Checking-Up on Branch-and-Check

Branch-and-Check, introduced ten years ago, is a generalization of logic-based Benders decomposition. The key extension is to solve the Benders sub-problems at each feasible solution of the master problem rather than only at an optimal solution. We perform the first systematic empirical comparison of logic-based Benders decomposition and branchand-check. On four problem types the results indicate that either Benders or branch-and-check may perform best, depending on the relative difficulty of solving the master problem and the sub-problems. We identify a characteristic of the logic-based Benders decomposition runs, the proportion of run-time spent solving the master problem, that is valuable in predicting the performance of branch-and-check. We also introduce a variation of branch-and-check to address difficult sub-problems. Empirical results show that this variation leads to more robust performance than both logic-based Benders decomposition and branch-and-check on the problems investigated.