  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 $(\epsilon, \delta)$-bound. Experiments
on extensive benchmarks show that our algorithm could solve polytopes with
dozens of dimensions, which significantly outperforms state-of-the-art
counters.
