The modularity is the best known and widely used quality function for community detection in graphs. We investigate the approximability of the modularity maximization problem and some related problems. We first design a polynomial-time 0.4209-additive approximation algorithm for the modularity maximization problem, which improves the current best additive approximation error of 0.4672. Our theoretical analysis also demonstrates that the proposed algorithm obtains a nearly-optimal solution for any instance with a high modularity value. We next design a polynomial-time 0.1660-additive approximation algorithm for the maximum modularity cut problem. Finally, we extend our algorithm to some related problems.

模块化是用于图形社区检测的最著名和广泛使用的质量函数。我们研究了模块化最大化问题和一些相关问题的逼近性。我们首先针对模块化最大化问题设计了多项式时间0.4209加法逼近算法，该算法提高了当前的最佳加法逼近误差0.4672。我们的理论分析还表明，对于任何具有高模块化值的实例，所提出的算法都能获得接近最优的解决方案。接下来，我们针对最大模块化切割问题设计多项式时间0.1660可加逼近算法。最后，我们将算法扩展到一些相关问题。