Incrementally Maintaining the Number of l-cliques

Abstract

The main contribution of this paper is an incremental algorithm to update the number of $l$-cliques, for $l \geq 3$, in which each node of a graph is contained, after the deletion of an arbitrary node. The initialization cost is $O(n^{\omega p+q})$, where $n$ is the number of nodes, $p=\lfloor \frac{l}{3} \rfloor$, $q=l \pmod{3}$, and $\omega=\omega(1,1,1)$ is the exponent of the multiplication of two $n x n$ matrices. The amortized updating cost is $O(n^{q}T(n,p,\epsilon))$ for any $\epsilon \in [0,1]$, where $T(n,p,\epsilon)=\min\{n^{p-1}(n^{p(1+\epsilon)}+n^{p(\omega(1,\epsilon,1)-\epsilon)}),n^{p \omega(1,\frac{p-1}{p},1)}\}$ and $\omega(1,r,1)$ is the exponent of the multiplication of an $n x n^{r}$ matrix by an $n^{r} x n$ matrix. The current best bounds on $\omega(1,r,1)$ imply an $O(n^{2.376p+q})$ initialization cost, an $O(n^{2.575p+q-1})$ updating cost for $3 \leq l \leq 8$, and an $O(n^{2.376p+q-0.532})$ updating cost for $l \geq 9$. An interesting application to constraint programming is also considered