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CP-AI-OR-02
Gomes & Shmoys
The Promise of LP to Boost CSP Techniques for Combinatorial Problems
The Promise of LP to Boost CSP Techniques for Combinatorial Problems
Carla P. [email protected]
David [email protected]
Department of Computer Science
School of Operations Research and Industrial Engineering
Cornell University
CP-AI-OR 2002
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MotivationMotivation
Increasing interest in combining Constraint Satisfaction Problem (CSP) formulations and Linear Programming (LP) based techniques for solving hard
computational problems.
Successful results for solving problems that are a mixture of linear constraints – where LP excels – and combinatorial constraints – where CSP excels.
However, surprisingly difficult to successfully integrate LP and CSP based techniques in a
purely combinatorial setting.
Example: Satisfiability
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Power of RandomizationPower of Randomization
Randomization is magic ---
we have some intuitions why it works.
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Outline of TalkOutline of Talk
• A purely combinatorial problem domain• Problem formulations
• CSP formulation• LP formulations
– Assignment formulation– Packing Formulation
• Randomization• Heavy-tailed behavior in combinatorial search• Approximation Algorithms for QCP
• A Hybrid Complete CSP/LP Randomized Rounding Backtrack Search Approach
• Empirical Results• Conclusions
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A purely combinatorial problem domain
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Quasigroups or Latin Squares:An Abstraction for Real World Applications
Quasigroups or Latin Squares:An Abstraction for Real World Applications
Gomes and Selman 97
Quasigroup or Latin Square
(Order 4)
A Quasigroup or Latin Square is an n-by-n matrix such that each row and
column is a permutation of the same n colors
68% holes
The Quasigroup or Latin Square Completion Problem (QCP):
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ComplexityComplexity
Better characterization
beyond worst case?
Critically constrained
area
42% 50%20%
Complexity of Latin Square Completion
EASY AREA EASY AREA
35% 42% 50%
Time: 150 1820 165
QCP is NP-Complete
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Problem Formulations
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QCP as a CSPQCP as a CSP
• Variables -
• Constraints -
}...,,2,1{, njix
....,,2,1,;,, njijicellofcolorjix
....,,2,1);,,...,2,
,1,
( ninixix
ixalldiff
....,,2,1);,,...,,2
,,1
( njjnxjx
jxalldiff
)2(nO
)(nO
row
column
kijPLStsjikjix ..,,
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Pure CSP approaches solve QCP instances up
to order 33 relatively well.
Higher orders (e.g.,critically constrained area)
are beyond the reach of CSP solvers.
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LP Formulations
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Assignment FormulationAssignment Formulation
Cubic representation of QCP
Columns
Rows
Colors
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QCPAssignment Formulation
QCPAssignment Formulation
}1,0{ijkx
....,,2,1,,1,
nkjii ijkx
kj
....,,2,1,,1,, nkjik ijkxji
....,,2,1,,1,
nkjij ijkx
ki
Row/color line
Column/color line
Row/column line
kijPLStskjikji
x ..,,
1,,
nijkxnn
j ki 1 11
max
....,,2,1,,;, nkjikcolorhasjicellijkx
Max number of colored cells
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Packing formulationPacking formulation
Max number of colored cells in the selected patterns
s.t. one pattern per family
a cell is covered at most by one pattern
Families of patterns
(partial patterns are not shown)
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QCPPacking Formulation
QCPPacking Formulation
MkMky
,},1,0{
,
1,
kMM Mk
yk one pattern per color
at most one pattern covering each cell
kMM MkyMn
k ,||1
max
1),:( ,1
,
Mji
kMM Mk
yn
kji
Max number of colored cells
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Any feasible solution to the packing LP relaxation is
also a solution to the assignment LP relaxation
The value of the assignment relaxation is at least the bound implied by the packing formulation => the packing formulation provides a tighter upper bound than the assignment formulation
Limitation – size of formulation is exponential in n. (one may apply column generation techniques)
k
MM Mky
ijkx
,
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Randomization
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BackgroundBackground
Stochastic strategies have been very successful in the area of local search.
Simulated annealingGenetic algorithmsTabu SearchWalksat and variants.
Limitation: inherent incomplete nature of local search methods.
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Randomized variable and/or value selection – lots of different ways.
Example: randomly breaking ties in variable and/or value selection.
Compare with standard lexicographic tie-breaking.
