Lattice-free Convex Sets Research Articles

A remark on perimeter–diameter and perimeter–circumradius inequalities under lattice constraints

In this note, we are concerned with planar convex sets whose interior is free from points of the integer lattice. Such lattice-free convex sets have been extensively studied with respect to determining sharp relations between any two of the geometric functionals: perimeter, diameter, circumradius, inradius, minimal width and area. The cases perimeter–diameter and perimeter–circumradius have been left open and are the subject of our investigations. We formulate precise conjectures on linear inequalities for these settings and prove them in special yet illustrative cases. Moreover, we obtain sharp non-linear inequalities that hold unconditionally.

The triangle closure is a polyhedron

Recently, cutting planes derived from maximal lattice-free convex sets have been studied intensively by the integer programming community. An important question in this research area has been to decide whether the closures associated with certain families of lattice-free sets are polyhedra. For a long time, the only result known was the celebrated theorem of Cook, Kannan and Schrijver who showed that the split closure is a polyhedron. Although some fairly general results were obtained by Andersen et al. (Math Oper Res 35(1):233---256, 2010) and Averkov (Discret Optimiz 9(4):209---215, 2012), some basic questions have remained unresolved. For example, maximal lattice-free triangles are the natural family to study beyond the family of splits and it has been a standing open problem to decide whether the triangle closure is a polyhedron. In this paper, we show that when the number of integer variables $$m=2$$ the triangle closure is indeed a polyhedron and its number of facets can be bounded by a polynomial in the size of the input data. The techniques of this proof are also used to give a refinement of necessary conditions for valid inequalities being facet-defining due to Cornuejols and Margot (Math Program 120:429---456, 2009) and obtain polynomial complexity results about the mixed integer hull.

Unique lifting of integer variables in minimal inequalities

This paper contributes to the theory of cutting planes for mixed integer linear programs (MILPs). Minimal valid inequalities are well understood for a relaxation of an MILP in tableau form where all the nonbasic variables are continuous; they are derived using the gauge function of maximal lattice-free convex sets. In this paper we study lifting functions for the nonbasic integer variables starting from such minimal valid inequalities. We characterize precisely when the lifted coefficient is equal to the coefficient of the corresponding continuous variable in every minimal lifting. The answer is a nonconvex region that can be obtained as a finite union of convex polyhedra. We then establish a necessary and sufficient condition for the uniqueness of the lifting function.

Maximal Lattice-Free Polyhedra: Finiteness and an Explicit Description in Dimension Three

A convex set with nonempty interior is maximal lattice-free if it is inclusion maximal with respect to the property of not containing integer points in its interior. Maximal lattice-free convex sets are known to be polyhedra. The precision of a rational polyhedron P in ℝd is the smallest natural number s such that sP is an integral polyhedron. In this paper we show that, up to affine mappings preserving ℤd, the number of maximal lattice-free rational polyhedra of a given precision s is finite. Furthermore, we present the complete list of all maximal lattice-free integral polyhedra in dimension three. Our results are motivated by recent research on cutting plane theory in mixed-integer linear optimization.

Inequalities for the lattice width of lattice-free convex sets in the plane

A closed, convex set K in \({\mathbb{R}^2}\) with non-empty interior is called lattice-free if the interior of K is disjoint with \({\mathbb{Z}^2}\). In this paper we study the relation between the area and the lattice width of a planar lattice-free convex set in the general and centrally symmetric case. A correspondence between lattice width on the one hand and covering minima on the other, allows us to reformulate our results in terms of covering minima introduced by Kannan and Lovász (Ann Math (2) 128(3):577–602, 1988). We obtain a sharp upper bound for the area for any given value of the lattice width. The lattice-free convex sets satisfying the upper bound are characterized. Lower bounds are studied as well. Parts of our results are applied in Averkov et al. (Maximal lattice-free polyhedra: finiteness and an explicit description in dimension three, http://arxiv.org/abs/1010.1077, 2010) for cutting plane generation in mixed integer linear optimization, which was the original inducement for this paper. We further rectify a result of Kannan and Lovász (Ann Math (2) 128(3):577–602, 1988) with a new proof.

