Geometric aspects related to solutions of #kSAT

Abstract

#kSAT is & complex problem equivalent to calculate the cardinalities of the null sets of conjunctive forms consisting of clauses with an uniform length. Each such null set is the union of linear varieties of uniform dimension in the hypercube. Here we study the class of sets in the hypercube that can be realized as such null sets. We look toward to characterize their cardinalities and the number of ways that they can be expressed as unions of linear varieties of uniform dimension. Using combinatorial and graph theory argumentations, we give such characterizations for very extremal values of k, either when it is very small or close to the hypercube dimension, and of the number of clauses appearing in an instance, either of value 2, or big enough to get a contradiction. © Springer-Verlag Berlin Heidelberg 2005.