Discrete Mathematics
Theory of linear and integer programming
Theory of linear and integer programming
A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed
Mathematics of Operations Research
Flows, view obstructions, and the lonely runner
Journal of Combinatorial Theory Series B
On the maximal width of empty lattice simplices
European Journal of Combinatorics - Special issue on combinatorics of polytopes
Hilbert Bases, Caratheodory's Theorem and Combinatorial Optimization
Proceedings of the 1st Integer Programming and Combinatorial Optimization Conference
On the Complexity of Selecting Disjunctions in Integer Programming
SIAM Journal on Optimization
Projecting Lattice Polytopes Without Interior Lattice Points
Mathematics of Operations Research
Maximal Lattice-Free Polyhedra: Finiteness and an Explicit Description in Dimension Three
Mathematics of Operations Research
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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.