Algorithms and Complexity for Counting Configurations in Steiner Triple Systems

TL;DR

This paper explores the computational complexity and algorithms for counting configurations in Steiner triple systems, balancing theoretical insights with practical methods for large or numerous systems.

Contribution

It provides new theoretical results and practical algorithms for counting configurations in Steiner triple systems, addressing challenges in large-scale or numerous instances.

Findings

Theoretical complexity bounds for counting configurations.

Efficient algorithms for specific configurations.

Analysis of practical performance on large systems.

Abstract

Steiner triple systems form one of the most studied classes of combinatorial designs. Configurations, including subsystems, play a central role in the investigation of Steiner triple systems. With sporadic instances of small systems, ad-hoc algorithms for counting or listing configurations are typically fast enough for practical needs, but with many systems or large systems, the relevance of computational complexity and algorithms of low complexity is highlighted. General theoretical results as well as specific practical algorithms for important configurations are presented.