Comparative algorithm performance evaluation and prediction for the maximum clique problem using instance space analysis

The maximum clique problem, a well-known graph-based combinatorial optimization problem, has been addressed through various algorithmic approaches, though systematic analyses of the problem instances remain sparse. This study employs the instance space analysis (ISA) methodology to systematically analyze the instance space of this problem and assess & predict the performance of state-of-the-art (SOTA) algorithms, including exact, heuristic, and graph neural network (GNN)-based methods. A dataset was compiled using graph instances from TWITTER, COLLAB and IMDB-BINARY benchmarks commonly used in graph machine learning research. A set of 33 generic and 2 problem-specific polynomial-time-computable graph-based features, including several spectral properties, was employed for the ISA. A composite performance measure incorporating both solution quality and algorithm runtime was utilized. The comparative analysis demonstrated that the exact algorithm Mixed Order Maximum Clique (MOMC) exhibited superior performance across approximately 74.7% of the instance space constituted by the compiled dataset. Gurobi & CliSAT accounted for superior performance in 13.8% and 11% of the instance space, respectively. The ISA-based algorithm performance prediction model run on 34 challenging test instances compiled from the BHOSLIB and DIMACS datasets yielded top-1 and top-2 best performing algorithm prediction accuracies of 88% and 97%, respectively.