Finding Connected Dense kk-Subgraphs

Given a connected graph $G$ on $n$ vertices and a positive integer $k\le n$, a subgraph of $G$ on $k$ vertices is called a $k$-subgraph in $G$. We design combinatorial approximation algorithms for finding a connected $k$-subgraph in $G$ such that its density is at least a factor $\Omega(\max\{n^{-2/5},k^2/n^2\})$ of the density of the densest $k$-subgraph in $G$ (which is not necessarily connected). These particularly provide the first non-trivial approximations for the densest connected $k$-subgraph problem on general graphs

Finding Connected-Dense-Connected Subgraphs and variants is NP-Hard

Finding Connected-Dense-Connected (CDC) subgraphs from Triple Networks is NP-Hard. finding One-Connected-Dense (OCD) sub- graphs from Triple Networks is also NP-Hard. We present formal proofs of these theorems hereby

Finding cliques and dense subgraphs using edge queries

We consider the problem of finding a large clique in an Erd\H{o}s--R\'enyi random graph where we are allowed unbounded computational time but can only query a limited number of edges. Recall that the largest clique in $G \sim G(n,1/2)$ has size roughly $2\log_{2} n$. Let $\alpha_{\star}(\delta,\ell)$ be the supremum over $\alpha$ such that there exists an algorithm that makes $n^{\delta}$ queries in total to the adjacency matrix of $G$, in a constant $\ell$ number of rounds, and outputs a clique of size $\alpha \log_{2} n$ with high probability. We give improved upper bounds on $\alpha_{\star}(\delta,\ell)$ for every $\delta \in [1,2)$ and $\ell \geq 3$. We also study analogous questions for finding subgraphs with density at least $\eta$ for a given $\eta$, and prove corresponding impossibility results.Comment: 19 pp, 5 figures, Focused Workshop on Networks and Their Limits held at the Erd\H{o}s Center, Budapest, Hungary in July 202

Finding the Hierarchy of Dense Subgraphs using Nucleus Decompositions

Finding dense substructures in a graph is a fundamental graph mining operation, with applications in bioinformatics, social networks, and visualization to name a few. Yet most standard formulations of this problem (like clique, quasiclique, k-densest subgraph) are NP-hard. Furthermore, the goal is rarely to find the "true optimum", but to identify many (if not all) dense substructures, understand their distribution in the graph, and ideally determine relationships among them. Current dense subgraph finding algorithms usually optimize some objective, and only find a few such subgraphs without providing any structural relations. We define the nucleus decomposition of a graph, which represents the graph as a forest of nuclei. Each nucleus is a subgraph where smaller cliques are present in many larger cliques. The forest of nuclei is a hierarchy by containment, where the edge density increases as we proceed towards leaf nuclei. Sibling nuclei can have limited intersections, which enables discovering overlapping dense subgraphs. With the right parameters, the nucleus decomposition generalizes the classic notions of k-cores and k-truss decompositions. We give provably efficient algorithms for nucleus decompositions, and empirically evaluate their behavior in a variety of real graphs. The tree of nuclei consistently gives a global, hierarchical snapshot of dense substructures, and outputs dense subgraphs of higher quality than other state-of-the-art solutions. Our algorithm can process graphs with tens of millions of edges in less than an hour