6,049 research outputs found
Detecting and Characterizing Small Dense Bipartite-like Subgraphs by the Bipartiteness Ratio Measure
We study the problem of finding and characterizing subgraphs with small
\textit{bipartiteness ratio}. We give a bicriteria approximation algorithm
\verb|SwpDB| such that if there exists a subset of volume at most and
bipartiteness ratio , then for any , it finds a set
of volume at most and bipartiteness ratio at most
. By combining a truncation operation, we give a local
algorithm \verb|LocDB|, which has asymptotically the same approximation
guarantee as the algorithm \verb|SwpDB| on both the volume and bipartiteness
ratio of the output set, and runs in time
, independent of the size of the
graph. Finally, we give a spectral characterization of the small dense
bipartite-like subgraphs by using the th \textit{largest} eigenvalue of the
Laplacian of the graph.Comment: 17 pages; ISAAC 201
Where Graph Topology Matters: The Robust Subgraph Problem
Robustness is a critical measure of the resilience of large networked
systems, such as transportation and communication networks. Most prior works
focus on the global robustness of a given graph at large, e.g., by measuring
its overall vulnerability to external attacks or random failures. In this
paper, we turn attention to local robustness and pose a novel problem in the
lines of subgraph mining: given a large graph, how can we find its most robust
local subgraph (RLS)?
We define a robust subgraph as a subset of nodes with high communicability
among them, and formulate the RLS-PROBLEM of finding a subgraph of given size
with maximum robustness in the host graph. Our formulation is related to the
recently proposed general framework for the densest subgraph problem, however
differs from it substantially in that besides the number of edges in the
subgraph, robustness also concerns with the placement of edges, i.e., the
subgraph topology. We show that the RLS-PROBLEM is NP-hard and propose two
heuristic algorithms based on top-down and bottom-up search strategies.
Further, we present modifications of our algorithms to handle three practical
variants of the RLS-PROBLEM. Experiments on synthetic and real-world graphs
demonstrate that we find subgraphs with larger robustness than the densest
subgraphs even at lower densities, suggesting that the existing approaches are
not suitable for the new problem setting.Comment: 13 pages, 10 Figures, 3 Tables, to appear at SDM 2015 (9 pages only
Fully Dynamic Algorithm for Top- Densest Subgraphs
Given a large graph, the densest-subgraph problem asks to find a subgraph
with maximum average degree. When considering the top- version of this
problem, a na\"ive solution is to iteratively find the densest subgraph and
remove it in each iteration. However, such a solution is impractical due to
high processing cost. The problem is further complicated when dealing with
dynamic graphs, since adding or removing an edge requires re-running the
algorithm. In this paper, we study the top- densest-subgraph problem in the
sliding-window model and propose an efficient fully-dynamic algorithm. The
input of our algorithm consists of an edge stream, and the goal is to find the
node-disjoint subgraphs that maximize the sum of their densities. In contrast
to existing state-of-the-art solutions that require iterating over the entire
graph upon any update, our algorithm profits from the observation that updates
only affect a limited region of the graph. Therefore, the top- densest
subgraphs are maintained by only applying local updates. We provide a
theoretical analysis of the proposed algorithm and show empirically that the
algorithm often generates denser subgraphs than state-of-the-art competitors.
Experiments show an improvement in efficiency of up to five orders of magnitude
compared to state-of-the-art solutions.Comment: 10 pages, 8 figures, accepted at CIKM 201
Detecting High Log-Densities -- an O(n^1/4) Approximation for Densest k-Subgraph
In the Densest k-Subgraph problem, given a graph G and a parameter k, one
needs to find a subgraph of G induced on k vertices that contains the largest
number of edges. There is a significant gap between the best known upper and
lower bounds for this problem. It is NP-hard, and does not have a PTAS unless
NP has subexponential time algorithms. On the other hand, the current best
known algorithm of Feige, Kortsarz and Peleg, gives an approximation ratio of
n^(1/3-epsilon) for some specific epsilon > 0 (estimated at around 1/60).
We present an algorithm that for every epsilon > 0 approximates the Densest
k-Subgraph problem within a ratio of n^(1/4+epsilon) in time n^O(1/epsilon). In
particular, our algorithm achieves an approximation ratio of O(n^1/4) in time
n^O(log n). Our algorithm is inspired by studying an average-case version of
the problem where the goal is to distinguish random graphs from graphs with
planted dense subgraphs. The approximation ratio we achieve for the general
case matches the distinguishing ratio we obtain for this planted problem.
At a high level, our algorithms involve cleverly counting appropriately
defined trees of constant size in G, and using these counts to identify the
vertices of the dense subgraph. Our algorithm is based on the following
principle. We say that a graph G(V,E) has log-density alpha if its average
degree is Theta(|V|^alpha). The algorithmic core of our result is a family of
algorithms that output k-subgraphs of nontrivial density whenever the
log-density of the densest k-subgraph is larger than the log-density of the
host graph.Comment: 23 page
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