4.4  Example of Branch and Bound applied to Unate covering

Let us consider the constraint matrix given below. It may be noted that this matrix cannot be cannot be further simplified. If we apply the quick lower bound estimate discussed in the last subsection we get Lower Bound = 4 comprising {1, 3,5,7}.

Now we start exploring the solution space. Let is start with selecting PI1 (i.e., PI1=1). With PI1 being taken, the following happens in the matrix

•  rows minterm1, minterm4, and minterm112 are covered,

•  columns PI2 (by PI3) and PI4 (by PI3) are dominated

•  column PI3 becomes essential.

The Binary search tree illustrating the search in the solution space by the branch and bound algorithm is shown in Figure 2. The left edge (right dotted edge) indicates the column being (not being) considered.

The left edge from root note indicates PI1 being taken. Also in the edge it is marked that “PI3, PI2',PI4”—this indicates that because of taking PI1, PI3 has to be considered and PI2 and PI4 get eliminated.

After reduction (i.e., taking PI1, PI3 and eliminating PI2, PI3) we get the following matrix. As of now, the cost of the solution is 2 (PI1, PI3).