New Results on Monotone Dualization and Generating Hypergraph Transversals

TL;DR

This paper advances the understanding of monotone dualization by providing new polynomial time results for key cases, and explores the complexity of duality checking with limited nondeterminism, shedding light on an important open problem.

Contribution

It introduces new polynomial and output-polynomial time results for monotone dualization and demonstrates that duality can be disproved with limited nondeterminism in polynomial time.

Findings

New polynomial time results for significant cases of monotone dualization.

Disproof of duality with limited nondeterminism in polynomial time.

Improved understanding of the complexity of the dualization problem.

Abstract

We consider the problem of dualizing a monotone CNF (equivalently, computing all minimal transversals of a hypergraph), whose associated decision problem is a prominent open problem in NP-completeness. We present a number of new polynomial time resp. output-polynomial time results for significant cases, which largely advance the tractability frontier and improve on previous results. Furthermore, we show that duality of two monotone CNFs can be disproved with limited nondeterminism. More precisely, this is feasible in polynomial time with O(chi(n) * log n) suitably guessed bits, where chi(n) is given by \chi(n)^chi(n) = n; note that chi(n) = o(log n). This result sheds new light on the complexity of this important problem.