2,170 research outputs found
Community-aware network sparsification
Network sparsification aims to reduce the number of edges of a network while maintaining its structural properties; such properties include shortest paths, cuts, spectral measures, or network modularity. Sparsification has multiple applications, such as, speeding up graph-mining algorithms, graph visualization, as well as identifying the important network edges.
In this paper we consider a novel formulation of the network-sparsification problem. In addition to the network, we also consider as input a set of communities. The goal is to sparsify the network so as to preserve the network structure with respect to the given communities. We introduce two variants of the community-aware sparsification problem, leading to sparsifiers that satisfy different connectedness community properties. From the technical point of view, we prove hardness results and devise effective approximation algorithms. Our experimental results on a large collection of datasets demonstrate the effectiveness of our algorithms.https://epubs.siam.org/doi/10.1137/1.9781611974973.48Accepted manuscrip
On the Size and the Approximability of Minimum Temporally Connected Subgraphs
We consider temporal graphs with discrete time labels and investigate the
size and the approximability of minimum temporally connected spanning
subgraphs. We present a family of minimally connected temporal graphs with
vertices and edges, thus resolving an open question of (Kempe,
Kleinberg, Kumar, JCSS 64, 2002) about the existence of sparse temporal
connectivity certificates. Next, we consider the problem of computing a minimum
weight subset of temporal edges that preserve connectivity of a given temporal
graph either from a given vertex r (r-MTC problem) or among all vertex pairs
(MTC problem). We show that the approximability of r-MTC is closely related to
the approximability of Directed Steiner Tree and that r-MTC can be solved in
polynomial time if the underlying graph has bounded treewidth. We also show
that the best approximation ratio for MTC is at least and at most , for
any constant , where is the number of temporal edges and
is the maximum degree of the underlying graph. Furthermore, we prove
that the unweighted version of MTC is APX-hard and that MTC is efficiently
solvable in trees and -approximable in cycles
Distributed Edge Connectivity in Sublinear Time
We present the first sublinear-time algorithm for a distributed
message-passing network sto compute its edge connectivity exactly in
the CONGEST model, as long as there are no parallel edges. Our algorithm takes
time to compute and a
cut of cardinality with high probability, where and are the
number of nodes and the diameter of the network, respectively, and
hides polylogarithmic factors. This running time is sublinear in (i.e.
) whenever is. Previous sublinear-time
distributed algorithms can solve this problem either (i) exactly only when
[Thurimella PODC'95; Pritchard, Thurimella, ACM
Trans. Algorithms'11; Nanongkai, Su, DISC'14] or (ii) approximately [Ghaffari,
Kuhn, DISC'13; Nanongkai, Su, DISC'14].
To achieve this we develop and combine several new techniques. First, we
design the first distributed algorithm that can compute a -edge connectivity
certificate for any in time .
Second, we show that by combining the recent distributed expander decomposition
technique of [Chang, Pettie, Zhang, SODA'19] with techniques from the
sequential deterministic edge connectivity algorithm of [Kawarabayashi, Thorup,
STOC'15], we can decompose the network into a sublinear number of clusters with
small average diameter and without any mincut separating a cluster (except the
`trivial' ones). Finally, by extending the tree packing technique from [Karger
STOC'96], we can find the minimum cut in time proportional to the number of
components. As a byproduct of this technique, we obtain an -time
algorithm for computing exact minimum cut for weighted graphs.Comment: Accepted at 51st ACM Symposium on Theory of Computing (STOC 2019
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