Sampling permutations satisfying constraints within the lopsided local lemma regime

2024-11-05

AI Summary

This work addresses efficient uniform sampling from multi-constraint random permutations—particularly in settings with stringent positional constraints, highly dependent variables, and failure of conventional methods. We propose a novel “dependency decomposition” sampling framework and introduce the notion of “inactive vertices” within path coupling analysis, enabling—for the first time—the joint sampling of multiple permutations under the lopsided Lovász Local Lemma (LLL) framework, without reliance on independent-variable assumptions. Our approach integrates Markov chain Monte Carlo, lopsided LLL, and structured coupling analysis to handle non-independent stochastic structures. Key theoretical contributions include: (i) an $O(n^2)$-time algorithm for approximating the permanent of an $n imes n$ dense 0–1 matrix—matching the optimal known complexity; and (ii) a near-linear-time, nearly uniform sampler for the Partial Rejection Sampling (PRS) paradigm under the Uniform Partial Decomposition Coupling (PDC) formulation.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $n imes n$ 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{'a}sz 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}.

Problem

• Sampling random permutations with restricted positions efficiently
• Handling multiple constrained permutations in combinatorial problems
• Overcoming long-range correlations in joint probability spaces

Innovation

• Optimal O(n^2) time for dense 0-1 matrix permanent
• Markov chain algorithm for disjunctive constraints permutations
• Correlated factorization framework for long-range dependencies