9,395 research outputs found
Counting MSTD Sets in Finite Abelian Groups
In an abelian group G, a more sums than differences (MSTD) set is a subset A
of G such that |A+A|>|A-A|. We provide asymptotics for the number of MSTD sets
in finite abelian groups, extending previous results of Nathanson. The proof
contains an application of a recently resolved conjecture of Alon and Kahn on
the number of independent sets in a regular graph.Comment: 17 page
On fractional realizations of graph degree sequences
We introduce fractional realizations of a graph degree sequence and a closely
associated convex polytope. Simple graph realizations correspond to a subset of
the vertices of this polytope. We describe properties of the polytope vertices
and characterize degree sequences for which each polytope vertex corresponds to
a simple graph realization. These include the degree sequences of pseudo-split
graphs, and we characterize their realizations both in terms of forbidden
subgraphs and graph structure.Comment: 18 pages, 4 figure
Efficient and exact sampling of simple graphs with given arbitrary degree sequence
Uniform sampling from graphical realizations of a given degree sequence is a
fundamental component in simulation-based measurements of network observables,
with applications ranging from epidemics, through social networks to Internet
modeling. Existing graph sampling methods are either link-swap based
(Markov-Chain Monte Carlo algorithms) or stub-matching based (the Configuration
Model). Both types are ill-controlled, with typically unknown mixing times for
link-swap methods and uncontrolled rejections for the Configuration Model. Here
we propose an efficient, polynomial time algorithm that generates statistically
independent graph samples with a given, arbitrary, degree sequence. The
algorithm provides a weight associated with each sample, allowing the
observable to be measured either uniformly over the graph ensemble, or,
alternatively, with a desired distribution. Unlike other algorithms, this
method always produces a sample, without back-tracking or rejections. Using a
central limit theorem-based reasoning, we argue, that for large N, and for
degree sequences admitting many realizations, the sample weights are expected
to have a lognormal distribution. As examples, we apply our algorithm to
generate networks with degree sequences drawn from power-law distributions and
from binomial distributions.Comment: 8 pages, 3 figure
Integer colorings with forbidden rainbow sums
For a set of positive integers , an -coloring of is
rainbow sum-free if it contains no rainbow Schur triple. In this paper we
initiate the study of the rainbow Erd\H{o}s-Rothchild problem in the context of
sum-free sets, which asks for the subsets of with the maximum number of
rainbow sum-free -colorings. We show that for , the interval is
optimal, while for , the set is optimal. We
also prove a stability theorem for . The proofs rely on the hypergraph
container method, and some ad-hoc stability analysis.Comment: 20 page
Conditional Lower Bounds for Space/Time Tradeoffs
In recent years much effort has been concentrated towards achieving
polynomial time lower bounds on algorithms for solving various well-known
problems. A useful technique for showing such lower bounds is to prove them
conditionally based on well-studied hardness assumptions such as 3SUM, APSP,
SETH, etc. This line of research helps to obtain a better understanding of the
complexity inside P.
A related question asks to prove conditional space lower bounds on data
structures that are constructed to solve certain algorithmic tasks after an
initial preprocessing stage. This question received little attention in
previous research even though it has potential strong impact.
In this paper we address this question and show that surprisingly many of the
well-studied hard problems that are known to have conditional polynomial time
lower bounds are also hard when concerning space. This hardness is shown as a
tradeoff between the space consumed by the data structure and the time needed
to answer queries. The tradeoff may be either smooth or admit one or more
singularity points.
We reveal interesting connections between different space hardness
conjectures and present matching upper bounds. We also apply these hardness
conjectures to both static and dynamic problems and prove their conditional
space hardness.
We believe that this novel framework of polynomial space conjectures can play
an important role in expressing polynomial space lower bounds of many important
algorithmic problems. Moreover, it seems that it can also help in achieving a
better understanding of the hardness of their corresponding problems in terms
of time
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