745 research outputs found
Shortest Paths and Distances with Differential Privacy
We introduce a model for differentially private analysis of weighted graphs
in which the graph topology is assumed to be public and the private
information consists only of the edge weights . This can
express hiding congestion patterns in a known system of roads. Differential
privacy requires that the output of an algorithm provides little advantage,
measured by privacy parameters and , for distinguishing
between neighboring inputs, which are thought of as inputs that differ on the
contribution of one individual. In our model, two weight functions are
considered to be neighboring if they have distance at most one.
We study the problems of privately releasing a short path between a pair of
vertices and of privately releasing approximate distances between all pairs of
vertices. We are concerned with the approximation error, the difference between
the length of the released path or released distance and the length of the
shortest path or actual distance.
For privately releasing a short path between a pair of vertices, we prove a
lower bound of on the additive approximation error for fixed
. We provide a differentially private algorithm that matches
this error bound up to a logarithmic factor and releases paths between all
pairs of vertices. The approximation error of our algorithm can be bounded by
the number of edges on the shortest path, so we achieve better accuracy than
the worst-case bound for vertex pairs that are connected by a low-weight path
with vertices.
For privately releasing all-pairs distances, we show that for trees we can
release all distances with approximation error for fixed
privacy parameters. For arbitrary bounded-weight graphs with edge weights in
we can release all distances with approximation error
-reconstruction of graphs
Let be a group of permutations acting on an -vertex set , and
and be two simple graphs on . We say that and are -isomorphic
if belongs to the orbit of under the action of . One can naturally
generalize the reconstruction problems so that when is , the symmetric
group, we have the usual reconstruction problems. In this paper, we study
-edge reconstructibility of graphs. We prove some old and new results on
edge reconstruction and reconstruction from end vertex deleted subgraphs.Comment: 8 page
Lipschitz Functions on Expanders are Typically Flat
This work studies the typical behavior of random integer-valued Lipschitz
functions on expander graphs with sufficiently good expansion. We consider two
families of functions: M-Lipschitz functions (functions that change by at most
M along edges) and integer-homomorphisms (functions that change by exactly 1
along edges). We prove that such functions typically exhibit very small
fluctuations. For instance, we show that a uniformly chosen M-Lipschitz
function takes only M+1 values on most of the graph, with a double exponential
decay for the probability to take other values.Comment: 26 page
Combinatorial Optimization Problems with Interaction Costs: Complexity and Solvable Cases
We introduce and study the combinatorial optimization problem with
interaction costs (COPIC). COPIC is the problem of finding two combinatorial
structures, one from each of two given families, such that the sum of their
independent linear costs and the interaction costs between elements of the two
selected structures is minimized. COPIC generalizes the quadratic assignment
problem and many other well studied combinatorial optimization problems, and
hence covers many real world applications. We show how various topics from
different areas in the literature can be formulated as special cases of COPIC.
The main contributions of this paper are results on the computational
complexity and approximability of COPIC for different families of combinatorial
structures (e.g. spanning trees, paths, matroids), and special structures of
the interaction costs. More specifically, we analyze the complexity if the
interaction cost matrix is parameterized by its rank and if it is a diagonal
matrix. Also, we determine the structure of the intersection cost matrix, such
that COPIC is equivalent to independently solving linear optimization problems
for the two given families of combinatorial structures
Combinatorics of Tripartite Boundary Connections for Trees and Dimers
A grove is a spanning forest of a planar graph in which every component tree
contains at least one of a special subset of vertices on the outer face called
nodes. For the natural probability measure on groves, we compute various
connection probabilities for the nodes in a random grove. In particular, for
"tripartite" pairings of the nodes, the probability can be computed as a
Pfaffian in the entries of the Dirichlet-to-Neumann matrix (discrete Hilbert
transform) of the graph. These formulas generalize the determinant formulas
given by Curtis, Ingerman, and Morrow, and by Fomin, for parallel pairings.
These Pfaffian formulas are used to give exact expressions for reconstruction:
reconstructing the conductances of a planar graph from boundary measurements.
