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.
REF determined that the complexity of selecting the general disjunction at each node is NP-hard.
On the Complexity of Selecting Disjunctions in Integer Programming
SIAM Journal on Optimization
Computer Science Mathematics