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Sampling Combinatorial Space Using Biased Random WalksPowerPoint Presentation

Sampling Combinatorial Space Using Biased Random Walks

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### Sampling Combinatorial Space Using Biased Random Walks

Jordan Erenrich, Wei Wei and Bart Selman

Dept. of Computer Science

Cornell University

- Many forms of probabilistic reasoning can be effectively reduced to sampling satisfying assignments from a Boolean formula (an instance of SAT).
- Question: Can state-of-the-art local search procedures for SAT sample effectively from the solution space? (as an alternative to standard Monte Carlo Markov Chain methods)

Characteristics of Solution space: Solution Clustering reduced to sampling satisfying assignments from a Boolean formula (an instance of SAT).

- Visualization with multi-dimensional scaling (MDS)
- Solutions to specific 75 variable, 325 clause 3-SAT instances
- 75 dimensional solution projected to two dimensions
- Distance between points approximates hamming distance

Solution Probability reduced to sampling satisfying assignments from a Boolean formula (an instance of SAT).

- Consider a simple 2-SAT problem
- x OR y

- Consider a simple SAT heuristic
- Starts with a random bit assignment
- Randomly flip a bit until a solution is found

- Consider the probability of finding each solution

Solution Probability Using WalkSat Algorithm reduced to sampling satisfying assignments from a Boolean formula (an instance of SAT).

- Empirically determined each solution’s probability (uf75-01 - 75 variable, 325 clause 3-SAT instance)
- WalkSat finds every solution, but with very large range of probabilities (1:104)
- Probability Clusters

Probability Ranges in Different Domains reduced to sampling satisfying assignments from a Boolean formula (an instance of SAT).

Improving the Uniformity of Sampling reduced to sampling satisfying assignments from a Boolean formula (an instance of SAT).

Mixed sampling strategy

- To reduce the range of probabilities, we propose a hybrid local search algorithm:
- With probability p, the algorithm makes a biased random walk move
- With probability 1-p, the algorithm makes a SA (simulated annealing) move

- In our experiment, we used
- 50% WalkSat + 50% SA at a fixed temperature

Results of the Hybrid Approach reduced to sampling satisfying assignments from a Boolean formula (an instance of SAT).

Our key figure.

Solution Clusters reduced to sampling satisfying assignments from a Boolean formula (an instance of SAT).

Results on a random 3-SAT instance (70 vars, 301 clauses, 2531 solutions).

Summary reduced to sampling satisfying assignments from a Boolean formula (an instance of SAT).

Proposal: Use SAT solvers to sample solutions from a

combinatorial space.

Findings:

- WalkSAT does sample all solutions.
- But, sampling can be highly biased.
- Using a new hybrid strategy, we can obtain
- effective near-uniform sampling.
- Lesson: Hybrid of SA and biased walk, is a
- promising alternative to MCMC methods
- for sampling.

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