A New Branching Rule for Range Minimization Problems

We consider range minimization problems featuring exponentially many variables, as frequently arising in fairness-oriented or bi-objective optimization. While branch-and-price is successful at solving cost-oriented problems with many variables, the performance of classical branch-and-price algorithms for range minimization is drastically impaired by weak linear programming relaxations. We propose range branching, a generic branching rule that directly tackles this issue and is compatible with any problem-specific branching scheme. We show several desirable properties of range branching and show its effectiveness on a series of fair capacitated vehicle routing instances. Range branching significantly improves multiple classical branching schemes in terms of computing time, optimality gap, and size of the branch-and-bound tree, allowing us to solve many more large instances than classical methods.