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The present paper develops a branching rule based on the flatness of the polyhedron. The base of this rule is the vector of minimal integer width. There is empirical evidence supporting the conjecture that the direction with the highest value of the components of this vector is a good direction for branching. In this paper, it is provided theoretical results demonstrating that the columns of the matrix A corresponding to the set of constraints may be used to estimate the vector of minimal integer width; this fact is used for building a new version of the branching rule reported earlier. In addition, the new rule is implemented by using a branching direction choosing the child node which is closest to the integer value (either up or down). Thus, a variable rule for descending the tree is used. Every time that a new sub-problem is solved, the list of yet not-solved sub-problems are analyzed and get priority those problems with minimum estimate of the objective function value. The conclusions of the work are based on the knapsack problems of the Knapsack OR-Library.  From the results, it is concluded that the rule designed presents low execution times and minimal number of nodes generated.

Accelerating the B&B algorithm for integer programming based on flatness information: an approach applied to the multidimensional knapsack problem