Note: No problem maintaining the completeness of the algorithm!
Randomized backtrack searchRandomized backtrack search
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Sa
mp
le m
ea
nErratic Behavior of Mean
Number runs
Empirical Evidence of Heavy-TailsEmpirical Evidence of Heavy-Tails
(*) no solution found - reached cutoff: 2000Time: (*)3011 (*)7
Easy instance – 15 % preassigned cells
Gomes et al. 97
500
2000
3500
Median = 1!
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Decay of DistributionsDecay of Distributions
Standard
Exponential Decay
e.g. Normal:
Heavy-Tailed
Power Law Decay
e.g. Pareto-Levy:
0,]Pr[ 2
CsomeforxCexX
Pr[ ] ,X x Cx x 0
Power Law Decay
Standard Distribution(finite mean & variance)
Exponential Decay
Infinite variance, infinite mean
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Exploiting Heavy-Tailed BehaviorExploiting Heavy-Tailed Behavior
Heavy Tailed behavior has been observed in several domains: QCP, Graph Coloring, Planning, Scheduling, Circuit synthesis, Decoding, etc.
Consequence for algorithm design:
Use restarts or parallel / interleaved runs to
exploit the extreme variance performance.
Restarts eliminate heavy-tailed behavior
70%unsolved
1-F
(x)
Un
solv
ed f
ract
ion
Number backtracks (log)
250 (62 restarts)
0.001%unsolved
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Randomized backtrack search – • active research area -> very effective when combined with
no-good learning!• solved open problems• different variants of randomization/restarts, e.g., biased
probability function for variable/value selection, “jumping” to different points in the search tree
State-of-the-art Sat Solvers incorporate randomized restarts:Chaff RelsatGrasp Goldberg’s SolverQuest SatZ, SATO, …
used to verify 1/7 of a Alpha chip (Pentium IV)
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Randomized Rounding
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Randomized RoundingRandomized Rounding
Randomized Rounding
Solve a relaxation of combinatorial problem;
Use randomization to go from the relaxed version to the original problem;
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Randomized Rounding of a 0-1 Integer Programming
Randomized Rounding of a 0-1 Integer Programming
Solve the LP relaxation;
Interpret the resulting fractional solution as providing the probability distribution over which to set the variables to 1.
Note: The resulting solution is not guaranteed to be feasible. Nevertheless, good intuition of why randomized rounding is a powerful tool.
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LP Based Approximations
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Approximation AlgorithmApproximation Algorithm
Assumption: Maximization problem
the value of the objective function delivered by algorithm A for input instance I.
the optimal value of the objective function for input instance I.
The performance ratio of an algorithm A is the infimum (supremum, for min) over all I of the ratio
A is an - approximation algorithm if it has performance ratio at least (at most, for min)
)(IAV
)(*
)(IV
IAV
)(*IV
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Approximation AlgorithmApproximation Algorithm
For randomized algorithms we replace by
in the definition of performance ratio.
(expectation is taken over the random choices performed by the algorithm).
Note: the only randomness in the performance guarantee stems from the randomization of the algorithm itself, and not due to any probabilistic assumptions on the instance.
In general, the term approximation algorithm will denote a polynomial-time algorithm.
)(IAV
)]([ IAVE
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Approximations Based on Assignment Formulation
Approximations Based on Assignment Formulation
Kumar et. al 99
Algorithm1 - at each iteration, the algorithm
solves the LP relaxation and sets to 1 the variable
closest to 1. This is an 1/3 approximation algorithm.
Algorithm 2 – at each iteration, the algorithm selects a compatible matching for a color, for which the LP relaxation places the greatest total weight.
This is an 1/2 approximation algorithm.
Experimental evaluation -> problems up to order 9.
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ApproximationBased on Packing Formulation
ApproximationBased on Packing Formulation
Randomization scheme:for each color K choose a pattern with probability (so that some matching is selected for each color)
As a result we have a pattern per color.
Problem: some patterns may overlap, even though in expectation, the constraints imply that the number of matchings in which a cell is involved is 1.
*,Mk
y
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Packing formulationPacking formulation
0.8
0.2
1
1 1
Max number of colored cells in the selected patterns
s.t. one pattern per family
a cell is covered at most by one pattern
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(1-1/e)- ApproximationBased on Packing Formulation
(1-1/e)- ApproximationBased on Packing Formulation
Let’s assume that the PLS is completable
Z*=h
What is the expected number of cells uncolored by our randomized procedure due to overlapping conflicts?
