The Integrality Number of an Integer Program

Abstract

We introduce the integrality number of an integer program (IP) in inequality form. 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 ∆ of the constraint matrix, our analysis allows us to make statements of the following form: there exist numbers τ (∆) and κ(∆) such that an IP with n ≥ τ (∆) many variables and n + κ(∆) · √ n many inequality constraints can be solved via a MIP relaxation with fewer than n integer constraints. A special instance of our results shows that IPs defined by only n constraints can be solved via a MIP relaxation with O( √ ∆) many integer constraints.1 Introduction.Let A ∈ Z m×n satisfy rank(A) = n and c ∈ Z n . We denote the integer linear program parameterized by right hand side b ∈ Z m by IP A,c (b) := max{c ⊺ x : Ax ≤ b and x ∈ Z n }.See [8] for more on parametric integer programs. We are interested in solving IP A,c (b) by relaxing it to have fewer integer constraints. In special cases we can solve IP A,c (b) with zero integer constraints by only solving its linear relaxation LP A,c (b) := max{c ⊺ x : Ax ≤ b}.However, these special cases require the underlying polyhedron to have optimal integral vertices, which occurs for instance when A is totally unimodular. Our target is to consider general matrices A and relaxations in the form of a mixed integer program:where k ∈ Z ≥0 and W ∈ Z k×n satisfies rank(W ) = k.One sufficient condition for solving IP A,c (b) using a mixed integer relaxation is that the vertices of W-MIP A,c (b) are integral. The vertices of W-MIP A,c (b) are the vertices of the polyhedron conv {x ∈ R n : Ax ≤ b, W x ∈ Z k } .

Integer programs with bounded subdeterminants and two nonzeros per row

“…, ∆}. In view of other work on linear and integer programs with bounded subdeterminants [42,44,3,11,13,34], it is tempting to believe that integer programs defined by totally ∆-modular matrices can be solved in polynomial time whenever ∆ is a constant. While this question is still open (even for ∆ = 3), we prove the following result.…”

Excluding a line from $\mathbb C$-representable matroids

“…Our final approximate result, which we prove at the end of Section 8, can be used to show that certain integer programs can be solved efficiently; see [34] and [17] for a detailed discussion. Given a positive integer ∆, we say that an integer matrix A is ∆-modular if the determinant of each rank(A) × rank(A) submatrix has absolute value at most ∆.…”

Complexity of optimizing over the integers

“…In 2009, Veselov and Chirkov showed that the feasibility problem can be solved in polynomial time when ∆ = 2 [102], and the optimization version was resolved using deep techniques from combinatorial optimization by Artmann, Weismantel and Zenkulsen in [5]. See also related results for general ∆ in [4,12,[57][58][59]89].…”