The densest subgraph of a large graph usually refers to some subgraph with the highest average degree, which has been extended to the family of $p$-means dense subgraph objectives by~\citet{veldt2021generalized}. The $p$-mean densest subgraph problem seeks a subgraph with the highest average $p$-th-power degree, whereas the standard densest subgraph problem seeks a subgraph with a simple highest average degree. It was shown that the standard peeling algorithm can perform arbitrarily poorly on generalized objective when $p>1$ but uncertain when $0<p<1$. In this paper, we are the first to show that a standard peeling algorithm can still yield $2^{1/p}$-approximation for the case $0<p < 1$. (Veldt 2021) proposed a new generalized peeling algorithm (GENPEEL), which for $p \geq 1$ has an approximation guarantee ratio $(p+1)^{1/p}$, and time complexity $O(mn)$, where $m$ and $n$ denote the number of edges and nodes in graph respectively. In terms of algorithmic contributions, we propose a new and faster generalized peeling algorithm (called GENPEEL++ in this paper), which for $p \in [1, +\infty)$ has an approximation guarantee ratio $(2(p+1))^{1/p}$, and time complexity $O(m(\log n))$, where $m$ and $n$ denote the number of edges and nodes in graph, respectively. This approximation ratio converges to 1 as $p \rightarrow \infty$.

翻译：大型图的稠密子图通常指具有最大平均度的子图，该概念已由~\citet{veldt2021generalized}扩展至$p$均值稠密子图目标族。$p$均值稠密子图问题旨在寻找具有最高平均$p$次幂度的子图，而标准稠密子图问题则寻求具有简单最高平均度的子图。已有研究表明，当$p>1$时，标准剥离算法在广义目标函数上可能表现极差，但当$0<p<1$时其效果尚不确定。本文首次证明，在$0<p<1$情况下，标准剥离算法仍可实现$2^{1/p}$近似比。Veldt (2021)提出了新型广义剥离算法(GENPEEL)，该算法在$p \geq 1$时具有$(p+1)^{1/p}$的近似保证比，时间复杂度为$O(mn)$，其中$m$和$n$分别表示图的边数和节点数。在算法贡献方面，我们提出了一种新的快速广义剥离算法(本文称为GENPEEL++)，对于$p \in [1, +\infty)$，其近似保证比为$(2(p+1))^{1/p}$，时间复杂度为$O(m(\log n))$（$m$和$n$分别为图的边数和节点数）。该近似比在$p \rightarrow \infty$时收敛于1。