From we can compute
So, the desired probability corresponds to the probability of a cell not be colored with any color, i.e.:
*,Mk
y
kMM
Mky
ijkx
*
,*
)*1()*2
1)(*1
1()1
*1( ijnxijx
ijx
n
k ijkx
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(1-1/e)- ApproximationBased on Packing Formulation
(1-1/e)- ApproximationBased on Packing Formulation
This expression is maximized when all the are equal therefore:
So the expected number of uncolored cells is at most at least holes are expected to be filled by this technique.
*ijkx
)*1()*2
1)(*1
1()1
*1( ijnxijx
ijx
n
k ijkx
enn
n
k ijkx 1)11()
1*1(
eh he)11(
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Putting all the pieces together
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CSP ModelLP Model + LP Randomized RoundingHeavy-tailsWe want to maintain completeness
How do we put all the pieces together?
A HYBRID COMPLETE CSP/LP RANDOMIZED ROUNDING BACKTRACK SEARCH
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HYBRID CSP/LP RANDOMIZED ROUNDING BACKTRACK SEARCH
HYBRID CSP/LP RANDOMIZED ROUNDING BACKTRACK SEARCH
Central features of algorithm:
• Complete Backtrack search algorithm
• It maintains two formulations
• CSP model
• Relaxed LP model
• LP Randomized rounding for setting values at the top of the tree
• CSP + LP inference
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Variable setting
controlled by LP Randomized
RoundingCSP & LP Inference
Search & Inferencecontrolled by CSP
%LP
Interleave-LP
HYBRID CSP/LP RANDOMIZED ROUNDING BACKTRACK SEARCH
HYBRID CSP/LP RANDOMIZED ROUNDING BACKTRACK SEARCH
•Populate CSP Model• Perform propagation
•Populate LP solver•Solve LP
Adaptive CUTOFF
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1. Initialize CSP model and perform propagation of constraints (Ilog Solver);
2. Solve LP model (Ilog Cplex Barrier)LP provides good heuristic guidance and pruning information for the search. However solving the LP is relatively expensive.
3. Two parameters control the LP effort%LP – this parameter controls the percentage of variables set based on the LP rounding (%LP=0 pure CSP strategy)Interleave-LP – sets the frequency in which we re-solve the LP.
4. Randomized rounding scheme: rank variables according to the LP value. Select the highest ranked variable and set its value to 1 with probability p given by its LP value. With probability (1-p), randomly select a color form the colors allowed in the CSP model.
5. Perform propagation CSP propagation after each variable setting. (A total of Interleave-LP variables is assigned this way without resolving the LP)
6. Use a cutoff value to restart the sercah (keep increasing it to maintain completeness)
HYBRID CSP/LP RANDOMIZED ROUNDING BACKTRACK SEARCH
HYBRID CSP/LP RANDOMIZED ROUNDING BACKTRACK SEARCH
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Empirical Results
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Time PerformanceTime Performance
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Performance in BacktracksPerformance in Backtracks
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PerformancePerformance
With the hybrid strategy we also solve instances of order 40 in critically constrained area – out of reach for pure CSP;
We even solved a few balanced instances of order 50 in the critically constrained order!
• more systematic experimentation is required to better understand limitations and strengths of approach.
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Conclusions
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ConclusionsConclusions
Approximations based on LP randomized rounding (variable/value setting) + CSP propagation --- very powerful.
Combating heavy-tails of backtrack search through randomization --- very effective.
Consequence: New ways of designing algorithms - aim for
strategies which have highly asymmetric distributions that can be exploited using restarts, portfolios of algorithms, and interleaved/parallel runs.
General approach holds promise for a range of combinatorial problems
Final TAKE HOME MESSAGE
Randomization does not incomplete search !!!
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www.cs.cornell.edu/gomeswww.orie.cornell.edu/~shmoys
Check also:
www.cis.cornell.edu/iisi
www.cs.cornell.edu/gomeswww.orie.cornell.edu/~shmoys
Check also:
www.cis.cornell.edu/iisi
Demos, papers, etc.
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Eighth International Conference on the
Principles and Practice of
Constraint Programming
September 7-13
Cornell, Ithaca NY
CP 2002
