 # C++ Programming Code Examples

## C++ > Computer Graphics Code Examples

### C++ Program to Find the Maximum Size Clique in a Graph

``` C++ Program to Find the Maximum Size Clique in a Graph This is a C++ Program to find the cliques of size k in a a graph. An undirected graph is formed by a finite set of vertices and a set of unordered pairs of vertices, which are called edges. By convention, in algorithm analysis, the number of vertices in the graph is denoted by n and the number of edges is denoted by m. A clique in a graph G is a complete subgraph of G; that is, it is a subset S of the vertices such that every two vertices in S are connected by an edge in G. A maximal clique is a clique to which no more vertices can be added; a maximum clique is a clique that includes the largest possible number of vertices, and the clique number ?(G) is the number of vertices in a maximum clique of G. #include <iostream> #include <fstream> #include <string> #include <vector> using namespace std; bool removable(vector<int> neighbor, vector<int> cover); int max_removable(vector<vector<int> > neighbors, vector<int> cover); vector<int> procedure_1(vector<vector<int> > neighbors, vector<int> cover); vector<int> procedure_2(vector<vector<int> > neighbors, vector<int> cover, int k); int cover_size(vector<int> cover); ifstream infile("graph.txt"); ofstream outfile("cliques.txt"); int main() { //Read Graph (note we work with the complement of the input graph) cout << "Clique Algorithm." << endl; int n, i, j, k, K, p, q, r, s, min, edge, counter = 0; infile >> n; vector<vector<int> > graph; for (i = 0; i < n; i++) { vector<int> row; for (j = 0; j < n; j++) { infile >> edge; if (edge == 0) row.push_back(1); else row.push_back(0); } graph.push_back(row); } //Find Neighbors vector<vector<int> > neighbors; for (i = 0; i < graph.size(); i++) { vector<int> neighbor; for (j = 0; j < graph[i].size(); j++) if (graph[i][j] == 1) neighbor.push_back(j); neighbors.push_back(neighbor); } cout << "Graph has n = " << n << " vertices." << endl; //Read maximum size of Clique wanted cout << "Find a Clique of size at least k = "; cin >> K; k = n - K; //Find Cliques bool found = false; cout << "Finding Cliques..." << endl; min = n + 1; vector<vector<int> > covers; vector<int> allcover; for (i = 0; i < graph.size(); i++) allcover.push_back(1); for (i = 0; i < allcover.size(); i++) { if (found) break; counter++; cout << counter << ". "; outfile << counter << ". "; vector<int> cover = allcover; cover[i] = 0; cover = procedure_1(neighbors, cover); s = cover_size(cover); if (s < min) min = s; if (s <= k) { outfile << "Clique (" << n - s << "): "; for (j = 0; j < cover.size(); j++) if (cover[j] == 0) outfile << j + 1 << " "; outfile << endl; cout << "Clique Size: " << n - s << endl; covers.push_back(cover); found = true; break; } for (j = 0; j < n - k; j++) cover = procedure_2(neighbors, cover, j); s = cover_size(cover); if (s < min) min = s; outfile << "Clique (" << n - s << "): "; for (j = 0; j < cover.size(); j++) if (cover[j] == 0) outfile << j + 1 << " "; outfile << endl; cout << "Clique Size: " << n - s << endl; covers.push_back(cover); if (s <= k) { found = true; break; } } //Pairwise Intersections for (p = 0; p < covers.size(); p++) { if (found) break; for (q = p + 1; q < covers.size(); q++) { if (found) break; counter++; cout << counter << ". "; outfile << counter << ". "; vector<int> cover = allcover; for (r = 0; r < cover.size(); r++) if (covers[p][r] == 0 && covers[q][r] == 0) cover[r] = 0; cover = procedure_1(neighbors, cover); s = cover_size(cover); if (s < min) min = s; if (s <= k) { outfile << "Clique (" << n - s << "): "; for (j = 0; j < cover.size(); j++) if (cover[j] == 0) outfile << j + 1 << " "; outfile << endl; cout << "Clique Size: " << n - s << endl; found = true; break; } for (j = 0; j < k; j++) cover = procedure_2(neighbors, cover, j); s = cover_size(cover); if (s < min) min = s; outfile << "Clique (" << n - s << "): "; for (j = 0; j < cover.size(); j++) if (cover[j] == 0) outfile << j + 1 << " "; outfile << endl; cout << "Clique Size: " << n - s << endl; if (s <= k) { found = true; break; } } } if (found) cout << "Found Clique of size at least " << K << "." << endl; else cout << "Could not find Clique of size at least " << K << "." << endl << "Maximum Clique size found is " << n - min << "." << endl; cout << "See cliques.txt for results." << endl; return 0; } bool removable(vector<int> neighbor, vector<int> cover) { bool check = true; for (int i = 0; i < neighbor.size(); i++) if (cover[neighbor[i]] == 0) { check = false; break; } return check; } int max_removable(vector<vector<int> > neighbors, vector<int> cover) { int r = -1, max = -1; for (int i = 0; i < cover.size(); i++) { if (cover[i] == 1 && removable(neighbors[i], cover) == true) { vector<int> temp_cover = cover; temp_cover[i] = 0; int sum = 0; for (int j = 0; j < temp_cover.size(); j++) if (temp_cover[j] == 1 && removable(neighbors[j], temp_cover) == true) sum++; if (sum > max) { max = sum; r = i; } } } return r; } vector<int> procedure_1(vector<vector<int> > neighbors, vector<int> cover) { vector<int> temp_cover = cover; int r = 0; while (r != -1) { r = max_removable(neighbors, temp_cover); if (r != -1) temp_cover[r] = 0; } return temp_cover; } vector<int> procedure_2(vector<vector<int> > neighbors, vector<int> cover, int k) { int count = 0; vector<int> temp_cover = cover; int i = 0; for (int i = 0; i < temp_cover.size(); i++) { if (temp_cover[i] == 1) { int sum = 0, index; for (int j = 0; j < neighbors[i].size(); j++) if (temp_cover[neighbors[i][j]] == 0) { index = j; sum++; } if (sum == 1 && cover[neighbors[i][index]] == 0) { temp_cover[neighbors[i][index]] = 1; temp_cover[i] = 0; temp_cover = procedure_1(neighbors, temp_cover); count++; } if (count > k) break; } } return temp_cover; } int cover_size(vector<int> cover) { int count = 0; for (int i = 0; i < cover.size(); i++) if (cover[i] == 1) count++; return count; } ``` 