On the lattice programming gap of the group problems

Abstract

Abstract. Given a full-dimensional lattice Λ ⊂ Z k and a cost vector l ∈ Q k >0 , we are concerned with the family of the group problemsThe lattice programming gap gap(Λ, l) is the largest value of the minima in (0.1) as r varies over Z k . We show that computing the lattice programming gap is NP-hard when k is a part of input. We also obtain lower and upper bounds for gap(Λ, l) in terms of l and the determinant of Λ. Introduction and statement of resultsConsider the integer programming problemGomory [11] defined a group relaxation of (1.1) as follows. Let B and N be the index sets of basic and nonbasic variables for an optimal basic solution to the linear programming relaxation min{c · x : Ax = b, x ≥ 0} of (1.1). Then the problem (1.1) can be written asand a relaxation of (1.2) is obtained by removing the restriction x B ≥ 0:Hence (1.3) is a lower bound for (1.1) and it can be used in any branch and bound procedure.The constraintsThus, given any nonnegative integral vector x N , the vector x B is integer if and only if (AThe program (1.4) is called the Gomory's group relaxation for (1.1).In this paper we fix a cost vector c ∈ Q n and for a matrix A ∈ Z d×n of rank d and b ∈ Sg(A) = {Au : u ∈ Z n ≥0 } consider the integer program IP c (A, b) = min{c · x : Ax = b, x ∈ Z n ≥0 } .

Distances to lattice points in knapsack polyhedra

“…Here b ranges over all integers such that (5) is feasible and bounded. Notice that computing Gap(c, a) when n is a part of input is NP-hard (see Aliev [1] and Eisenbrand et al [9]). For any fixed n, the integer programming gap can be computed in polynomial time due to results of Hoşten and Sturmfels [16] (see also Eisenbrand and Shmonin [10]).…”

“…Here b ranges over all integers such that (5) is feasible and bounded. Notice that computing Gap(c, a) when n is a part of input is NP-hard (see Aliev [1] and Eisenbrand et al [9]).…”

Integrality Gaps of Integer Knapsack Problems

“…Appendix: Proof of Theorem 5We will denote for A ∈ Q(T ) the index of a maximum coordinate by i(A) and we set c A = −e i(A) . The tuples (A, c A ) are generic and in view of Theorem 1.8 we findGap c A (A) ≥ ρ n−1 a n T n−1 T 1−1/(n−1) ≤ 2 n T n−1/(n−1) …”

“…Note that for n = 2 we have Gap(Λ A , l) = l 1 (|A τ | − 1) and thus (4.4) implies (1.8). For n > 2, the bound (1.8) immediately follows from (4.4) and Theorem 1.2(i) in [1].…”

“…We will say that the tuple (A, c) is generic if for any positive b ∈ Z the linear programming relaxation (1.2) has a unique optimal solution. An optimal lower bound for Gap c (A) with generic (A, c) can be obtained using recent results [1] on the lattice programming gaps associated with the group relaxations to (1.1).…”