A Geometric Perspective on Lifting

Recently it has been shown that minimal inequalities for a continuous relaxation of mixed-integer linear programs are associated with maximal lattice-free convex sets. In this paper, we show how to lift these inequalities for integral nonbasic variables by considering maximal lattice-free convex sets in a higher dimensional space. We apply this approach to several examples. In particular, we identify cases in which the lifting is unique.

Maximal Lattice-Free Convex Sets in Linear Subspaces

We consider a model that arises in integer programming and show that all irredundant inequalities are obtained from maximal lattice-free convex sets in an affine subspace. We also show that these sets are polyhedra. The latter result extends a theorem of Lovász characterizing maximal lattice-free convex sets in ℝn.

Two row mixed-integer cuts via lifting

Recently Andersen et al. [1], Borozan and Cornuejols [6] and Cornuejols and Margot [9] have characterized the extreme valid inequalities of a mixed integer set consisting of two equations with two free integer variables and non-negative continuous variables. These inequalities are either split cuts or intersection cuts derived using maximal lattice-free convex sets. In order to use these inequalities to obtain cuts from two rows of a general simplex tableau, one approach is to extend the system to include all possible non-negative integer variables (giving the two row mixed-integer infinite-group problem), and to develop lifting functions giving the coefficients of the integer variables in the corresponding inequalities. In this paper, we study the characteristics of these lifting functions. We show that there exists a unique lifting function that yields extreme inequalities when starting from a maximal lattice-free triangle with multiple integer points in the relative interior of one of its sides, or a maximal lattice-free triangle with integral vertices and one integer point in the relative interior of each side. In the other cases (maximal lattice-free triangles with one integer point in the relative interior of each side and non-integral vertices, and maximal lattice-free quadrilaterals), non-unique lifting functions may yield distinct extreme inequalities. For the latter family of triangles, we present sufficient conditions to yield an extreme inequality for the two row mixed-integer infinite-group problem.

Minimal Inequalities for an Infinite Relaxation of Integer Programs

We show that maximal S-free convex sets are polyhedra when S is the set of integral points in some rational polyhedron of $\mathbb{R}^n$. This result extends a theorem of Lovász characterizing maximal lattice-free convex sets. Our theorem has implications in integer programming. In particular, we show that maximal S-free convex sets are in one-to-one correspondence with minimal inequalities.

Minimal Valid Inequalities for Integer Constraints

In this paper, we consider a semi-infinite relaxation of mixed-integer linear programs. We show that minimal valid inequalities for this relaxation correspond to maximal lattice-free convex sets, and that they arise from nonnegative, piecewise linear, positively homogeneous, convex functions.

On an Analysis of the Strength of Mixed-Integer Cutting Planes from Multiple Simplex Tableau Rows

In this paper we derive valid inequalities for general mixed-integer linear programs (MILPs) by considering several simplex tableau rows simultaneously. We explore links between these cutting planes and lattice-free convex sets which have a representation as the sum of a polytope and a linear space. The codimension of the linear space of such a set is called split-dimension. We show that in terms of a strength-measure introduced by Goemans [Math. Programming, 69 (1995), pp. 335-349], only lattice-free convex sets with full split-dimension give rise to a good approximation of the mixed-integer hull, whereas lattice-free convex sets with low split-dimension might approximate the mixed-integer hull arbitrarily badly, in general. However, we also show that ordinary split cuts approximate the mixed-integer hull to within a constant factor when the size of the input data is given.

On degenerate multi-row Gomory cuts

Borozan and Cornuéjols show that valid inequalities for an infinite relaxation for MIPs, relative to some vertex f of the linear relaxation, are determined by maximal lattice-free convex sets containing f . We show that cuts for the original MIP are given by such sets with f in the interior.