1,106 research outputs found
Optimal Query Complexity for Reconstructing Hypergraphs
In this paper we consider the problem of reconstructing a hidden weighted
hypergraph of constant rank using additive queries. We prove the following: Let
be a weighted hidden hypergraph of constant rank with n vertices and
hyperedges. For any there exists a non-adaptive algorithm that finds the
edges of the graph and their weights using
additive queries. This solves the open problem in [S. Choi, J. H. Kim. Optimal
Query Complexity Bounds for Finding Graphs. {\em STOC}, 749--758,~2008].
When the weights of the hypergraph are integers that are less than
where is the rank of the hypergraph (and therefore for
unweighted hypergraphs) there exists a non-adaptive algorithm that finds the
edges of the graph and their weights using additive queries.
Using the information theoretic bound the above query complexities are tight
Strong Products of Hypergraphs: Unique Prime Factorization Theorems and Algorithms
It is well-known that all finite connected graphs have a unique prime factor
decomposition (PFD) with respect to the strong graph product which can be
computed in polynomial time. Essential for the PFD computation is the
construction of the so-called Cartesian skeleton of the graphs under
investigation.
In this contribution, we show that every connected thin hypergraph H has a
unique prime factorization with respect to the normal and strong (hypergraph)
product. Both products coincide with the usual strong graph product whenever H
is a graph. We introduce the notion of the Cartesian skeleton of hypergraphs as
a natural generalization of the Cartesian skeleton of graphs and prove that it
is uniquely defined for thin hypergraphs. Moreover, we show that the Cartesian
skeleton of hypergraphs can be determined in O(|E|^2) time and that the PFD can
be computed in O(|V|^2|E|) time, for hypergraphs H = (V,E) with bounded degree
and bounded rank
Risk-Averse Matchings over Uncertain Graph Databases
A large number of applications such as querying sensor networks, and
analyzing protein-protein interaction (PPI) networks, rely on mining uncertain
graph and hypergraph databases. In this work we study the following problem:
given an uncertain, weighted (hyper)graph, how can we efficiently find a
(hyper)matching with high expected reward, and low risk?
This problem naturally arises in the context of several important
applications, such as online dating, kidney exchanges, and team formation. We
introduce a novel formulation for finding matchings with maximum expected
reward and bounded risk under a general model of uncertain weighted
(hyper)graphs that we introduce in this work. Our model generalizes
probabilistic models used in prior work, and captures both continuous and
discrete probability distributions, thus allowing to handle privacy related
applications that inject appropriately distributed noise to (hyper)edge
weights. Given that our optimization problem is NP-hard, we turn our attention
to designing efficient approximation algorithms. For the case of uncertain
weighted graphs, we provide a -approximation algorithm, and a
-approximation algorithm with near optimal run time. For the case
of uncertain weighted hypergraphs, we provide a
-approximation algorithm, where is the rank of the
hypergraph (i.e., any hyperedge includes at most nodes), that runs in
almost (modulo log factors) linear time.
We complement our theoretical results by testing our approximation algorithms
on a wide variety of synthetic experiments, where we observe in a controlled
setting interesting findings on the trade-off between reward, and risk. We also
provide an application of our formulation for providing recommendations of
teams that are likely to collaborate, and have high impact.Comment: 25 page
Boxicity and separation dimension
A family of permutations of the vertices of a hypergraph is
called 'pairwise suitable' for if, for every pair of disjoint edges in ,
there exists a permutation in in which all the vertices in one
edge precede those in the other. The cardinality of a smallest such family of
permutations for is called the 'separation dimension' of and is denoted
by . Equivalently, is the smallest natural number so that
the vertices of can be embedded in such that any two
disjoint edges of can be separated by a hyperplane normal to one of the
axes. We show that the separation dimension of a hypergraph is equal to the
'boxicity' of the line graph of . This connection helps us in borrowing
results and techniques from the extensive literature on boxicity to study the
concept of separation dimension.Comment: This is the full version of a paper by the same name submitted to
WG-2014. Some results proved in this paper are also present in
arXiv:1212.6756. arXiv admin note: substantial text overlap with
arXiv:1212.675
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