On the Complexity of Selecting Disjunctions in Integer Programming

Abstract

The imposition of general disjunctions of the form "πx ≤ π 0 ∨ πx ≥ π 0 + 1", where π, π 0 are integer valued, is a fundamental operation in both the branch-and-bound and cuttingplane algorithms for solving mixed integer linear programs. Such disjunctions can be used for branching at each iteration of the branch-and-bound algorithm or to generate split inequalities for the cutting-plane algorithm. We first consider the problem of selecting a general disjunction and show that the problem of selecting an optimal such disjunction, according to specific criteria described herein, is N P-hard. We further show that the problem remains N P-hard even for binary programs or when considering certain restricted classes of disjunctions. We observe that the problem of deciding whether a given inequality is a split inequality can be reduced to one of the above problems, which leads to a proof that the problem is N P-complete.

Branching on Multi-aggregated Variables

“…On the theoretical side, Mahajan and Ralphs proved that the problem of finding a general disjunction with maximal objective gain is N P-hard [26]. Finally, Local Branching by Fischetti and Lodi [27] is a strategy to interleave variablebased branching with branching on general {−1, 0, 1}-disjunctions.…”

Improved branching disjunctions for branch-and-bound: An analytic center approach

“…Computational experiments show a significant reduction in the number of branching nodes. The mixed integer program that is used to generate the disjunctions, however, is N P-hard and could be computationally expensive to solve in practice (Mahajan & Ralphs, 2010).…”

Achieving MILP feasibility quickly using general disjunctions

“…There is an infinite number of ways to construct a general disjunction, and it has been shown that selecting the best one is NP-hard [Mahajan and Ralphs 2010]. For this reason we concentrate on heuristic methods that can quickly construct a general disjunction that is likely to be effective.…”

“…Further, it can be time-consuming to construct an effective general disjunction. Mahajan and Ralphs [2010] determined that the complexity of selecting the general disjunction at each node is NP-hard. Even selecting a general disjunction in which all the coefficients of the disjunction function are in the set {-1,0,1} is an NP-complete problem.…”