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# Sampling Combinatorial Space Using Biased Random Walks - PowerPoint PPT Presentation

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).

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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.