The integrality number of an integer program

Joseph Paat, Miriam Schlöter, Robert Weismantel

Abstract: 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 $$\varDelta $$ Δ of the constraint matrix, our analysis allows us to make statements of the following form: there exists a number $$\tau (\varDelta )$$ τ ( Δ ) such that an IP with n many variables and $$n + \sqrt{n /\tau (\varDelta )}$$ 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 $$O(\sqrt{\varDelta })$$ O ( Δ ) many integer constraints.