We prove similar theorems for the double-dimer model on bipartite planar
graphs.Comment: 29 pages, 7 figure
Tree 3-spanners of diameter at most 5
Tree spanners approximate distances within graphs; a subtree of a graph is a
tree -spanner of the graph if and only if for every pair of vertices their
distance in the subtree is at most times their distance in the graph. When
a graph contains a subtree of diameter at most , then trivially admits a
tree -spanner. Now, determining whether a graph admits a tree -spanner of
diameter at most is an NP complete problem, when , and it is
tractable, when . Although it is not known whether it is tractable to
decide graphs that admit a tree 3-spanner of any diameter, an efficient
algorithm to determine graphs that admit a tree 3-spanner of diameter at most 5
is presented. Moreover, it is proved that if a graph of diameter at most 3
admits a tee 3-spanner, then it admits a tree 3-spanner of diameter at most 5.
Hence, this algorithm decides tree 3-spanner admissibility of diameter at most
3 graphs
Families of -weights of some particular graphs
Let be a positive-weighted graph, that is a graph
endowed with a function from the edge set of to the set of positive
real numbers; for any distinct vertices , we define
to be the weight of the path in joining and with minimum weight. In
this paper we fix a particular class of graphs and we give a criterion to
establish whether, given a family of positive real numbers , there exists a positive-weighted graph in the class we have fixed, with vertex set equal to
and such that for any .
In particular, the classes of graphs we consider are the following: snakes,
caterpillars, polygons, bipartite graphs, complete graphs, planar graphs.Comment: 14 page
Quantum algorithms for graph problems with cut queries
Let be an -vertex graph with edges. When asked a subset of
vertices, a cut query on returns the number of edges of that have
exactly one endpoint in . We show that there is a bounded-error quantum
algorithm that determines all connected components of after making
many cut queries. In contrast, it follows from results in
communication complexity that any randomized algorithm even just to decide
whether the graph is connected or not must make at least
many cut queries. We further show that with many cut queries a
quantum algorithm can with high probability output a spanning forest for .
En route to proving these results, we design quantum algorithms for learning
a graph using cut queries. We show that a quantum algorithm can learn a graph
with maximum degree after many cut queries, and can learn
a general graph with many cut queries. These two
upper bounds are tight up to the poly-logarithmic factors, and compare to
and lower bounds on the number of cut queries
needed by a randomized algorithm for the same problems, respectively.
The key ingredients in our results are the Bernstein-Vazirani algorithm,
approximate counting with "OR queries", and learning sparse vectors from inner
products as in compressed sensing.Comment: Corrected an error in Lemma 1. This led to an extra log factor in the
complexity of the connectivity and spanning forest algorithm
Topological Graphic Passwords And Their Matchings Towards Cryptography
Graphical passwords (GPWs) are convenient for mobile equipments with touch
screen. Topological graphic passwords (Topsnut-gpws) can be saved in computer
by classical matrices and run quickly than the existing GPWs. We research
Topsnut-gpws by the matching of view, since they have many advantages. We
discuss: configuration matching partition, coloring/labelling matching
partition, set matching partition, matching chain, etc. And, we introduce new
graph labellings for enriching Topsnut-matchings and show that these labellings
can be realized for trees or spanning trees of networks. In theoretical works
we explore Graph Labelling Analysis, and show that every graph admits our
extremal labellings and set-type labellings in graph theory. Many of the graph
labellings mentioned are related with problems of set matching partitions to
number theory, and yield new objects and new problems to graph theory
Uniform hypergraphs and dominating sets of graphs
A (simple) hypergraph is a family H of pairwise incomparable sets of a finite
set. We say that a hypergraph H is a domination hypergraph if there is at least
a graph G such that the collection of minimal dominating sets of G is equal to
H. Given a hypergraph, we are interested in determining if it is a domination
hypergraph and, if this is not the case, we want to find domination hypergraphs
in some sense close to it, the domination completions. Here we will focus on
the family of hypergraphs containing all the subsets with the same cardinality,
the uniform hypergraphs of maximum size. Specifically, we characterize those
hypergraphs H in this family that are domination hypergraphs and, in any other
case, we prove that the hypergraph H is uniquely determined by some of its
domination completions and that H can be recovered from them by using a
suitable hypergraph operation.Comment: 24 pages, 3 figure
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