New Korkin-Zolotarev Inequalities

Korkin and Zolotarev showed that if $$\sum_i A_i(x_i-\sum_{j>i} \alpha_{ij}x_j)^2$$ is the Lagrange expansion of a Korkin–Zolotarev reduced positive definite quadratic form, then $A_{i+1}\geq \frac{3}{4} A_i$ and $A_{i+2}\geq \frac{2}{3}A_i$. They showed that the implied bound $A_{5}\geq \frac{4}{9}A_1$ is not attained by any KZ-reduced form. We propose a method to optimize numerically over the set of Lagrange … Read more

On generalized branching methods for mixed integer programming

In this paper we present a restructuring of the computations in Lenstra’s methods for solving mixed integer linear programs. We show that the problem of finding a good branching hyperplane can be formulated on an adjoint lattice of the Kernel lattice of the equality constraints without requiring any dimension reduction. As a consequence the short … Read more

Hard equality constrained integer knapsacks

We consider the following integer feasibility problem: “Given positive integer numbers $a_0,a_1,\dots,a_n,$ with $\gcd(a_1,\dots,a_n)=1$ and $\va=(a_1,\dots,a_n)$, does there exist a vector $\vx\in\bbbz^n_{\ge \zero}$ satisfying $\va\vx = a_0$?” Some instances of this type have been found to be extremely hard to solve by standard methods such as branch-and-bound, even if the number of variables is as … Read more