The (not so) trivial lifting in two dimensions

Abstract

When generating cutting-planes for mixed-integer programs from multiple rows of the simplex tableau, the usual approach has been to relax the integrality of the non-basic variables, compute an intersection cut, then strengthen the cut coefficients corresponding to integral non-basic variables using the so-called trivial lifting procedure. Although of polynomial-time complexity in theory, this lifting procedure can be computationally costly in practice. For the case of two-row relaxations, we present a practical algorithm that computes trivial lifting coefficients in constant time, for arbitrary maximal lattice-free sets. Computational experiments confirm that the algorithm works well in practice.

A Source code

The source code for the trivial lifting algorithm presented in Sect. 2 has been made available online [13] under the GNU General Public License, version 3. The source code is written in ANSI C, having only IBM ILOG CPLEX as its external dependency (required only for benchmarking mip and mip-pre methods). The structure of the project is presented in Table 3. Complete instructions to compile the source code, run the computational experiments and generate the tables have been included in the file README.md.

Table 3 Directory structure and main files

Let $S \subseteq \mathbb {R}^2$ be a lattice-free set containing $f \in \mathbb {R}^2 \setminus \mathbb {Z}^2$ in its interior. In our code, we make the assumption that S is a bounded set, and therefore can be represented either as the convex combination of vertices $v^1,\ldots ,v^k \in \mathbb {R}^2$, or as the as the intersection of half-spaces $\{x \in \mathbb {R}^2 : h^i(x-f) \le 1, \text {for } i=1,\ldots ,q\}$. For maximal lattice-free sets, our code also assumes that a list of lattice points in the boundary of the set is given. This data is stored in the structure LFreeSet2D. The halfspace representation is needed for evaluating the gauge function and performing the trivial lifting. If this representation is not readily available, the source code includes the auxiliary function LFREE_2D_compute_halfspaces, which computes it, given the interior point f and the list of vertices. The source code also includes the function LFREE_2D_preprocess, which implements the preprocessing step described in Sect. 3. This function receives an LFreeSet2D, modifies it in-place and returns the transformation matrix. After the set is preprocessed, the trivial lifting function can be efficiently computed by calling LIFTING_2D_bound. A sample usage is presented below. For brevity, no preprocessing is performed.

Finally, for benchmarking purposes, the source code also includes the functions LIFTING_2D_naive and LIFTING_2D_mip, which implement the naive and the MIP methods, respectively. Other auxiliary functions include LFREE_2D_get_bounding_box, which computes the bounding box for a given bounded lattice-free set, and LIFTING_2D_psi, which simply evaluates the gauge function. More complete examples, showing the usages of all these functions, can be found in the unit tests, which are included in the source code package.