Computing the k densest subgraphs of a graph

ACM SIGMOD Record, 2021

Given a directed graph G, the directed densest subgraph (DDS) problem refers to the finding of a subgraph from G, whose density is the highest among all the subgraphs of G. The DDS problem is fundamental to a wide range of applications, such as fraud detection, community mining, and graph compression. However, existing DDS solutions suffer from efficiency and scalability problems: on a threethousand- edge graph, it takes three days for one of the best exact algorithms to complete. In this paper, we develop an efficient and scalable DDS solution. We introduce the notion of [x, y]-core, which is a dense subgraph for G, and show that the densest subgraph can be accurately located through the [x, y]-core with theoretical guarantees. Based on the [x, y]-core, we develop both exact and approximation algorithms. We have performed an extensive evaluation of our approaches on eight real large datasets. The results show that our proposed solutions are up to six orders of magnitude faster than...

2016

This thesis studies the concept of dense subgraphs, speci cally for graphs with multiple edge sets. Our work improves the running time of an existing Linear Program (LP) for solving the Densest Common Subgraph problem. This LP was originally created by Moses Charikar for a single graph and extended to multiple graphs (DCS LP) by Vinay Jethava and Niko Beerenwinkel. This thesis shows that the run time of the DCS LP can be shortened considerably by using an interior-point method instead of the simplex method. The DCS LP is also compared to a greedy algorithm and a Lagrangian relaxation of DCS LP.We conclude that the greedy algorithm is a good heuristic while the LP is well suited to problems where a closer approximation is important.

2020

Given an undirected graph $G$, the Densest $k$-subgraph problem (DkS) asks to compute a set $S \subset V$ of cardinality $\left\lvert S\right\rvert \leq k$ such that the weight of edges inside $S$ is maximized. This is a fundamental NP-hard problem whose approximability, inspite of many decades of research, is yet to be settled. The current best known approximation algorithm due to Bhaskara et al. (2010) computes a $\mathcal{O}\left({n^{1/4 + \epsilon}}\right)$ approximation in time $n^{\mathcal{O}\left(1/\epsilon\right)}$. We ask what are some "easier" instances of this problem? We propose some natural semi-random models of instances with a planted dense subgraph, and study approximation algorithms for computing the densest subgraph in them. These models are inspired by the semi-random models of instances studied for various other graph problems such as the independent set problem, graph partitioning problems etc. For a large range of parameters of these models, we get si...

Lecture Notes in Computer Science, 2013

Exact and approximation algorithms for densest k-subgraph * N. Bourgeois (a) A. Giannakos (b) G. Lucarelli (c) I. Milis (d) V. Th. Paschos (b)(e)

Information Processing Letters, 2014

Partial vertex cover Exact algorithm Design of algorithms Many graph concepts such as cliques, k-clubs, and k-plexes are used to define cohesive subgroups in a social network. The concept of densest k-set is one of them. A densest k-set in an undirected graph G = (V , E) is a vertex set S ⊆ V of size k such that the number of edges in the subgraph of G induced by S is maximum among all subgraphs of G induced by vertex sets of size k. One can obtain a densest k-set of G in O (k 2 n k) time by exhaustivesearch technique for an undirected graph of n vertices and a number k < n. However, if the value of k approaches n/2, the running time of the exhaustive-search algorithm is O * (2 n). Whether there exists an O * (c n)-time algorithm with the fixed constant c < 2 to find a densest k-set in an undirected graph of n vertices remains open in the literature. In this paper, we point out that the densest k-set problem and a class of problems related to the concept of densest k-sets can be solved in time O * (1.7315 n).

2012

Abstract. Dense subgraph discovery is a key issue in graph mining, due to its importance in several applications, such as correlation analysis, community discovery in the Web, gene co-expression and protein-protein interactions in bioinformatics. In this work, we study the discovery of the top-k dense subgraphs in a set of graphs. After the investigation of the problem in its static case, we extend the methodology to work with dy-namic graph collections, where the graph collection changes over time. Our methodology is based on lower and upper bounds of the density, resulting in a reduction of the number of exact density computations. Our algorithms do not rely on user-dened threshold values and the only input required is the number of dense subgraphs in the result (k). In addition to the exact algorithms, an approximation algorithm is pro-vided for top-k dense subgraph discovery, which trades result accuracy for speed. We show that a signicant number of exact density compu-tations i...

2007 IEEE/ACS International Conference on Computer Systems and Applications, 2007

In the maximum common subgraph (MCS) problem, we are given a pair of graphs and asked to find the largest induced subgraph common to them both. With its plethora of applications, MCS is a familiar and challenging problem. Many algorithms exist that can deliver optimal MCS solutions, but whose asymptotic worst-case run times fail to do better than mere brute-force, which is exponential in the order of the smaller graph. In this paper, we present a faster solution to MCS. We transform an essential part of the search process into the task of enumerating maximal independent sets in only a part of only one of the input graphs. This is made possible by exploiting an efficient decomposition of a graph into a minimum vertex cover and the maximum independent set in its complement. The result is an algorithm whose run * time is bounded by a function exponential in the order of the smaller cover rather than in the order of the smaller graph.

Proceedings of the 2022 International Conference on Management of Data

Given a graph, densest subgraph search reports a single subgraph that maximizes the density (i.e., average degree). To diversify the search results without imposing rigid constraints, this paper studies the problem of anchored densest subgraph search (ADS). Given a graph, a reference node set and an anchored node set with ⊆ , ADS reports a supergraph of that maximizes thesubgraph density-a density that favors nodes that are close to and are not over-popular compared to nodes in. These two levels of localities to bring wide applications, as demonstrated by our use cases. For ADS, we propose an algorithm that is local since the complexity is only related to the nodes in as opposed to the entire graph. Extensive experiments show that our local algorithm for ADS outperforms the global algorithm by up to three orders of magnitude in time and space consumption; moreover, our local algorithm outperforms existing local community detection solutions in locality, result density, and query processing time and space. CCS CONCEPTS • Mathematics of computing → Graph algorithms; • Information systems → Clustering.

Journal of Algorithms, 1987

l the minimum dominating set problem for trees , l the minimum total dominating set problem for trees , l the maximum disjoint triangle problem for series-parallel graphs Pa l the traveling salesman problem for Halin networks [6], l the Steiner problem for Hahn networks [22], l the traveling salesman problem for bandwidth-limited networks [15], and a great variety of similar problems.

2009

This paper presents a method for identifying a set of dense subgraphs of a given sparse graph. Within the main applications of this "dense subgraph problem", the dense subgraphs are interpreted as communities, as in, e.g., social networks. The problem of identifying dense subgraphs helps analyze graph structures and complex networks and it is known to be challenging. It bears some similarities with the problem of reordering/blocking matrices in sparse matrix techniques. We exploit this link and adapt the idea of recognizing matrix column similarities, in order to compute a partial clustering of the vertices in a graph, where each cluster represents a dense subgraph. In contrast to existing subgraph extraction techniques which are based on a complete clustering of the graph nodes, the proposed algorithm takes into account the fact that not every participating node in the network needs to belong to a community. Another advantage is that the method does not require to specify the number of clusters; this number is usually not known in advance and is difficult to estimate. The computational process is very efficient, and the effectiveness of the proposed method is demonstrated in a few real-life examples.