP
The rings are indistinguishable (only numbered for the purposes of the CSP model)
We can eliminate the symmetry using SBDS just by describing all transpositions of pairs of rings:
e.g. r12(x[i,j] = v)  x[i,2]=v if j = x[i,1]=v if j = x[i,j]=v otherwise
That’s all – we can forget about symmetry (e.g. when choosing the variable ordering)
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15. Three Alternative Models
Whether a given node is on a given ring:
xij = 1 if node i is on ring j
Which ring(s) each node is on:
Ni = set of rings node i is on
Which nodes are on each ring
Rj = set of nodes on ring j
In principle, any of these 3 sets of variables could be the basis of a complete CSP model
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16. Dual Models from Boolean Variables
Given a CSP with Boolean variables xijkl… we can form a new set of variables with one less subscript:
e.g. xijkl… = 1 corresponds to yjkl… = i (an integer variable) or i  Yjkl… (a set variable), depending on whether one or several possible values i are associated with each combination of j,k,l,…
if the Boolean variables have n subscripts, we can derive n sets of dual variables
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Which Model to Choose?
We don’t need to choose just one set of variables – we can use them all at once
We then need new channelling constraints to link the sets of variables:
(xij = 1) = (i  Rj) = (j  Ni)
(see Cheng, Choi, Lee & Wu, Constraints, 1999)
But we should not combine all 3 complete models
Adding variables & channelling constraints doesn’t cost much – duplicated constraints do
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Why add more variables?
Express each problem constraint in whichever way is easiest /most natural/ propagates best
e.g gives better results than
(Often easiest /most natural/ propagates best are the same thing)
We can observe effects of search on different aspects of the model, & express them
develop implied constraints, search strategies, etc.
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19. Possible Variables in the SONET Problem
Whether a given node is on a given ring 
Which ring(s) each node is on 
Which nodes are on each ring 
How many nodes are on each ring 
How many rings each node is on 
The total number of nodes on all rings 
The total of the number of rings each node is on 
Which demand pairs are on each ring
Which ring(s) each demand pair is on
How many rings are used
………
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20. Choosing the Search Variables
We need to choose a set of variables such that an assignment to each one, satisfying the constraints, is a complete solution to the problem
Assume we pass the search variables to the search algorithm in a list or array
the order determines a static variable ordering
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Possible Choices
Use just one set of variables, e.g. xij – the others are just for constraint propagation
Use two (or more) sets of variables (of the same type) e.g. Rj ,Ni
interleave them in a sensible (static) order
or use a dynamic ordering applied to both sets of variables
Use an incomplete set of variables first, to reduce the search space before assigning a complete set
e.g. decide how many rings each node is on (search variables |Ni|) and then which rings each node is on (xij )
All three possibilities are useful – the first will be used in the SONET problem (and later the third)
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Implied Constraints
Constraints which can be derived from the existing constraints, and so don’t eliminate any solutions
We only want useful implied constraints:
they reduce search: i.e. at some point during search, a partial assignment must be tried which is inconsistent with the implied constraint but would otherwise not fail immediately
they reduce running time – the overhead of propagating extra constraints must be less than the savings in search
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23. How to Find Useful Implied Constraints
Identify obviously wrong partial assignments that may/do occur during search
Try to predict them by contemplation/intuition
Observe the search in progress
Having many variables in the model enables observing/thinking about many possible aspects of the search
But we still need to check empirically that new constraints do reduce both search and running time
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24. Implied Constraints: SONET
A node with degree in the demand graph > 4 must be on more than 1 ring (|Ni| > 1)
If a pair of connected nodes have more than 3 neighbours in total, at least one of the pair must be on more than 1 ring (|Nk|+|Nl| > 2)
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25. Optimality Constraints
In optimizing, if we know that for any solution with a particular characteristic, there must be another solution at least as good, we can add constraints forbidding it
e.g.
no ring should have just one node on it
any two rings must have more than 5 nodes in total (otherwise we could merge them)
Derive these in the same way as implied constraints
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26. Variable Ordering Heuristics
Armies of Queens:
Place a white queen next where it will attack fewest additional squares
SONET problem:
add a node to a ring with spare capacity; choose the node connected to most nodes already on the ring; of those, the node connected to most nodes still to be placed
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27. Finding a Good Solution v. Proving Optimality
Armies of Queens
the heuristic finds an optimal solution for every case where this is known
but… it’s hopeless for proving optimality
the `anti-heuristic’ (place a white queen where it will attack most additional squares) is much better!
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28. Finding a Good Solution v. Proving Optimality: SONET
The heuristic finds near-optimal solutions quite quickly, but is no good for proving optimality
How could we prove this solution is optimal: {2,3,4,9,12}, {1,3,7,8,10}, {4,5,6,7,10}, {1,8,11,12,13} ?
e.g. show that we cannot reduce the number of times that any of 3, 4, 7, 8, 12 appear without having another node appear twice instead
Introduce variables ni = |Ni| i.e. the number of rings that node i is on
 Two-stage search: find a good solution, then start search again, assigning ni variables first and then xij variables
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What do I mean by Faster?
The basic model (just the xij variables, no symmetry-breaking) can only solve small problems (7 nodes, 8 demand pairs)
The final model can solve the problems in the Sherali & Smith paper (13 nodes, 24 demand pairs)
3 full sets of variables + others
SBDS
implied constraints
variable ordering heuristic
two-stage search process
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Some Advice
Eliminate symmetry
Use lots of variables, with channelling constraints
But don’t express the same problem constraints twice
Add constraints that make explicit what you know about satisfying/optimal solutions
But only if they reduce search and running time
Learn from solving the problem by hand and observing the search
If finding good solutions is easy and proving optimality is hard, consider using a different strategies for each stage
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Conclusions
Can we list “10 Steps to Successful Modelling”?
New problems still often lead to new ideas about modelling
But some patterns do recur frequently
e.g. models representing dual viewpoints
We are beginning to automate some aspects of modelling
e.g. symmetry, implied constraints
Still a long way to go before building a good model of a problem is straightforward
e.g. we often can’t tell if model A is better than model B without trying them both
More research is still needed...
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Selected References
Symmetry Breaking During Search in Constraint Programming, I. P. Gent and B. M. Smith, Proceedings ECAI'2000, pp , 2000.
Models and Symmetry Breaking for `Peaceable Armies of Queens’, K.E Petrie, B. M. Smith & I. P. Gent, APES Report APES , May 2002.
Improving Discrete Model Representations via Symmetry Considerations, H. D. Sherali & J. C. Smith, Man.Sci. (47) pp , 2001.
Increasing Constraint Propagation by Redundant Modeling: an Experience Report, B. M. W. Cheng, K. M. F. Choi, J. H. M. Lee & J. C. K. Wu, Constraints (4) pp ,  
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