Concepts of consistency have long played a key role in constraint programming but never developed in integer programming (IP). Consistency nonetheless plays a role in IP as well. For example, cutting planes can reduce backtracking by achieving various forms of consistency as well as by tightening the linear programming (LP) relaxation. We introduce a type of consistency that is particularly suited for 0-1 programming and develop the associated theory. We define a 0-1 constraint set as LP-consistent when any partial assignment that is consistent with its linear programming relaxation is consistent with the original 0-1 constraint set. We prove basic properties of LP-consistency, including its relationship with Chvátal-Gomory cuts and the integer hull. We show that a weak form of LP-consistency can reduce or eliminate backtracking in a way analogous to k-consistency but is easier to achieve. In so doing, we identify a class of valid inequalities that can be more effective than traditional cutting planes at cutting off infeasible 0-1 partial assignments.

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We study network models where flows cannot be split or merged when passing through certain nodes, i.e., for such nodes, each incoming arc flow must be matched to an outgoing arc flow of identical value. This requirement, which we call no-split no-merge (NSNM), appears in railroad applications where train compositions can only be modified at yards where necessary equipment is available. This combinatorial requirement is crucial when formulating problems occurring in the unit train business. We propose modeling approaches to represent the NSNM requirement. In particular, we give a linear formulation of the requirement on a single node that describes the convex hull in a lifted space. We present a cut-generating linear program to obtain valid inequalities in the original space of variables, and introduce a polynomial-time procedure to lift strong inequalities of lower-dimensional models into strong inequalities of the original model. In addition, we identify an exponential family of facet-defining inequalities that can be separated efficiently. To evaluate our results computationally, we study a stylized unit train problem. We compare a solution approach based on our results with one that relies on column generation. We then show that our results significantly reduce relaxation times and gaps when compared to leading commercial branch-and-cut software.