Abstract Implied-integer detection is a well-known presolving technique that is used by many Mixed-Integer Linear Programming solvers. Informally, a variable is said to be implied integer if its integrality is enforced implicitly by integrality of other variables and the constraints of a problem. In this case, algorithms used in MILP software can choose whether they treat it as an integer or continuous variable. In this work we formalize the definition of implied integrality by taking a polyhedral perspective. Our main result characterizes implied integrality as occurring when a subset of integer variables is fixed to integer values and the polyhedron on the remaining variables is integral. While integral polyhedra are well-understood theoretically, existing detection methods infer implied integrality only for one variable at a time. We introduce new detection methods based on the detection of integral polyhedra, extending existing techniques to multiple variables. Additionally, we discuss the computational complexity of recognizing implied integers. We conduct experiments using a new detection method that uses totally unimodular submatrices to identify implied integrality. For the MIPLIB 2017 collection dataset our results indicate that, on average, 18.8 % of the variables are classified as implied integer after presolving, compared to just 3.3 % identified by state-of-the-art techniques. Moreover, we are able to reduce the average percentage of variables whose integrality needs to be enforced after presolving from 70.2% to 59.0%.

The SCIP Optimization Suite provides a collection of software packages for mathematical optimization centered around the constraint integer programming framework SCIP . The focus of this article is on the role of the SCIP Optimization Suite in supporting research. SCIP ’s main design principles are discussed, followed by a presentation of the latest performance improvements and developments in version 8.0, which serve both as examples of SCIP ’s application as a research tool and as a platform for further developments. Furthermore, this article gives an overview of interfaces to other programming and modeling languages, new features that expand the possibilities for user interaction with the framework, and the latest developments in several extensions built upon SCIP .