NP$^{\#P}$ = $\exists$PP and other remarks about maximized counting

TL;DR

This paper explores the complexity of a decision problem related to counting solutions in propositional formulas, connecting it to the class NP#P and discussing its implications for maximized counting problems.

Contribution

It establishes the complexity relationship between MAX#SAT decision problems and the class NP#P, providing new insights into counting and maximization in propositional logic.

Findings

DMAX#SAT is a central problem in counting complexity.

The paper shows the equivalence of NP#P to certain maximization problems.

Results have implications for understanding the complexity of counting solutions.

Abstract

We consider the following decision problem DMAX#SAT, and generalizations thereof: given a quantifier-free propositional formula $F(\mathbf{x},\mathbf{y})$, where $\mathbf{x},\mathbf{y}$ are tuples of variables, and a bound $B$, determine if there is $\vec{x}$ such that $\#\{\mathbf{y} \mid F(\mathbf{x},\mathbf{y})\} \geq B$. This is the decision version of the problem of MAX#SAT: finding $\mathbf{x}$ and $B$ for maximal $B$.