Dense subgraph discovery is an important problem in graph mining and network analysis with several applications. Two canonical problems here are to find a maxcore (subgraph of maximum min degree) and to find a densest subgraph (subgraph of maximum average degree). Both of these problems can be solved in polynomial time. Veldt, Benson, and Kleinberg [VBK21] introduced the generalized $p$-mean densest subgraph problem which captures the maxcore problem when $p=-\infty$ and the densest subgraph problem when $p=1$. They observed that the objective leads to a supermodular function when $p \ge 1$ and hence can be solved in polynomial time; for this case, they also developed a simple greedy peeling algorithm with a bounded approximation ratio. In this paper, we make several contributions. First, we prove that for any $p \in (-\frac{1}{8}, 0) \cup (0, \frac{1}{4})$ the problem is NP-Hard and for any $p \in (-3,0) \cup (0,1)$ the weighted version of the problem is NP-Hard, partly resolving a question left open in [VBK21]. Second, we describe two simple $1/2$-approximation algorithms for all $p < 1$, and show that our analysis of these algorithms is tight. For $p > 1$ we develop a fast near-linear time implementation of the greedy peeling algorithm from [VBK21]. This allows us to plug it into the iterative peeling algorithm that was shown to converge to an optimum solution [CQT22]. We demonstrate the efficacy of our algorithms by running extensive experiments on large graphs. Together, our results provide a comprehensive understanding of the complexity of the $p$-mean densest subgraph problem and lead to fast and provably good algorithms for the full range of $p$.

翻译：稠密子图发现是图挖掘和网络分析中的一个重要问题，具有多种应用。其中两个经典问题是寻找最大核（最大最小度子图）和寻找最稠密子图（最大平均度子图）。这两个问题都可以在多项式时间内求解。Veldt、Benson和Kleinberg [VBK21] 引入了广义p-均值最稠密子图问题，该问题在p=-∞时捕捉到最大核问题，在p=1时捕捉到最稠密子图问题。他们观察到，当p≥1时，目标函数导致一个超模函数，因此可以在多项式时间内求解；针对这种情况，他们还开发了一种简单的贪心剥离算法，并具有有界的近似比。在本文中，我们做出若干贡献。首先，我们证明对于任意p∈(-1/8,0)∪(0,1/4)，该问题是NP困难的，并且对于任意p∈(-3,0)∪(0,1)，该问题的加权版本是NP困难的，部分解决了[VBK21]中遗留的一个问题。其次，我们描述了两种简单的1/2近似算法，适用于所有p<1的情况，并证明我们对这些算法的分析是紧的。对于p>1，我们开发了[VBK21]中贪心剥离算法的快速近线性时间实现。这使得我们能够将其嵌入迭代剥离算法中，该算法已被证明收敛到最优解[CQT22]。我们通过在大型图上进行大量实验，展示了我们算法的有效性。综合起来，我们的结果提供了对p-均值最稠密子图问题复杂性的全面理解，并为整个p取值区间提供了快速且可证明性能良好的算法。