Sampling permutations satisfying constraints within the lopsided local lemma regime

TL;DR

This paper develops efficient algorithms for sampling constrained permutations, achieving optimal time for dense matrices and introducing a new model and framework for handling multiple permutations under local lemma conditions.

Contribution

It presents the first optimal $O(n^2)$ algorithm for dense matrix permanent approximation and introduces the PDC model and correlated factorization framework for sampling multiple constrained permutations.

Findings

Optimal $O(n^2)$ time algorithm for dense matrix permanent approximation.

Nearly linear time sampling algorithm for uniform PDC formulas.

Novel sampling framework addressing long-range correlations in permutation spaces.

Abstract

Sampling a random permutation with restricted positions, or equivalently approximating the permanent of a 0-1 matrix, is a fundamental problem in computer science, with several notable results achieved over the years. However, existing algorithms typically exhibit high computational complexity. Achieving the optimal running time remains elusive, even for nontrivial subsets of the problem. Furthermore, existing algorithms primarily focus on a single permutation, leaving many combinatorial problems involving multiple constrained permutations unaddressed. For a single permutation, we achieve the optimal running time $O(n^2)$ for approximating the permanent of a very dense $n \times n$ 0-1 matrix, where each row and column contains at most $\sqrt{(n-2)/20}$ zeros. This result serves as a fundamental building block in our sampling algorithm for multiple permutations. We further introduce a general model called permutations with disjunctive constraints (PDC) for handling multiple constrained permutations. We propose a novel Markov chain-based algorithm for sampling nearly uniform solutions of PDC within a lopsided Lov\'asz Local Lemma (LLL) regime. For uniform PDC formulas, where all constraints are of the same width and all permutations are of the same size, our algorithm runs in nearly linear time with respect to the number of variables. Previous approaches for sampling LLL relied on the variable model. In contrast, the sampling problem of PDC encounters a fundamental challenge: the random variables within each permutation in the joint probability space are not mutually independent, leading to long-range correlations. To tackle this challenge, we introduce a novel sampling framework called correlated factorization and a new concept in the path coupling analysis, termed the inactive vertex.