Approximately Sampling Elements with Fixed Rank in Graded Posets

TL;DR

This paper introduces a new method called balanced bias for approximately sampling fixed-rank elements in graded posets efficiently, bypassing traditional log-concavity proofs.

Contribution

It presents a novel approach to approximate sampling in graded posets using biased Markov chains and introduces the first provably efficient chain for sampling restricted integer partitions.

Findings

Efficient polynomial-time sampling algorithms for fixed-rank elements in certain posets.

First provably efficient Markov chain for sampling restricted integer partitions.

Improved algorithms requiring O(n^{1/2} log n) space and expected O(n^{9/4}) time for unrestricted partitions.

Abstract

Graded posets frequently arise throughout combinatorics, where it is natural to try to count the number of elements of a fixed rank. These counting problems are often $\#\textbf{P}$-complete, so we consider approximation algorithms for counting and uniform sampling. We show that for certain classes of posets, biased Markov chains that walk along edges of their Hasse diagrams allow us to approximately generate samples with any fixed rank in expected polynomial time. Our arguments do not rely on the typical proofs of log-concavity, which are used to construct a stationary distribution with a specific mode in order to give a lower bound on the probability of outputting an element of the desired rank. Instead, we infer this directly from bounds on the mixing time of the chains through a method we call $\textit{balanced bias}$. A noteworthy application of our method is sampling restricted classes of integer partitions of $n$. We give the first provably efficient Markov chain algorithm to uniformly sample integer partitions of $n$ from general restricted classes. Several observations allow us to improve the efficiency of this chain to require $O(n^{1/2}\log(n))$ space, and for unrestricted integer partitions, expected $O(n^{9/4})$ time. Related applications include sampling permutations with a fixed number of inversions and lozenge tilings on the triangular lattice with a fixed average height.