Counting linear extensions with volume computation and applications in AI

This project will implement a new volume-based approach for approximately counting linear extensions of partially ordered sets in volesti. The core idea is to exploit the special geometry of order polytopes by designing optimized exact Hamiltonian Monte Carlo boundary oracles, rather than relying on generic polytope routines. The work will include implementing specialized HMC walks, extending them to general Gaussian settings, integrating them into volesti’s volume-estimation pipeline, and evaluating their performance against existing approximation methods. Expected deliverables include the new C++ implementation, tests, examples, documentation, and a benchmark study on representative poset families.