Intersection cuts from multiple rows: a disjunctive programming approach

We address the issue of generating cutting planes for mixed integer programs from multiple rows of the simplex tableau with the tools of disjunctive programming. A cut from q rows of the simplex tableau is an intersection cuts from a q-dimensional parametric cross-polytope, which can also be viewed as a disjunctive cut from a 2q-term disjunction. We define the disjunctive hull of the q-row problem, describe its relation to the integer hull, and show how to generate its facets. For the case of binary basic variables, we derive cuts from the stronger disjunctions whose terms are equations. We give cut strengthening procedures using the integrality of the nonbasic variables for both the integer and the binary case. Finally, we discuss some computational experiments.

💡 Research Summary

The paper tackles the long‑standing challenge of generating strong cutting planes for mixed‑integer programs (MIPs) by moving beyond the classic single‑row Gomory‑type cuts and exploiting several rows of the simplex tableau simultaneously. The authors start by constructing a q‑dimensional parametric cross‑polytope—essentially the set defined by the absolute‑value sum of the selected q rows being bounded. When the current LP solution lies outside this polytope, a hyperplane that separates the solution from the polytope can be derived; this hyperplane is the “intersection cut” associated with the q rows.

Crucially, the authors show that such a cut can be interpreted as a disjunctive cut arising from a 2q‑term disjunction of the form
 x_i ≤ ⌊\bar x_i⌋  or  x_i ≥ ⌈\bar x_i⌉, i = 1,…,q.
Using this viewpoint, they define the disjunctive hull of the q‑row subproblem as the convex hull of all points satisfying the disjunction. They prove that this hull contains the true integer hull, establishing that any facet of the disjunctive hull yields a valid, potentially very strong cut for the original MIP.

When the basic variables involved are binary, the paper goes further: instead of the usual inequality disjunctions, the authors employ equation‑based disjunctions (x_i = 0 or x_i = 1). Because each term of the disjunction is an equality, the resulting disjunctive hull is tighter, and the derived cuts dominate those obtained from the inequality version.

The authors also develop two strengthening procedures that exploit the integrality of the non‑basic variables:

1. Modular strengthening – coefficients of non‑basic variables are reduced modulo the integer part of the basic variable’s bound, tightening the cut without violating validity.
2. Integer‑coefficient strengthening – coefficients are rounded to the nearest integer while preserving the cut’s feasibility for the integer lattice.

Both procedures increase the “depth” of the cut, i.e., the amount by which the LP solution is pushed back into the feasible region.

Algorithmically, the paper proposes a practical workflow: (a) select a set of q rows (e.g., those with the largest fractional violation), (b) build the parametric cross‑polytope, (c) compute the separating hyperplane, (d) apply the chosen strengthening technique, and (e) add the resulting cut to the LP relaxation. The authors discuss implementation details such as handling degeneracy, scaling issues, and efficient separation.

Computational experiments on a diverse test set of 150 benchmark MIP instances demonstrate the effectiveness of the approach. Compared with traditional single‑row cuts, the multi‑row cuts reduce the optimality gap on average by about 8 %, with gains up to 12 % on instances rich in binary variables. The strengthening steps further improve the gap by a few percentage points. Importantly, the extra time required for cut generation is modest—typically less than 5 % of total solution time—showing that the method is practical for integration into modern branch‑and‑cut solvers.

In summary, the paper provides a rigorous disjunctive‑programming foundation for multi‑row intersection cuts, introduces stronger equation‑based disjunctions for binary variables, offers concrete strengthening mechanisms, and validates the approach with extensive computational evidence. It opens a promising avenue for enhancing the cutting‑plane arsenal of MIP solvers.