Graph admissibility: Case generation and analysis by learning models

Abstract

A graph G is t-admissible if it has a spanning tree T such that any adjacent vertices in G are at a distance at most t in T, in which case T is called a tree t-spanner of G. We denote as the stretch index ofG, or σ(G), the smallest value t such that G is t-admissible. Despite having a polynomial decider for t=2, the problem is NP-complete for t≥4, while the t=3 variant remains open after decades since its proposal.The present work elaborates algorithms to create graphs with known stretch indexes and/or admissibility factors. Indeed, we show how to build a dataset providing the parameter σ of all 12,111 isomorphism classes of connected graphs from 3 to 8 vertices and discuss random generation methods for t-admissible and non-t-admissible graphs of arbitrary sizes. These techniques give us the ability to study the problem from machine learning’s perspective. We show how both lazy and eager models can have high performance under interesting testing conditions, even for instances in the NP-complete spectrum of the problem.