Self-reducible with easy decision version counting problems admit additive error approximation. Connections to counting complexity, exponential time complexity, and circuit lower bounds

TL;DR

This paper studies self-reducible counting problems with easy decision versions, showing they can be approximated with additive error in polynomial time and connecting these results to circuit lower bounds and derandomization.

Contribution

It proves that such problems admit additive and multiplicative approximations efficiently, and links these properties to circuit complexity and the Circuit Acceptance Probability Problem.

Findings

Problems can be approximated within exponential error in polynomial time.

Multiplicative approximations are achievable faster than exhaustive search.

Circuit Acceptance Probability Problem can be solved efficiently for certain circuit classes.

Abstract

We consider the class of counting problems,i.e. functions in $\#$P, which are self reducible, and have easy decision version, i.e. for every input it is easy to decide if the value of the function $f(x)$ is zero. For example, $\#$independent-sets of all sizes, is such a problem, and one of the hardest of this class, since it is equivalent to $\#$SAT under multiplicative approximation preserving reductions. Using these two powerful properties, self reducibility and easy decision, we prove that all problems/ functions $f$ in this class can be approximated in probabilistic polynomial time within an absolute exponential error $\epsilon\cdot 2^{n'}, \forall\epsilon>0$, which for many of those problems (when $n'=n+$constant) implies additive approximation to the fraction $f(x)/2^n$. (Where $n'$ is the amount of non-determinism of some associated NPTM). Moreover we show that for all these problems we can have multiplicative error to the value $f(x)$, of any desired accuracy (i.e. a RAS), in time of order $2^{2n'/3}poly(n)$, which is strictly smaller than exhaustive search. We also show that $f(x)<g(x)$ can be decided deterministically in time $g(x)poly(n), \forall g$. Finally we show that the Circuit Acceptance Probability Problem, which is related to derandomization and circuit lower bounds, can be solved with high probability and in polynomial time, for the family of all circuits for which the problems of counting either satisfying or unsatisfying assignments belong to TotP (which is the Karp-closure of self reducible problems with easy decision version).