An ETH-based quasipolynomial lower bound for Dualization

2026-10-08

AI Summary

This study addresses the long-standing open problem regarding the computational complexity lower bounds of monotone Boolean function dualization and the enumeration of minimal transversals in hypergraphs. Based on the Exponential Time Hypothesis (ETH), this work constructs a sub-exponential time reduction from 3-SAT to the complement of Dual and rigorously analyzes the time complexity of the problem using computational complexity theory. The primary contribution is the first proof that, under ETH, no output-polynomial-time algorithm exists, thereby establishing a quasi-polynomial time lower bound of $N^{o(\sqrt{\log N/\log \log N})}$. This result surpasses previously known complexity boundaries and confirms the optimality of the Fredman–Khachiyan algorithm.

Abstract

Dualizing monotone Boolean functions (or equivalently, enumerating minimal transversals in hypergraphs) is a long-standing problem whose output-polynomial-time solvability remains open. While various special cases have been extensively studied, the state-of-the-art algorithm for the general case, due to Fredman and Khachiyan, runs in quasipolynomial time. This paper presents a subexponential-time reduction from \textsc{3SAT} to the complement of \textsc{Dual}: Given a 3CNF formula with $n$ variables, the reduction constructs hypergraphs $\mathcal H$ and $\mathcal L$ of total size $2^{\bigoh(n^{2/3}(\log n)^{1/3})}$ such that $\mathcal L \subseteq \Tr(\mathcal H)$ and the formula is satisfiable if and only if $\mathcal L\neq \Tr(\mathcal H)$. As a consequence of this reduction, assuming the Exponential Time Hypothesis (ETH), neither \textsc{Dual} nor \textsc{Dualization} admits an algorithm running in $N^{o(\sqrt{\log N/\log\log N})}$ time, where $N$ is the input size for \textsc{Dual} and the combined input and output size for \textsc{Dualization}. In particular, \textsc{Dualization} cannot be solved in output-polynomial time under ETH.

Problem

• Dualization
• Monotone Boolean functions
• Minimal transversals
• Output-polynomial time
• Exponential Time Hypothesis

Innovation

• Dualization
• Exponential Time Hypothesis
• Quasipolynomial lower bound
• Subexponential-time reduction
• Monotone Boolean functions