# Identifying Maximal Non-Redundant Integer Cone Generators

## TL;DR

This paper determines exact values of the maximum size of non-redundant integer cone generators in dimensions 4 to 6, improving bounds and enabling more efficient solutions for related logical and algebraic problems.

## Contribution

It provides the first exact values for N(d) for d > 3, advancing the understanding of integer cone generators and their applications in logic and software verification.

## Key findings

- Exact values: N(4)=5, N(5)=7, N(6)=9.
- Lower bounds established for N(7) to N(10).
- Developed specialized search algorithms for exploration.

## Abstract

A non-redundant integer cone generator (NICG) of dimension $d$ is a set $S$ of vectors from $\{0,1\}^d$ whose vector sum cannot be generated as a positive integer linear combination of a proper subset of $S$. The largest possible cardinality of NICG of a dimension $d$, denoted by $N(d)$, provides an upper bound on the sparsity of systems of integer equations with a large number of integer variables. A better estimate of $N(d)$ means that we can consider smaller sub-systems of integer equations when solving systems with many integer variables. Furthermore, given that we can reduce constraints on set algebra expressions to constraints on cardinalities of Venn regions, tighter upper bound on $N(d)$ yields a more efficient decision procedure for a logic of sets with cardinality constraints (BAPA), which has been applied in software verification. Previous attempts to compute $N(d)$ using SAT solvers have not succeeded even for $d=3$. The only known values were computed manually: $N(d)=d$ for $d < 4$ and $N(4) > 4$. We provide the first exact values for $d > 3$, namely, $N(4)=5$, $N(5)=7$, and $N(6)=9$, which is a significant improvement of the known asymptotic bound (which would give only e.g. $N(6) \le 29$, making a decision procedure impractical for $d=6$). We also give lower bounds for $N(7)$, $N(8)$, $N(9)$, and $N(10)$, which are: $11$, $13$, $14$, and $16$, respectively. We describe increasingly sophisticated specialized search algorithms that we used to explore the space of non-redundant generators and obtain these results.