Combinatorial Benders' Cuts for Mixed-Integer Linear Programming

TLDR

MIPs with big‑M logical implications are among the hardest to solve. The paper proposes an automatic reformulation to eliminate dependence on big‑M coefficients. The method uses a master ILP and a slave LP to generate combinatorial cuts from minimal infeasible subsystems, yielding a Benders‑style algorithm independent of big‑M values. The reformulation produces a tighter LP relaxation and solves hard MIPs orders of magnitude faster than the original model.

Abstract

Mixed-integer programs (MIPs) involving logical implications modeled through big-M coefficients are notoriously among the hardest to solve. In this paper, we propose and analyze computationally an automatic problem reformulation of quite general applicability, aimed at removing the model dependency on the big-M coefficients. Our solution scheme defines a master integer linear problem (ILP) with no continuous variables, which contains combinatorial information on the feasible integer variable combinations that can be “distilled” from the original MIP model. The master solutions are sent to a slave linear program (LP), which validates them and possibly returns combinatorial inequalities to be added to the current master ILP. The inequalities are associated to minimal (or irreducible) infeasible subsystems of a certain linear system, and can be separated efficiently in case the master solution is integer. The overall solution mechanism closely resembles the Benders' one, but the cuts we produce are purely combinatorial and do not depend on the big-M values used in the MIP formulation. This produces an LP relaxation of the master problem which can be considerably tighter than the one associated with original MIP formulation. Computational results on two specific classes of hard-to-solve MIPs indicate that the new method produces a reformulation which can be solved some orders of magnitude faster than the original MIP model.