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The Promise of LP to Boost CSP Techniques for Combinatorial Problems

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The Promise of LP to Boost CSP Techniques for Combinatorial Problems

Carla P. Gomes

David Shmoys

Department of Computer Science

School of Operations Research and Industrial Engineering

Cornell University

CP-AI-OR 2002

- 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

- Randomization is magic ---
- we have some intuitions why it works.

- 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

- A purely combinatorial problem domain

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

The Quasigroup or Latin Square Completion Problem (QCP):

68% holes

Gomes and Selman 97

Critically

constrained area

EASY AREA

EASY AREA

Complexity of Latin Square Completion

Time: 150

1820

165

20%

42%

50%

35%

42%

50%

QCP is NP-Complete

Better

characterization

beyond worst case?

- Problem Formulations

- Variables -
- Constraints -

row

column

- 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.

- LP Formulations

Rows

Colors

Columns

Cubic representation of QCP

Max number

of colored cells

Row/color line

Column/color line

Row/column line

Families of patterns

(partial patterns are not shown)

Max number of colored cells in the selected patterns

s.t. one pattern per family

a cell is covered at most by one pattern

Max number of

colored cells

one pattern per color

at most one pattern

covering each cell

- 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)

- Randomization

- Stochastic strategies have been very successful in the area of local search.
- Simulated annealing
- Genetic algorithms
- Tabu Search
- Walksat and variants.

- Limitation: inherent incomplete nature of local search methods.

Randomized backtrack search

- 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!

Time:

7

11

30

(*)

(*)

(*) no solution found - reached cutoff: 2000

Erratic Behavior of Mean

Sample mean

Number runs

Easy instance – 15 % preassigned cells

3500

2000

Median = 1!

500

Gomes et al. 97

Power Law Decay

Exponential Decay

Standard Distribution

(finite mean & variance)

Standard

Exponential Decay

e.g. Normal:

Heavy-Tailed

Power Law Decay

e.g. Pareto-Levy:

Infinite variance, infinite mean

70%

unsolved

1-F(x)

Unsolved fraction

0.001%

unsolved

250 (62 restarts)

Number backtracks (log)

- 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

- 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:
- ChaffRelsat
- GraspGoldberg’s Solver
- QuestSatZ, SATO, …
- used to verify 1/7 of a Alpha chip (Pentium IV)

- Randomized Rounding

- Randomized Rounding
- Solve a relaxation of combinatorial problem;
- Use randomization to go from the relaxed version to the original problem;

- 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.

- LP Based Approximations

- 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)

- 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.

- 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.

- 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.

1

0.8

1

1

0.2

Max number of colored cells in the selected patterns

s.t. one pattern per family

a cell is covered at most by one pattern

- 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.:

- 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.

- Putting all the pieces together

- CSP Model
- LP Model + LP Randomized Rounding
- Heavy-tails
- We want to maintain completeness

How do we put all the pieces together?

A HYBRID COMPLETE 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

- Populate CSP Model
- Perform propagation
- Populate LP solver
- Solve LP

Variable

setting

controlled

by LP Randomized

Rounding

CSP & LP Inference

%LP

Interleave-LP

Search & Inference

controlled by CSP

Adaptive CUTOFF

- Initialize CSP model and perform propagation of constraints (Ilog Solver);
- Solve LP model (Ilog Cplex Barrier)
- LP provides good heuristic guidance and pruning information for the search. However solving the LP is relatively expensive.

- 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.

- 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.
- Perform propagation CSP propagation after each variable setting. (A total of Interleave-LP variables is assigned this way without resolving the LP)
- Use a cutoff value to restart the sercah (keep increasing it to maintain completeness)

- Empirical Results

- 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.

- Conclusions

- 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 !!!

Demos, papers, etc.

www.cs.cornell.edu/gomeswww.orie.cornell.edu/~shmoysCheck also:www.cis.cornell.edu/iisi

- Eighth International Conference on the
- Principles and Practice of
- Constraint Programming
- September 7-13
- Cornell, Ithaca NY
- CP 2002