[Paper Review] Maximal lattice free bodies, test sets and the Frobenius problem

Anders Jensen, Niels Lauritzen+1 more|ArXiv.org|May 30, 2007

graph theory and CDMA systems13 references4 citations

TL;DR

This paper presents an efficient algorithm to compute maximal lattice-free bodies for an integral $(d+1)\times d$ matrix $A$, using a test set derived via algebraic methods. The method generalizes the Scarf-Shallcross algorithm for the 3D Frobenius problem and enables efficient computation of the Frobenius number by enumerating maximal lattice-free polytopes via integer programming and Gröbner basis techniques.

ABSTRACT

Maximal lattice free bodies are maximal polytopes without interior integral points. Scarf initiated the study of maximal lattice free bodies relative to the facet normals in a fixed matrix. In this paper we give an efficient algorithm for computing the maximal lattice free bodies of an integral matrix A. An important ingredient is a test set for a certain integer program associated with A. This test set may be computed using algebraic methods. As an application we generalize the Scarf-Shallcross algorithm for the three-dimensional Frobenius problem to arbitrary dimension. In this context our method is inspired by the novel algorithm by Einstein, Lichtblau, Strzebonski and Wagon and the Groebner basis approach by Roune.

Motivation & Objective

• To develop an efficient algorithm for computing all maximal lattice-free bodies relative to the facet normals of a fixed integral $(d+1)\times d$ matrix $A$.
• To establish a connection between maximal lattice-free bodies and the Frobenius problem in higher dimensions.
• To generalize the Scarf-Shallcross algorithm for the 3D Frobenius problem to arbitrary dimensions using algebraic and geometric techniques.
• To compute a test set for an integer program associated with $A$ using algebraic methods, enabling efficient optimization over lattice-free polytopes.
• To apply the framework to compute the Frobenius number $g(a_1,\dots,a_n)$ via maximization over lattice-free bodies, leveraging integral translations and polyhedral geometry.

Proposed method

• Use the matrix $A$ to define a family of polyhedra $P_A(b) = \{x \in \mathbb{R}^d \mid Ax \leq b\}$, where $b \in \mathbb{Z}^{d+1}$, and identify maximal lattice-free bodies among them.
• Construct a test set for the integer program $\min\{-x \mid Bx \leq b, x \in \mathbb{Z}^2\}$ associated with $A$, using unimodular transformations and Gröbner basis techniques.
• Apply a reduction procedure inspired by the Scarf-Shallcross algorithm to transform the integer program into a canonical form with known test sets $\{e_1, e_2, e_1+e_2\}$.
• Use the inverse of the unimodular transformation to lift the test set back to the original problem, enabling efficient enumeration of maximal lattice-free bodies.
• Compute the Frobenius number as $g(a_1,\dots,a_n) = \max\{ab \mid K_b \text{ is maximal lattice-free}\} - \sum a_i$, where $K_b$ is defined via $A$ and $b$.
• Leverage the fact that all maximal lattice-free bodies are integral translates of finitely many representatives, reducing the search space to a finite set of $b$-vectors.

Experimental results

Research questions

• RQ1 How can maximal lattice-free bodies be efficiently enumerated for a given integral $(d+1)\times d$ matrix $A$?
• RQ2 What is the structure of the test set for the integer program associated with $A$, and how can it be computed algebraically?
• RQ3 Can the Scarf-Shallcross algorithm for the 3D Frobenius problem be generalized to higher dimensions using lattice-free polytopes?
• RQ4 How does the use of Gröbner bases and unimodular transformations improve the efficiency of computing the Frobenius number?
• RQ5 What is the relationship between the maximal lattice-free bodies and the solution of the Frobenius problem in terms of optimization over integer points?

Key findings

• The algorithm efficiently computes all maximal lattice-free bodies for a given $(d+1)\times d$ integral matrix $A$ by leveraging a test set derived from algebraic methods.
• The test set for the associated integer program is computed via unimodular transformations and Gröbner basis techniques, enabling optimization over lattice-free polytopes.
• The method generalizes the Scarf-Shallcross algorithm to arbitrary dimensions, extending its applicability beyond the 3D case.
• For the example $g(12,13,17)$, the algorithm identifies two representative $b$-vectors: $b_1 = (0,5,2)^T$ and $b_2 = (0,2,3)^T$, leading to the Frobenius number $g(12,13,17) = 57$.
• The Frobenius number is computed as $\max\{ab \mid K_b \text{ is maximal lattice-free}\} - \sum a_i$, with the maximum achieved at $ab = -12 + 4\cdot13 + 1\cdot17 = 57$.
• The framework enables record-breaking computations of the Frobenius number due to the efficient enumeration of lattice-free bodies and the use of algebraic test sets.