Computational complexity of reconstruction and isomorphism testing for designs and line graphs

TL;DR

This paper investigates the computational complexity of testing isomorphism for line graphs of highly regular t-designs, providing algorithms with polynomial and quasi-polynomial time complexities, and extending previous results on Steiner systems.

Contribution

It introduces a polynomial-time reconstruction method for t-designs from their line graphs and analyzes the isomorphism testing complexity for these highly symmetric graphs.

Findings

Worst-case running time of O(v^{log v + O(1)}) for isomorphism testing.

Polynomial-time reconstruction of t-designs from line graphs.

Extension of complexity results to broader classes of designs.

Abstract

Graphs with high symmetry or regularity are the main source for experimentally hard instances of the notoriously difficult graph isomorphism problem. In this paper, we study the computational complexity of isomorphism testing for line graphs of $t$-$(v,k,\lambda)$ designs. For this class of highly regular graphs, we obtain a worst-case running time of $O(v^{\log v + O(1)})$ for bounded parameters $t,k,\lambda$. In a first step, our approach makes use of the Babai--Luks algorithm to compute canonical forms of $t$-designs. In a second step, we show that $t$-designs can be reconstructed from their line graphs in polynomial-time. The first is algebraic in nature, the second purely combinatorial. For both, profound structural knowledge in design theory is required. Our results extend earlier complexity results about isomorphism testing of graphs generated from Steiner triple systems and block designs.