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.

翻译：线性约束的整数解计数已在多个领域展现出有趣的应用，这等价于计算多面体内格点的数量。然而，最先进的算法即使对于变量数量较少的情况也显得过于缓慢。本文提出了一种新框架，通过新的随机游走采样方法近似计算多面体内的格点数量。我们方法计算的计数已被证明具有$(\epsilon, \delta)$-近似边界。在广泛基准测试上的实验表明，该算法可求解数十维的多面体问题，显著优于现有最先进的计数器。