Integer and combinatorial optimization
Integer and combinatorial optimization
Chva´tal closures for mixed integer programming problems
Mathematical Programming: Series A and B
On the complexity of choosing the branching literal in DPLL
Artificial Intelligence
The Flatness Theorem for Nonsymmetric Convex Bodies Via the Local Theory of Banach Spaces
Mathematics of Operations Research
Computational Optimization and Applications
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
An Introduction to Empty Lattice Simplices
Proceedings of the 7th International IPCO Conference on Integer Programming and Combinatorial Optimization
A Computational Study of Search Strategies for Mixed Integer Programming
INFORMS Journal on Computing
The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
Active-constraint variable ordering for faster feasibility of mixed integer linear programs
Mathematical Programming: Series A and B
Valid inequalities for mixed integer linear programs
Mathematical Programming: Series A and B
Optimizing over the split closure
Mathematical Programming: Series A and B
Information-theoretic approaches to branching in search
IJCAI'07 Proceedings of the 20th international joint conference on Artifical intelligence
Branching on general disjunctions
Mathematical Programming: Series A and B
Column basis reduction and decomposable knapsack problems
Discrete Optimization
Operations Research Letters
Achieving MILP feasibility quickly using general disjunctions
Computers and Operations Research
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The imposition of general disjunctions of the form “$\pi x\leq\pi_0\vee\pi x\geq\pi_0+1$,” where $\pi,\pi_0$ are integer-valued, is a fundamental operation in both the branch-and-bound and cutting-plane 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 $\mathcal{NP}$-hard. We further show that the problem remains $\mathcal{NP}$-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 $\mathcal{NP}$-complete.