An Introduction to Hollow Simplices (draft)

We study simplices with integer vertices and no other integer points. Such`hollow' simplices come up in the theory of integer programming, and provide a common formulation of some combinatorial problems, including particular cases of the knapsack or the partition problem. They have been investigated in various contexts and under varying terminology Can thèhollowness' of simplices bèwell-characterized' ? Is their`width' small ? Do the integer points of the parallelepiped they generate have a particular structure ? Thègood characterization' of hollow simplices occurs to be open in general ! We provide a polynomial algorithm for deciding when a given`knapsack' or`partition' simplex is hollow, and study structural properties of some hollow intersections of parallelepipeds with half-spaces. We show polynomially checkable conditions, suucient for this kind of polytope to be hollow, and possibly also necessary. We prove that they are necessary and suucient for n 4, and deduce several variants of classical results of Reeve, White and Scarf about small dimensional simplices. For instance, a three dimensional simplex is hollow if and only if all its faces have width 1. The diierent results can be easily proved from one another using the Hermite normal form. In xed dimension the width of polytopes can be computed in polynomial time (see the simple integer programming formulation of Haase and Ziegler). We prove that it is already NP-complete to decide whether the width of a very special class of simplices is 1, and we also provide for every n 3 a simple example of n-dimensional hollow simplices of width n ? 2, improving on earlier bounds.