On Almost-Uniform Generation of SAT Solutions: The power of 3-wise independent hashing

Abstract

Given a Boolean formula φ and a distribution parameter ε, the problem of almost-uniform generation seeks to design a randomized generator such that every solution of φ is output with probability within (1 + ε)-factor of where sol(φ) is the set of all the solutions of φ. The prior state of the art scheme due to Jerrum, Valiant, and Vazirani, makes calls to a SAT oracle and employs 2 − wise independent hash functions. In this work, we design a new randomized algorithm that makes calls to a SAT oracle and employs 3 − wise independent hash functions. The widely used 2 − wise independent hashing is tabulation hashing proposed by Carter and Wegman. Since this classical scheme is also 3 − wise independent, we observe that practical implementation of our technique does not incur additional overhead. We demonstrate that theoretical improvements translate to practice; in particular, we conduct a comprehensive study over 562 benchmarks and demonstrate that while JVV would time out for 544 out of 562 instances, our proposed scheme can handle all the 562 instances. To the best of our knowledge, this is the first almost-uniform generation scheme that can handle practical instances from real-world applications. We also present a nuanced analysis focusing on the both the size of SAT queries as well as the number of queries.