Algorithm Outline

Suppose we are given G(V,E) corrupted clique graph, that is $G\in\Omega(H,p)$ for some clique graph H with $\gamma$-cluster structure. Because H has $\gamma$-cluster structure the maximum number of cliques in H is $m=\lceil\frac{1}{\gamma}\rceil$.

  

Figure 12.1: Algorithm steps are shown schematically.

The algorithm will perform the following steps (please refer to figure 12.1):

1.

Uniformly draw $S_0 \subset V$, such that $\vert S_0\vert = O(\log \log (n))$;

2.

Uniformly draw $S_1 \subset V \backslash S_0$, such that $\vert S_1\vert = O(\log (n))$;

3.

For each clustering of S₀ into m clusters $\{C^0_1, ..., C^0_m\}$, perform:

(a)

Extend the clustering $\{C^0_1, ..., C^0_m\}$ of S₀ into a clustering $\{C^1_1, ..., C^1_m\}$ of $S_0 \cup S_1$;

(b)

Extend the clustering $\{C^1_1, ..., C^1_m\}$ into a clustering $\{C_1, ..., C_m\}$ of V;

4.

Each clustering $\{C_1, ..., C_m\}$ of V from the previous step determines a clique graph C(V, E(C)); from all such clusterings $\{C_1, ..., C_m\}$ of V, output the one which minimizes $\vert E \Delta E(C)\vert$;