Branching on Multi-aggregated Variables

Abstract

Abstract. In mixed-integer programming, the branching rule is a key component to a fast convergence of the branch-and-bound algorithm. The most common strategy is to branch on simple disjunctions that split the domain of a single integer variable into two disjoint intervals. Multi-aggregation is a presolving step that replaces variables by an affine linear sum of other variables, thereby reducing the problem size. While this simplification typically improves the performance of MIP solvers, it also restricts the degree of freedom in variable-based branching rules. We present a novel branching scheme that tries to overcome the above drawback by considering general disjunctions defined by multi-aggregated variables in addition to the standard disjunctions based on single variables. This natural idea results in a hybrid between variable-and constraint-based branching rules. Our implementation within the constraint integer programming framework SCIP incorporates this into a full strong branching rule and reduces the number of branch-and-bound nodes on a general test set of publicly available benchmark instances. For a specific class of problems, we show that the solving time decreases significantly.

Integer Programming and Combinatorial Optimization

“…Many of the well-known cuts generated by MIP solvers can be derived as split cuts [14] and they are effective in closing the integrality gap in practice [18]. Branching typically uses only simple split disjunctions (where the a above is a unit vector), although some studies have considered the computational performance of branching on general disjunctions [15,20,30,38].…”