Algorithms and complexity for counting configurations in Steiner triple 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.

Enumerating Steiner triple systems

“…for large v. By [13], the number of isomorphism classes of STS (21)s that contain at least one sub-STS( 7) is 116,635,963,205,551. The real proportion for STS (21) which approximately coincides with the real proportion of α 1.3 in the case of STS (19)s and is in line with the experimental results of [14].…”

“…The graphs with degree sequence 5 14 are regular graphs and their number can be found in the literature [32,35]. The main computations were carried out in a cluster, and the computing times announced here are for computations on the equivalent of one core of a 2.4-GHz Intel Xeon E5-2665.…”

“…The general framework considered here builds on that of [13, 22], where STSs with subsystems are considered. Actually, the approach for classifying STS(19)s used in [19] can also be put into this framework.…”

“…In [13, 19, 22], the blocks are partitioned into three sets: a defining set $${\rm{ {\mathcal B} }}^{\prime} $$ that forms a subsystem and sets $${\rm{ {\mathcal F} }}$$ and $${\mathscr{D}}$$ whose blocks intersect the point set of $${\rm{ {\mathcal B} }}^{\prime} $$ in one point and zero points, respectively. As $${\rm{ {\mathcal B} }}^{\prime} $$ contains the blocks of a subsystem—which in [19] is just a sub‐STS(3), a single block—no block can intersect the point set of $${\rm{ {\mathcal B} }}^{\prime} $$ in exactly two points.…”

“…Indeed, for all these orders, representatives of the isomorphism classes have been determined. For the smallest open case of order 21, however, such an approach does not seem feasible at the moment as the number of isomorphism classes has been estimated [13] to be greater than $$1{0}^{16}$$. There are numerous studies on the classification of subclasses of STSs of order 21 [3, 10, 16–18, 23, 25–29, 36, 37].…”