Mathematical Programming

, Volume 145, Issue 1–2, pp 199–222

# Design and verify: a new scheme for generating cutting-planes

• Santanu S. Dey
• Sebastian Pokutta
Full Length Paper Series A

## Abstract

A cutting-plane procedure for integer programming (IP) problems usually involves invoking a black-box procedure (such as the Gomory–Chvátal procedure) to compute a cutting-plane. In this paper, we describe an alternative paradigm of using the same cutting-plane black-box. This involves two steps. In the first step, we design an inequality $$cx \le d$$ where $$c$$ and $$d$$ are integral, independent of the cutting-plane black-box. In the second step, we verify that the designed inequality is a valid inequality by verifying that the set $$P \cap \{x\in \mathbb R ^n \mid cx \ge d + 1\} \cap \mathbb Z ^n$$ is empty using cutting-planes from the black-box. Here $$P$$ is the feasible region of the linear-programming relaxation of the IP. We refer to the closure of all cutting-planes that can be verified to be valid using a specific cutting-plane black-box as the verification closure of the considered cutting-plane black-box. This paper undertakes a systematic study of properties of verification closures of various cutting-plane black-box procedures.

## Keywords

Verification scheme Cutting planes Integer programming

90C10 90C57

## Notes

### Acknowledgments

The authors are most grateful to the anonymous referees for their detailed remarks that considerably improved the presentation and simplified some of the proofs.

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