2,702 research outputs found
Scalable Kernelization for Maximum Independent Sets
The most efficient algorithms for finding maximum independent sets in both
theory and practice use reduction rules to obtain a much smaller problem
instance called a kernel. The kernel can then be solved quickly using exact or
heuristic algorithms---or by repeatedly kernelizing recursively in the
branch-and-reduce paradigm. It is of critical importance for these algorithms
that kernelization is fast and returns a small kernel. Current algorithms are
either slow but produce a small kernel, or fast and give a large kernel. We
attempt to accomplish both of these goals simultaneously, by giving an
efficient parallel kernelization algorithm based on graph partitioning and
parallel bipartite maximum matching. We combine our parallelization techniques
with two techniques to accelerate kernelization further: dependency checking
that prunes reductions that cannot be applied, and reduction tracking that
allows us to stop kernelization when reductions become less fruitful. Our
algorithm produces kernels that are orders of magnitude smaller than the
fastest kernelization methods, while having a similar execution time.
Furthermore, our algorithm is able to compute kernels with size comparable to
the smallest known kernels, but up to two orders of magnitude faster than
previously possible. Finally, we show that our kernelization algorithm can be
used to accelerate existing state-of-the-art heuristic algorithms, allowing us
to find larger independent sets faster on large real-world networks and
synthetic instances.Comment: Extended versio
A Novel Approach to Finding Near-Cliques: The Triangle-Densest Subgraph Problem
Many graph mining applications rely on detecting subgraphs which are
near-cliques. There exists a dichotomy between the results in the existing work
related to this problem: on the one hand the densest subgraph problem (DSP)
which maximizes the average degree over all subgraphs is solvable in polynomial
time but for many networks fails to find subgraphs which are near-cliques. On
the other hand, formulations that are geared towards finding near-cliques are
NP-hard and frequently inapproximable due to connections with the Maximum
Clique problem.
In this work, we propose a formulation which combines the best of both
worlds: it is solvable in polynomial time and finds near-cliques when the DSP
fails. Surprisingly, our formulation is a simple variation of the DSP.
Specifically, we define the triangle densest subgraph problem (TDSP): given
, find a subset of vertices such that , where is the number of triangles induced
by the set . We provide various exact and approximation algorithms which the
solve the TDSP efficiently. Furthermore, we show how our algorithms adapt to
the more general problem of maximizing the -clique average density. Finally,
we provide empirical evidence that the TDSP should be used whenever the output
of the DSP fails to output a near-clique.Comment: 42 page
On a registration-based approach to sensor network localization
We consider a registration-based approach for localizing sensor networks from
range measurements. This is based on the assumption that one can find
overlapping cliques spanning the network. That is, for each sensor, one can
identify geometric neighbors for which all inter-sensor ranges are known. Such
cliques can be efficiently localized using multidimensional scaling. However,
since each clique is localized in some local coordinate system, we are required
to register them in a global coordinate system. In other words, our approach is
based on transforming the localization problem into a problem of registration.
In this context, the main contributions are as follows. First, we describe an
efficient method for partitioning the network into overlapping cliques. Second,
we study the problem of registering the localized cliques, and formulate a
necessary rigidity condition for uniquely recovering the global sensor
coordinates. In particular, we present a method for efficiently testing
rigidity, and a proposal for augmenting the partitioned network to enforce
rigidity. A recently proposed semidefinite relaxation of global registration is
used for registering the cliques. We present simulation results on random and
structured sensor networks to demonstrate that the proposed method compares
favourably with state-of-the-art methods in terms of run-time, accuracy, and
scalability
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