Tight bounds and output sensitive algorithms for maximal clique enumeration in link streams

Abstract

A link stream is a set of interactions between pairs of vertices, each one lasting during some time interval, and a clique of a link stream is a set of vertices together with a time interval during which all of them interact. A clique is maximal if neither its vertex set nor its interval can be enlarged. All known algorithms listing the maximal cliques of a link stream may spend time exponential in the size of a clique for each clique they output. Here, we approach the question under the light of two parameters of the instantaneous graphs of the stream, their maximum degree $Δ$ and their degeneracy $k$. We prove that a link stream with $m$ links has $O(mΔ^2 3^{Δ/3})$ maximal cliques, and that a link stream with $n$ vertices and $|\mathcal{T}|$ distinct end times has $O(n|\mathcal{T}|k^2 3^{k/3})$ maximal cliques. Both bounds are tight up to a factor polynomial in $Δ$ and $k$ respectively, and the second one improves the factor $2^k$ of previous bounds to $3^{k/3}$. Then we present two algorithms. The first one has setup time $O(m\log m)$ and polynomial time delay $\mathrm{poly}(Δ)\log m$. The second one has setup time $O(m\log m+mk^2\log^3 n)$ and polynomial time delay $\mathrm{poly}(k)\log m$. To the best of our knowledge, these are the first algorithms with polynomial time delay for this problem. We also give an online version of the second algorithm, and we show that our results apply to the $Δ$-cliques and $(Δ,γ)$-cliques of temporal graphs.