# Approximate Integer Solution Counts over Linear Arithmetic Constraints

arXiv: https://arxiv.org/abs/2312.08776 · HTML: https://mlsanity.com/paper/2312.08776/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 29 references that are in the map.
- First posted on arXiv 14 Dec 2023.

## Builds on (most cited first, 1 of 1)

- [Gaussian Cooling and O*(n^3) Algorithms for Volume and Gaussian Volume](https://mlsanity.com/paper/1409.6011/) (2014)

## Built on by (most cited first, 2 of 2)

- [Approximate SMT Counting Beyond Discrete Domains](https://mlsanity.com/paper/2507.18612/) (2025)
- [An Exhaustive DPLL Approach to Model Counting over Integer Linear Constraints with Simplification Techniques](https://mlsanity.com/paper/2509.13880/) (2025)

## Abstract (arXiv)

Counting integer solutions of linear constraints has found interesting applications in various fields. It is equivalent to the problem of counting lattice points inside a polytope. However, state-of-the-art algorithms for this problem become too slow for even a modest number of variables. In this paper, we propose a new framework to approximate the lattice counts inside a polytope with a new random-walk sampling method. The counts computed by our approach has been proved approximately bounded by a (ε, δ)-bound. Experiments on extensive benchmarks show that our algorithm could solve polytopes with dozens of dimensions, which significantly outperforms state-of-the-art counters.