The integrality number of an integer program

• Published in: Mathematical programming Vol. 192; no. 1-2; pp. 271 - 291
• Main Authors: Paat, Joseph, Schlöter, Miriam, Weismantel, Robert
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2022; Springer; Springer Nature B.V
• Subjects: Calculus of Variations and Optimal Control; Optimization; Combinatorics; Full Length Paper; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Mixed integer; Numerical Analysis; Theoretical; 90C10; 90C11
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-021-01651-0

We introduce the integrality number of an integer program (IP). Roughly speaking, the integrality number is the smallest number of integer constraints needed to solve an IP via a mixed integer (MIP) relaxation. One notable property of this number is its invariance under unimodular transformations of the constraint matrix. Considering the largest minor Δ of the constraint matrix, our analysis allows us to make statements of the following form: there exists a number τ ( Δ ) such that an IP with n many variables and n + n / τ ( Δ ) many inequality constraints can be solved via a MIP relaxation with fewer than n integer constraints. From our results it follows that IPs defined by only n constraints can be solved via a MIP relaxation with O ( Δ ) many integer constraints.