An Introduction to Empty Lattice Simplices

Abstract

We study simplices whose vertices lie on a lattice and have no other lattice points. Such ‘empty lattice simplices’ come up in the theory of integer programming, and in some combinatorial problems. They have been investigated in various contexts and under varying terminology by Reeve, White, Scarf, Kannan and Lovász, Reznick, Kantor, Haase and Ziegler, etc.

Can the ‘emptiness’ of lattice simplices be ‘well-characterized’? Is their ‘lattice-width’ small? Do the integer points of the parallelepiped they generate have a particular structure?

The ‘good characterization’ of empty lattice simplices occurs to be open in general ! We provide a polynomial algorithm for deciding when a given integer ‘knapsack’ or ‘partition’ lattice simplex is empty. More generally, we ask for a characterization of linear inequalities satisfied by the lattice points of a lattice parallelepiped. We state a conjecture about such inequalities, prove it for n ≤ 4, and deduce several variants of classical results of Reeve, White and Scarf characterizing the emptiness of small dimensional lattice simplices. For instance, a three dimensional integer simplex is empty if and only if all its faces have width 1. Seemingly different characterizations can be easily proved from one another using the Hermite normal form.

In fixed 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 integer simplices is 1, and we also provide for every n ≥ 3 a simple example of n-dimensional empty integer simplices of width n − 2, improving on earlier bounds.

Research developed partly during a visit in the Research Institute for Mathematical Sciences, Kyoto University.