Constructing an Indeterminate String from Its Associated Graph

As discussed at length in Christodoulakis et al. (2015) [3], there is a natural one-many correspondence between simple undirected graphs G with vertex set V={1,2,…,n} and indeterminate strings x=x[1..n] — that is, sequences of subsets of some alphabet Σ. In this paper, given G, we consider the “reverse engineering” problem of computing a corresponding x on an alphabet Σmin of minimum cardinality. This turns out to be equivalent to the NP-hard problem of computing the intersection number of G, thus in turn equivalent to the clique cover problem. We describe a heuristic algorithm that computes an approximation to Σmin and a corresponding x. We give various properties of our algorithm, including some experimental evidence that on average it requires O(n2log⁡n) time. We compare it with other heuristics, and state some conjectures and open problems.