On the Number of Path Systems
Daniel Cizma, Nati Linial
TL;DR
The paper investigates the enumeration of path systems on $n$-vertex graphs, distinguishing consistent path systems from those realizable as (unique) metric geodesics. It develops a combinatorial framework using résumés and a triple-based betweenness characterization to connect path-system metrizability with the metric cone $MET_n$, and employs probabilistic constructions, plane partitions, and VC-dimension theory to derive tight asymptotics. The key results show that the number of consistent path systems is $n^{\frac{n^2}{2}(1-o(1))}$, while strictly metric ones grow as $2^{\Theta(n^2)}$, and they provide refined bounds on the faces of the metric cone and on maximum VC classes. These findings illuminate the intrinsic complexity gap between general consistent path systems and those induced by a metric, with implications for geometry of path systems, betweenness theory, and combinatorial VC theory. The work also introduces novel tools (résumés, signatures, and LP-based tests) and raises open questions about exact constants, graph-specific counts, and the geometry of the metric cone.
Abstract
A path system in a graph $G$ is a collection of paths, with exactly one path between any two vertices in $G$. A path system is said to be consistent if it is intersection-closed. We show that the number of consistent path systems on $n$ vertices is $n^{\frac{n^2}{2}(1-o(1))}$, whereas the number of consistent path systems which are realizable as the unique geodesics w.r.t. some metric is only $2^{Θ(n^2)}$. In addition, these insights allow us to improve known bounds on the face-count of the metric cone and shed new light on enumerating maximum-VC-classes.