Draft:MaxCliqueDyn algorithm

MaxCliqueDyn algorithm

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Class	maximum clique algorithm

The MaxCliqueDyn algorithm is an algorithm for finding a maximum clique in an undirected graph. It is based on a basic algorithm which finds a maximum clique of bounded size. The bound is found using improved coloring algorithm. The MaxCliqueDyn is the basic algorithm extended to include dynamically varying bounds. This algorithm was designed by Janez Konc and description was published in 2007. In comparison to earlier algorithms described in the article the MaxCliqueDyn algorithm is improved by an improved approximate coloring algorithm (ColorSort) and by applying tighter, more computationally expensive upper bounds on a fraction of the search space. Both improvements reduce time to find maximum clique. In addition to reducing time improved coloring algorithm also reduces the number of steps needed to find a maximum clique.

Coloring algorithm (ColorSort)

The ColorSort algorithm is an improved algorithm of the approximate coloring algorithm

ColorSort algorithm

   Procedure ColorSort(R, C)
       max_no := 1;
       kmin := |Qmax| − |Q| + 1;
       if kmin ≤ 0 then kmin := 1;
       j := 0;
       C1 := Ø; C2 := Ø;
       for i := 0 to |R| − 1 do
           p := R[i]; {the i-th vertex in R}
           k := 1;
           while Ck ⋂ Γ(p) ≠ Ø do
               k := k+1;
           if k > max_no then
               max_no := k;
               Cmax_no+1 := Ø;
           end if
           Ck := Ck ⋃ {p};
           if k < kmin then
               R[j] := R[i];
               j := j+1;
           end if
       end for
       C[j−1] := 0;
       for k := kmin to max_no do
           for i := 1 to |Ck| do
               R[j] := Ck[i];
               C[j] := k;
               j := j+1;
           end for
       end for

MaxCliqueDyn algorithm

MaxCliqueDyn algorithm

   Procedure MaxCliqueDyn(R, C, level)
       S[level] := S[level] + S[level−1] − Sold[level];
       Sold[level] := S[level−1];
       while R ≠ Ø do
           choose a vertex p with maximum C(p) (last vertex) from R;
           R := R\{p};
           if |Q| + C[index of p in R] > |Qmax| then
               Q := Q ⋃ {p};
               if R ⋂ Γ(p) ≠ Ø then
                   if S[level]/ALL STEPS < Tlimit then
                       calculate the degrees of vertices in G(R ⋂ Γ(p));
                       sort vertices in R ⋂ Γ(p) in a descending order
                       with respect to their degrees;
                   end if
                   ColorSort(R ⋂ Γ(p), C')
                   S[level] := S[level] + 1;
                   ALL STEPS := ALL STEPS + 1;
                   MaxCliqueDyn(R ⋂ Γ(p), C', level + 1);
               else if |Q| > |Qmax| then Qmax := Q;
               Q := Q\{p};
           else return
       end while

External links

• www.example.com