Candidate range limitation for efficient rainbow clique detection

Multiple deterministic and probabilistic solutions have been proposed for the max-clique (MC) and max-planted-clique (MPC) problems. However, those remain prohibitively expensive in large graphs. The problem can be simplified if one assumes a coloring of the graph and a regular coloring of the clique (a rainbow clique). However, to date, no efficient rainbow clique algorithm has been proposed. We propose SPHERA (Search Space Limitation Efficient Rainbow Clique Algorithm) to find rainbow cliques using a combination of greedy growth, backtracking, and an efficient minimization of the search space using colored k−Cores. We show in G(n,p) and real-world colored graphs that SPHERA detects the rainbow clique with a much higher probability and much faster than current non-rainbow clique algorithms. We further propose multiple heuristics for the initial vertex selection in real-world graphs and show that those improve the clique detection speed in SPHERA. The code is available in GitHub at https://github.com/louzounlab/SPHERA.