Approximate Counting of Linear Extensions in Practice

We investigate the problem of computing the number of linear extensions of a given partial order on n elements. The problem has applications in numerous areas, such as sorting, planning, and learning graphical models. The problem is #P-hard but admits fully polynomial-time approximation schemes. However, the polynomial complexity bounds of the known schemes involve high degrees and large constant factors, rendering the schemes only feasible when n is some dozens. We present novel schemes, which stem from the idea of not requiring provable polynomial worst-case running time bounds. Using various new algorithmic techniques and implementation optimizations, we discover schemes that yield speedups by several orders of magnitude, enabling accurate approximations even when n is in several hundreds.