Large-scale clique cover of real-world networks

The edge clique cover (ECC ) problem deals with discovering a set of (possibly overlapping) cliques in a given graph that covers each of the graph's edges. This problem finds applications ranging from social networks to compiler optimization and stringology. We consider several variants of the ECC problem, using classical quality measures (like the number of cliques) and new ones. We describe efficient heuristic algorithms, the fastest one taking O(mdG) time for a graph with m edges, degeneracy dG (also known as k-core number). For large real-world networks with millions of nodes, like social networks, an algorithm should have (almost) linear running time to be practical: Our algorithm for finding ECCs of large networks has linear-time performance in practice because dG is small, as our experiments show, on real-world networks with thousands to several million nodes

Constructing an indeterminate string from its associated graph

As discussed at length in Christodoulakis et al. (2015) , 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