Effective Asymptotics of Combinatorial Systems

2025-08-27

AI Summary

This paper addresses the fundamental challenge of extending asymptotic analysis of combinatorial systems from basic constructions (Cartesian product, disjoint union) to general symbolic constructions (e.g., sets, cycles). Methodologically, it integrates analytic combinatorics, complex analysis, and symbolic computation, employing singularity analysis of generating functions—systematically handling algebraic-logarithmic singularities—and invoking the Schanuel conjecture to resolve transcendental singularities arising from set and cycle constructions. The main contribution is a near-complete algorithmic pipeline that automatically derives asymptotic expansions directly from combinatorial specifications, covering a broad class of constructions including sets and cycles for the first time. Asymptotics for Cartesian products and disjoint unions are rigorously established without number-theoretic assumptions; results for other constructions hold conditionally under the Schanuel conjecture. This work significantly extends both the scope and automation level of combinatorial asymptotics.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Analytic combinatorics studies asymptotic properties of families of combinatorial objects using complex analysis on their generating functions. In their reference book on the subject, Flajolet and Sedgewick describe a general approach that allows one to derive precise asymptotic expansions starting from systems of combinatorial equations. In the situation where the combinatorial system involves only cartesian products and disjoint unions, the generating functions satisfy polynomial systems with positivity constraints for which many results and algorithms are known. We extend these results to the general situation. This produces an almost complete algorithmic chain going from combinatorial systems to asymptotic expansions. Thus, it is possible to compute asymptotic expansions of all generating functions produced by the symbolic method of Flajolet and Sedgewick when they have algebraic-logarithmic singularities (which can be decided), under the assumption that Schanuel's conjecture from number theory holds. That conjecture is not needed for systems that do not involve the constructions of sets and cycles.

Problem

• Extends analytic combinatorics to general combinatorial systems
• Provides algorithmic chain from systems to asymptotic expansions
• Handles algebraic-logarithmic singularities under Schanuel's conjecture

Innovation

• Extends analytic combinatorics to general systems
• Algorithmic chain from combinatorial systems to asymptotics
• Handles algebraic-logarithmic singularities under Schanuel's conjecture