Probabilistic Model Counting with Short XORs

TL;DR

This paper introduces a method for probabilistic model counting using shorter XOR constraints, maintaining rigorous guarantees by leveraging the geometry of solution sets akin to error-correcting codes.

Contribution

It demonstrates that shorter parity constraints can be used effectively for model counting, improving efficiency while preserving theoretical guarantees.

Findings

Shorter XOR constraints are sufficient for accurate probabilistic counting.

The solution space geometry resembles error-correcting codes, enabling rigorous guarantees.

Method improves computational efficiency over traditional approaches.

Abstract

The idea of counting the number of satisfying truth assignments (models) of a formula by adding random parity constraints can be traced back to the seminal work of Valiant and Vazirani, showing that NP is as easy as detecting unique solutions. While theoretically sound, the random parity constraints in that construction have the following drawback: each constraint, on average, involves half of all variables. As a result, the branching factor associated with searching for models that also satisfy the parity constraints quickly gets out of hand. In this work we prove that one can work with much shorter parity constraints and still get rigorous mathematical guarantees, especially when the number of models is large so that many constraints need to be added. Our work is based on the realization that the essential feature for random systems of parity constraints to be useful in probabilistic model counting is that the geometry of their set of solutions resembles an error-correcting code.