Renyi entropies as a measure of the complexity of counting problems

Abstract

Counting problems such as determining how many bit strings satisfy a given Boolean logic formula are notoriously hard. In many cases, even getting an approximate count is difficult. Here we propose that entanglement, a common concept in quantum information theory, may serve as a telltale of the difficulty of counting exactly or approximately. We quantify entanglement by using Renyi entropies S(q), which we define by bipartitioning the logic variables of a generic satisfiability problem. We conjecture that S(q\rightarrow 0) provides information about the difficulty of counting solutions exactly, while S(q>0) indicates the possibility of doing an efficient approximate counting. We test this conjecture by employing a matrix computing scheme to numerically solve #2SAT problems for a large number of uniformly distributed instances. We find that all Renyi entropies scale linearly with the number of variables in the case of the #2SAT problem; this is consistent with the fact that neither exact nor approximate efficient algorithms are known for this problem. However, for the negated (disjunctive) form of the problem, S(q\rightarrow 0) scales linearly while S(q>0) tends to zero when the number of variables is large. These results are consistent with the existence of fully polynomial-time randomized approximate algorithms for counting solutions of disjunctive normal forms and suggests that efficient algorithms for the conjunctive normal form may not exist.