On the complexity of probabilistic trials for hidden satisfiability problems

TL;DR

This paper explores the complexity of solving hidden satisfiability problems using probabilistic and quantum methods, demonstrating polynomial-time solutions for 1SAT and 2SAT under certain models and conditions.

Contribution

It introduces a probabilistic oracle model for hidden SAT problems and shows polynomial-time solutions for 1SAT and 2SAT, extending results to the quantum regime.

Findings

1SAT can be solved in polynomial time with probabilistic access.

2SAT can be learned and solved in polynomial time without repeated clauses.

Quantum algorithms can solve 1QSAT and learn 2QSAT efficiently.

Abstract

What is the minimum amount of information and time needed to solve 2SAT? When the instance is known, it can be solved in polynomial time, but is this also possible without knowing the instance? Bei, Chen and Zhang (STOC '13) considered a model where the input is accessed by proposing possible assignments to a special oracle. This oracle, on encountering some constraint unsatisfied by the proposal, returns only the constraint index. It turns out that, in this model, even 1SAT cannot be solved in polynomial time unless P=NP. Hence, we consider a model in which the input is accessed by proposing probability distributions over assignments to the variables. The oracle then returns the index of the constraint that is most likely to be violated by this distribution. We show that the information obtained this way is sufficient to solve 1SAT in polynomial time, even when the clauses can be repeated. For 2SAT, as long as there are no repeated clauses, in polynomial time we can even learn an equivalent formula for the hidden instance and hence also solve it. Furthermore, we extend these results to the quantum regime. We show that in this setting 1QSAT can be solved in polynomial time up to constant precision, and 2QSAT can be learnt in polynomial time up to inverse polynomial precision.