Greedily finding a dense subgraph

Decomposing a graph into a hierarchical structure via kcore analysis is a standard operation in any modern graphmining toolkit. k-core decomposition is a simple and efficient method that allows to analyze a graph beyond its mere degree distribution. More specifically, it is used to identify areas in the graph of increasing centrality and connectedness, and it allows to reveal the structural organization of the graph.Despite the fact that k-core analysis relies on vertex degrees, k-cores do not satisfy a certain, rather natural, density property. Simply put, the most central k-core is not necessarily the densest subgraph. This inconsistency between k-cores and graph density provides the basis of our study.We start by defining what it means for a subgraph to be locally-dense, and we show that our definition entails a nested chain decomposition of the graph, similar to the one given by k-cores, but in this case the components are arranged in order of increasing density. We show that such a locally-dense decomposition for a graph G = (V, E) can be computed in polynomial time. The running time of the exact decomposition algorithm is O(|V | 2 |E|) but is significantly faster in practice. In addition, we develop a lineartime algorithm that provides a factor-2 approximation to the optimal locally-dense decomposition. Furthermore, we show that the k-core decomposition is also a factor-2 approximation, however, as demonstrated by our experimental evaluation, in practice k-cores have different structure than locally-dense subgraphs, and as predicted by the theory, kcores are not always well-aligned with graph density.

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“…4 We apply Core, GreedyLD, and ExactLD to every dataset. We use a computer equipped with 3GHz Intel Core i7 and 8GB of RAM.…”

Section: Methodsmentioning

confidence: 99%

Density-friendly Graph Decomposition

Tatti

Gionis

2015

Proceedings of the 24th International Conference on World Wide Web

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“…The currently best approximation guarantee for the latter problem is O(n −δ ), for some universal constant δ < 1/3, due to Feige, Kortsarz and Peleg [23], superseding an earlier O(n −0.3885 ) factor given by Kortsarz and Peleg [41]. Additional approaches whose performance depends on the ratio k/n have emerged over the years, for example, a greedy heuristic proposed by Asahiro, Iwama, Tamaki and Tokuyama [8], and SDP-based algorithms developed by Feige and Langberg [22] and Han, Ye and Zhang [36]. For the case k = (n), Arora, Karger and Karpinski [6] devised a PTAS in dense graphs.…”

Section: Related Workmentioning

confidence: 96%

Approximate k-Steiner Forests via the Lagrangian Relaxation Technique with Internal Preprocessing

Segev

2008

Algorithmica

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An instance of the k-Steiner forest problem consists of an undirected graph G = (V , E), the edges of which are associated with non-negative costs, and a collection D = {(s 1 , t 1 ), . . . , (s d , t d )} of distinct pairs of vertices, interchangeably referred to as demands. We say that a forest F ⊆ G connects a demand (s i , t i ) when it contains an s i -t i path. Given a profit k i for each demand (s i , t i ) and a requirement parameter k, the goal is to find a minimum cost forest that connects a subset of demands whose combined profit is at least k. This problem has recently been studied by Hajiaghayi and Jain (SODA '06), whose main contribution in this context was to relate the inapproximability of k-Steiner forest to that of the dense k-subgraph problem. However, Hajiaghayi and Jain did not provide any algorithmic result for the respective settings, and posed this objective as an important direction for future research.In this paper, we present the first non-trivial approximation algorithm for the k-Steiner forest problem, which is based on a novel extension of the Lagrangian relaxation technique. Specifically, our algorithm constructs a feasible forest whose cost is within a factor of O(min{n 2/3 , √ d} · log d) of optimal, where n is the number of vertices in the input graph and d is the number of demands. We believe that the approach illustrated in the current writing is of independent interest, and may be applicable in other settings as well.

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“…Until recently, when Bhaskara et al [7] presented an O(n 1/4+ )-approximation algorithm for any > 0, it has been a long-standing open problem whether the factor O(n 1/3 ) can be significantly beaten. Ashahiro et al [8] showed that a simple greedy strategy yields an approximation ratio of O(n/k), and Feige and Langberg [9] showed that n/k is achievable using semidefinite programming. On the other hand, using a random sampling technique, Arora et al [10] presented a polynomial-time approximation scheme (PTAS) for dense graphs with k = (n), that is, if the number of edges is (n 2 ) and k = (n), then there is an (1 + )-approximation algorithm for any > 0.…”

Section: Introductionmentioning

confidence: 98%

PTAS for Densest $$k$$ k -Subgraph in Interval Graphs

Nonner

2014

Algorithmica

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Given an interval graph and integer k, we consider the problem of finding a subgraph of size k with a maximum number of induced edges, called densest ksubgraph problem in interval graphs. This problem is NP-hard even for chordal graphs (Perl and Corneil in Discret Appl Math 9(1): [27][28][29][30][31][32][33][34][35][36][37][38][39] 1984), and there is probably no PTAS for general graphs (Khot and Subhash in SIAM J Comput 36 (4): 2006). However, the exact complexity status for interval graphs is a long-standing open problem (Perl and Corneil in Discret Appl Math 9(1): [27][28][29][30][31][32][33][34][35][36][37][38][39] 1984), and the best known approximation result is a 3-approximation algorithm (Liazi et al. in Inf Process Lett 108(1):29-32, 2008). We shed light on the approximation complexity of finding a densest k-subgraph in interval graphs by presenting a polynomial-time approximation scheme (PTAS), that is, we show that there is an (1 + )-approximation algorithm for any > 0, which is the first such approximation scheme for the densest k-subgraph problem in an important graph class without any further restrictions.

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Greedily finding a dense subgraph

Cited by 102 publications

References 6 publications

Density-friendly Graph Decomposition

Density-friendly Graph Decomposition

Approximate k-Steiner Forests via the Lagrangian Relaxation Technique with Internal Preprocessing

PTAS for Densest $$k$$ k -Subgraph in Interval Graphs

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