21,574 research outputs found
Fast Distributed Approximation for Max-Cut
Finding a maximum cut is a fundamental task in many computational settings.
Surprisingly, it has been insufficiently studied in the classic distributed
settings, where vertices communicate by synchronously sending messages to their
neighbors according to the underlying graph, known as the or
models. We amend this by obtaining almost optimal
algorithms for Max-Cut on a wide class of graphs in these models. In
particular, for any , we develop randomized approximation
algorithms achieving a ratio of to the optimum for Max-Cut on
bipartite graphs in the model, and on general graphs in the
model.
We further present efficient deterministic algorithms, including a
-approximation for Max-Dicut in our models, thus improving the best known
(randomized) ratio of . Our algorithms make non-trivial use of the greedy
approach of Buchbinder et al. (SIAM Journal on Computing, 2015) for maximizing
an unconstrained (non-monotone) submodular function, which may be of
independent interest
Approximating Subdense Instances of Covering Problems
We study approximability of subdense instances of various covering problems
on graphs, defined as instances in which the minimum or average degree is
Omega(n/psi(n)) for some function psi(n)=omega(1) of the instance size. We
design new approximation algorithms as well as new polynomial time
approximation schemes (PTASs) for those problems and establish first
approximation hardness results for them. Interestingly, in some cases we were
able to prove optimality of the underlying approximation ratios, under usual
complexity-theoretic assumptions. Our results for the Vertex Cover problem
depend on an improved recursive sampling method which could be of independent
interest
Fast approximation of centrality and distances in hyperbolic graphs
We show that the eccentricities (and thus the centrality indices) of all
vertices of a -hyperbolic graph can be computed in linear
time with an additive one-sided error of at most , i.e., after a
linear time preprocessing, for every vertex of one can compute in
time an estimate of its eccentricity such that
for a small constant . We
prove that every -hyperbolic graph has a shortest path tree,
constructible in linear time, such that for every vertex of ,
. These results are based on an
interesting monotonicity property of the eccentricity function of hyperbolic
graphs: the closer a vertex is to the center of , the smaller its
eccentricity is. We also show that the distance matrix of with an additive
one-sided error of at most can be computed in
time, where is a small constant. Recent empirical studies show that
many real-world graphs (including Internet application networks, web networks,
collaboration networks, social networks, biological networks, and others) have
small hyperbolicity. So, we analyze the performance of our algorithms for
approximating centrality and distance matrix on a number of real-world
networks. Our experimental results show that the obtained estimates are even
better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author
Inapproximability of Combinatorial Optimization Problems
We survey results on the hardness of approximating combinatorial optimization
problems
On the equivalence between graph isomorphism testing and function approximation with GNNs
Graph neural networks (GNNs) have achieved lots of success on
graph-structured data. In the light of this, there has been increasing interest
in studying their representation power. One line of work focuses on the
universal approximation of permutation-invariant functions by certain classes
of GNNs, and another demonstrates the limitation of GNNs via graph isomorphism
tests.
Our work connects these two perspectives and proves their equivalence. We
further develop a framework of the representation power of GNNs with the
language of sigma-algebra, which incorporates both viewpoints. Using this
framework, we compare the expressive power of different classes of GNNs as well
as other methods on graphs. In particular, we prove that order-2 Graph
G-invariant networks fail to distinguish non-isomorphic regular graphs with the
same degree. We then extend them to a new architecture, Ring-GNNs, which
succeeds on distinguishing these graphs and provides improvements on real-world
social network datasets
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