In this paper, we present a collection of novel and scalable algorithms designed to tackle the challenges inherent in the $k$-clique densest subgraph problem (\kcdsp) within network analysis. We propose \psctl, a novel algorithm based on the Frank-Wolfe approach for addressing \kcdsp, effectively solving a distinct convex programming problem. \textcolor{black}{\psctl is able to approximate \kcdsp with near optimal guarantees.} The notable advantage of \psctl lies in its time complexity, which is independent of the count of $k$-cliques, resulting in remarkable efficiency in practical applications. Additionally, we present \spath, a sampling-based algorithm with the capability to handle networks on an unprecedented scale, reaching up to $1.8\times 10^9$ edges. By leveraging the \ccpath algorithm as a uniform $k$-clique sampler, \spath ensures the efficient processing of large-scale network data, accompanied by a detailed analysis of accuracy guarantees. Together, these contributions represent a significant advancement in the field of $k$-clique densest subgraph discovery. In experimental evaluations, our algorithms demonstrate orders of magnitude faster performance compared to the current state-of-the-art solutions.

翻译：本文提出了一系列新颖且可扩展的算法，旨在解决网络分析中$k$-团最密子图问题（\kcdsp）所固有的挑战。我们提出了\psctl，一种基于Frank-Wolfe方法解决\kcdsp的新算法，有效求解了一个独特的凸规划问题。\textcolor{black}{\psctl能够以接近最优的保证近似求解\kcdsp。} \psctl的显著优势在于其时间复杂度独立于$k$-团的个数，从而在实际应用中实现了卓越的效率。此外，我们还提出了\spath，一种基于采样的算法，能够处理规模空前的网络，最高可达$1.8\times 10^9$条边。通过利用\ccpath算法作为均匀$k$-团采样器，\spath确保了大规模网络数据的高效处理，并附有精度保证的详细分析。这些贡献共同代表了$k$-团最密子图发现领域的重大进步。在实验评估中，我们的算法相比当前最先进的解决方案，实现了数量级的性能提升。