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Dense subgraph discovery is an important problem in graph mining and network analysis with several applications. Two canonical polynomial-time solvable 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 [Veldt et al., 2021] introduced the generalized p-mean densest subgraph problem which captures the maxcore problem when p = -∞ and the densest subgraph problem when p = 1. They observed that for p ≥ 1, the objective function is supermodular and hence the problem can be solved in polynomial time. In this work, we focus on the p-mean densest subgraph problem for p ∈ (-∞, 1). We prove that for every p ∈ (-∞,1), the problem is NP-hard, thus resolving an open question from [Veldt et al., 2021]. We also show that for every p ∈ (0,1), the weighted version of the problem is APX-hard. On the algorithmic front, we describe two simple 1/2-approximation algorithms for every p ∈ (-∞, 1). We complement the approximation algorithms by exhibiting non-trivial instances on which the algorithms simultaneously achieve an approximation factor of at most 1/2. 

The densest subgraph problem in a graph (\dsg), in the simplest form, is the following. Given an undirected graph $G=(V,E)$ find a subset $$S \subseteq V$$ of vertices that maximizes the ratio $|E(S)|/|S|$ where $E(S)$ is the set of edges with both endpoints in $$S$$. \dsg and several of its variants are well-studied in theory and practice and have many applications in data mining and network analysis. In this paper we study fast algorithms and structural aspects of \dsg via the lens of \emph{supermodularity}. For this we consider the densest supermodular subset problem (\dssp): given a non-negative supermodular function $$f: 2^V \rightarrow \mathbb{R}_+$, maximize $f(S)/|S|$$. For \dsg we describe a simple flow-based algorithm that outputs a $$(1-\eps)$-approximation in deterministic $$\tilde{O}(m/\eps)$$ time where $$m$$ is the number of edges. Our algorithm is the first to have a near-linear dependence on $$m$$ and $$1/\eps$$ and improves previous methods based on an LP relaxation. It generalizes to hypergraphs, and also yields a faster algorithm for directed \dsg. Greedy peeling algorithms have been very popular for \dsg and several variants due to their efficiency, empirical performance, and worst-case approximation guarantees. We describe a simple peeling algorithm for \dssp and analyze its approximation guarantee in a fashion that unifies several existing results. Boob et al.\ \cite{bgpstww-20} developed an \emph{iterative} peeling algorithm for \dsg which appears to work very well in practice, and made a conjecture about its convergence to optimality. We affirmatively answer their conjecture, and in fact prove that a natural generalization of their algorithm converges to a $$(1-\eps)$$-approximation for \emph{any} supermodular function $$f$$; the key to our proof is to consider an LP formulation that is derived via the \Lovasz extension of a supermodular function. For \dsg the bound on the number of iterations we prove is $$O(\frac{\Delta \ln |V|}{\lambda^*}\cdot \frac{1}{\eps^2})$ where $$\Delta$$ is the maximum degree and $$\lambda^*$$ is the optimum value. Our work suggests that iterative peeling can be an effective heuristic for several objectives considered in the literature. Finally, we show that the $$2$$-approximation for densest-at-least-$$k$$ subgraph \cite{ks-09} extends to the supermodular setting. We also give a unified analysis of the peeling algorithm for this problem, and via this analysis derive an approximation guarantee for a generalization of \dssp to maximize $$f(S)/g(|S|)$ for a concave function $$g$$.