This is a self-contained exposition of several core aspects of the theory of rational polyhedra
with a view towards algorithmic applications to efficient counting of integer points, a problem …

Research efforts of the past fifty years have led to a development of linear integer
programming as a mature discipline of mathematical optimization. Such a level of maturity …

It is undeniable that geometric ideas have been very important to the foundations of modern
discrete optimization. The influence that geometric algorithms have in optimization was …

Combinatorial reciprocity is a very interesting phenomenon, which can be described as
follows: A polynomial, whose values at positive integers count combinatorial objects of some …

This chapter surveys a selection of results from the interplay of integer programming and the
geometry of numbers. Apart from being a survey, the text is also intended as an entry point …

We investigate arithmetic, geometric and combinatorial properties of symmetric edge
polytopes. We give a complete combinatorial description of their facets. By combining …

Semi-linear sets, which are rational subsets of the monoid (Z^ d,+), have numerous
applications in theoretical computer science. Although semi-linear sets are usually given …

The Ehrhart polynomial of a lattice polytope $ P $ encodes information about the number of
integer lattice points in positive integral dilates of $ P $. The $ h^\ast $-polynomial of $ P $ is …

We describe the use of pyramid decomposition in Normaliz, a software tool for the
computation of Hilbert bases and enumerative data of rational cones and affine monoids …

In this paper, we consider the counting function\({{\,\mathrm {{{\,\mathrm {\mathcal {E}}\,}} _
{{{\,\mathrm {\mathcal {P}}\,}}}}\,}}(y)=|{{\,\mathrm {\mathcal {P}}\,}} _ {y}\cap {{\,\mathrm …