# A Sequential Importance Sampling Algorithm for Estimating Linear Extensions

## TL;DR

This paper introduces a practical sequential importance sampling algorithm to estimate the number of linear extensions of a partially ordered set, improving upon traditional methods by incorporating importance functions for better efficiency.

## Contribution

The paper presents a novel SIS algorithm for counting linear extensions, enhancing the practicality of probabilistic sampling methods in this domain.

## Key findings

- Implemented two importance functions and evaluated their performance.
- Achieved more efficient estimation compared to traditional algorithms.
- Demonstrated the method's applicability to complex counting problems.

## Abstract

In recent decades, a number of profound theorems concerning approximation of hard counting problems have appeared. These include estimation of the permanent, estimating the volume of a convex polyhedron, and counting (approximately) the number of linear extensions of a partially ordered set. All of these results have been achieved using probabilistic sampling methods, specifically Monte Carlo Markov Chain (MCMC) techniques. In each case, a rapidly mixing Markov chain is defined that is guaranteed to produce, with high probability, an accurate result after only a polynomial number of operations. Although of polynomial complexity, none of these results lead to a practical computational technique, nor do they claim to. The polynomials are of high degree and a non-trivial amount of computing is required to get even a single sample. Our aim in this paper is to present practical Monte Carlo methods for one of these problems, counting linear extensions. Like related work on estimating the coefficients of the reliability polynomial, our technique is based on improving the so-called Knuth counting algorithm by incorporating an importance function into the node selection technique giving a sequential importance sampling (SIS) method. We define and report performance on two importance functions.