Experimental algorithms for the dualization problem
Mauro Mezzini, Fernando Cuartero Gomez, Jose Javier Paulet Gonzalez, Hernan Indibil de la Cruz Calvo, Vicente Pascual, Fernando L. Pelayo
TL;DR
This work tackles the dualization problem for positive Boolean functions represented in PIDNF, introducing a practical classical algorithm (Dual) with scalable performance in relevant hypergraph regimes and contrasting it with prior methods whose complexities scale as $N^{O(\,\log^2 N\,)}$ or $N^{o(\log N)}$, where $N=|F|+|G|$. It develops two constructive primitives: counting all hitting sets of a hypergraph using a subset-removal approach and a simple scheme to locate a minimal hitting set via the Dual framework. Extensive experiments demonstrate that counting via subset removal and the minimal hitting set search outperform several alternatives, including Algorithm A, across generated hypergraphs; results are made reproducible via an online Colab notebook. The authors also sketch potential quantum-speedup avenues and plan future work to test on hypergraphs with exponential edge counts and to compare with quantum computing platforms, highlighting practical impact for understanding dualization in combinatorial structures and Boolean function analysis.
Abstract
In this paper, we present experimental algorithms for solving the dualization problem. We present the results of extensive experimentation comparing the execution time of various algorithms.