133 research outputs found
Swap-Robust and Almost Supermagic Complete Graphs for Dynamical Distributed Storage
To prevent service time bottlenecks in distributed storage systems, the
access balancing problem has been studied by designing almost supermagic edge
labelings of certain graphs to balance the access requests to different
servers. In this paper, we introduce the concept of robustness of edge
labelings under limited-magnitude swaps, which is important for studying the
dynamical access balancing problem with respect to changes in data popularity.
We provide upper and lower bounds on the robustness ratio for complete graphs
with vertices, and construct -almost supermagic labelings that are
asymptotically optimal in terms of the robustness ratio.Comment: 27 pages, no figur
Rapid Mixing of Gibbs Sampling on Graphs that are Sparse on Average
In this work we show that for every and the Ising model defined
on , there exists a , such that for all with probability going to 1 as , the mixing time of the
dynamics on is polynomial in . Our results are the first
polynomial time mixing results proven for a natural model on for where the parameters of the model do not depend on . They also provide
a rare example where one can prove a polynomial time mixing of Gibbs sampler in
a situation where the actual mixing time is slower than n \polylog(n). Our
proof exploits in novel ways the local treelike structure of Erd\H{o}s-R\'enyi
random graphs, comparison and block dynamics arguments and a recent result of
Weitz.
Our results extend to much more general families of graphs which are sparse
in some average sense and to much more general interactions. In particular,
they apply to any graph for which every vertex of the graph has a
neighborhood of radius in which the induced sub-graph is a
tree union at most edges and where for each simple path in
the sum of the vertex degrees along the path is . Moreover, our
result apply also in the case of arbitrary external fields and provide the
first FPRAS for sampling the Ising distribution in this case. We finally
present a non Markov Chain algorithm for sampling the distribution which is
effective for a wider range of parameters. In particular, for it
applies for all external fields and , where is the critical point for decay of correlation for the Ising model on
.Comment: Corrected proof of Lemma 2.
Algebraic Combinatorics of Magic Squares
We describe how to construct and enumerate Magic squares, Franklin squares,
Magic cubes, and Magic graphs as lattice points inside polyhedral cones using
techniques from Algebraic Combinatorics. The main tools of our methods are the
Hilbert Poincare series to enumerate lattice points and the Hilbert bases to
generate lattice points. We define polytopes of magic labelings of graphs and
digraphs, and give a description of the faces of the Birkhoff polytope as
polytopes of magic labelings of digraphs.Comment: Ph.D. Thesi
On Sum Graphs over Some Magmas
We consider the notions of sum graph and of relaxed sum graph over a magma,
give several examples and results of these families of graphs over some natural
magmas. We classify the cycles that are sum graphs for the magma of the subsets
of a set with the operation of union, determine the abelian groups that provide
a sum labelling of , and show that is a sum graph over the
abelian group , where is the
corresponding Fibonacci number. For integral sum graphs, we give a linear upper
bound for the radius of matchings, improving Harary's labelling for this family
of graphs, and give the exact radius for the family of totally disconnected
graphs.
We found integer labellings for the 4D-cube, giving a negative answer to a
question of Melnikov and Pyatikin, actually showing that the 4D-cube has
infinitely many primitive labellings. We have also obtained some new results on
mod sum graphs and relaxed sum graphs. Finally, we show that the direct product
operation is closed for strong integral sum graphs
Optimal Deterministic Massively Parallel Connectivity on Forests
We show fast deterministic algorithms for fundamental problems on forests in
the challenging low-space regime of the well-known Massive Parallel Computation
(MPC) model. A recent breakthrough result by Coy and Czumaj [STOC'22] shows
that, in this setting, it is possible to deterministically identify connected
components on graphs in rounds, where is the
diameter of the graph and the number of nodes. The authors left open a
major question: is it possible to get rid of the additive factor
and deterministically identify connected components in a runtime that is
completely independent of ?
We answer the above question in the affirmative in the case of forests. We
give an algorithm that identifies connected components in
deterministic rounds. The total memory required is words, where is
the number of edges in the input graph, which is optimal as it is only enough
to store the input graph. We complement our upper bound results by showing that
time is necessary even for component-unstable algorithms,
conditioned on the widely believed 1 vs. 2 cycles conjecture. Our techniques
also yield a deterministic forest-rooting algorithm with the same runtime and
memory bounds.
Furthermore, we consider Locally Checkable Labeling problems (LCLs), whose
solution can be verified by checking the -radius neighborhood of each
node. We show that any LCL problem on forests can be solved in
rounds with a canonical deterministic algorithm, improving over the
runtime of Brandt, Latypov and Uitto [DISC'21]. We also show that there is no
algorithm that solves all LCL problems on trees asymptotically faster.Comment: ACM-SIAM Symposium on Discrete Algorithms (SODA) 202
On the Troll-Trust Model for Edge Sign Prediction in Social Networks
In the problem of edge sign prediction, we are given a directed graph
(representing a social network), and our task is to predict the binary labels
of the edges (i.e., the positive or negative nature of the social
relationships). Many successful heuristics for this problem are based on the
troll-trust features, estimating at each node the fraction of outgoing and
incoming positive/negative edges. We show that these heuristics can be
understood, and rigorously analyzed, as approximators to the Bayes optimal
classifier for a simple probabilistic model of the edge labels. We then show
that the maximum likelihood estimator for this model approximately corresponds
to the predictions of a Label Propagation algorithm run on a transformed
version of the original social graph. Extensive experiments on a number of
real-world datasets show that this algorithm is competitive against
state-of-the-art classifiers in terms of both accuracy and scalability.
Finally, we show that troll-trust features can also be used to derive online
learning algorithms which have theoretical guarantees even when edges are
adversarially labeled.Comment: v5: accepted to AISTATS 201
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