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

TL;DR

The paper applies Instance Space Analysis to the Maximum Clique Problem to systematically compare five MCP algorithms (MOMC, Gurobi, CliSAT, FastWClq, HGS) across a large, diverse instance set derived from standard graph benchmarks. It uses a 35-feature graph representation (primarily generic, with two MCP-specific features) and a composite hardness metric to map instances into a 2D space, revealing regions where each algorithm excels. An ISA-based predictive model (SVC) trained on 6138 instances accurately forecasts the best-performing algorithm on 34 challenging, larger test graphs (88% top-1, 97% top-2), indicating strong generalization and substantial potential time savings. The work also identifies avenues for extending ISA to other graph COPs, incorporating quantum methods, and refining features and performance metrics for broader applicability and explainability.

Abstract

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.

Figures (6)

• Figure 1: Instance Space Analysis methodological framework that extends Rice’s Algorithm Selection Problem
• Figure 2: Node count, edge count, and density distribution of 6138 graph instances used for performing the initial instance space analysis
• Figure 3: The features selected by the SIFTED routine of the ISA plotted on the $Z_{1}$-$Z_{2}$ plane for the existing dataset comprising of 6138 graph instances and the estimate of the boundaries of the instance space obtained by the CLOISTER routine (red boundary lines)
• Figure 4: Instance space containing existing dataset containing instances from COLLAB, IMDB-BINARY and TWITTER graphs (6138 instances)
• Figure 5: Performance of the five algorithms across all the instances of the existing dataset. The performance measure has been scaled from 0 to 1 and the scale applies across all the plots. A lower value of the performance measure indicates better performance.