An inference algorithm for monotone Boolean functions associated with undirected graphs

Boolean functions are a modelling tool useful in many applications; monotone Boolean functions make up an important class of these functions. For instance, monotone Boolean functions can be used for describing the structure of the feasible subsystems of an infeasible system of constraints, because feasibility is a monotone feature. In this paper we consider monotone Boolean functions (MBFs), associated with undirected graphs, whose upper zeros are defined as binary tuples for which the corresponding subgraph of the original undirected graphs is either the empty graph, or it has no edges.
For this class of MBFs, we present the settings of problems which are related to the search for upper zeros and maximal upper zeros of these functions. The notion of $k$-vertices and $(k, m)$-vertices in a graph is introduced. It is shown that for any $k$-vertices of the original graph there exists a maximal upper zero of an MBF associated with the graph, in which the component $x_{i}$ corresponding to this $k$-vertex takes the value $1$.
Based on this statement, we construct an algorithm of searching for a maximal upper zero, for the class of MBFs under consideration, which allows one to find, under certain conditions, the solution to the problem of searching for a maximal upper zero, or to substantially reduce the dimension of the original problem.
The proposed algorithm was extended for the case of $(k, m)$-vertices. This extended algorithm allows one to fix a bound on the deviation of an upper zero of the MBF from the maximal upper zeros, in the sense of the number of units in these tuples. The algorithm has the complexity $O(n^2p)$, where $n$ is a number of vertices and $p$ is a number of edges of the original graph.