74 research outputs found
Nondeterministic graph property testing
A property of finite graphs is called nondeterministically testable if it has
a "certificate" such that once the certificate is specified, its correctness
can be verified by random local testing. In this paper we study certificates
that consist of one or more unary and/or binary relations on the nodes, in the
case of dense graphs. Using the theory of graph limits, we prove that
nondeterministically testable properties are also deterministically testable.Comment: Version 2: 11 pages; we allow orientation in the certificate,
describe new application
Testing Uniformity of Stationary Distribution
A random walk on a directed graph gives a Markov chain on the vertices of the
graph. An important question that arises often in the context of Markov chain
is whether the uniform distribution on the vertices of the graph is a
stationary distribution of the Markov chain. Stationary distribution of a
Markov chain is a global property of the graph. In this paper, we prove that
for a regular directed graph whether the uniform distribution on the vertices
of the graph is a stationary distribution, depends on a local property of the
graph, namely if (u,v) is an directed edge then outdegree(u) is equal to
indegree(v).
This result also has an application to the problem of testing whether a given
distribution is uniform or "far" from being uniform. This is a well studied
problem in property testing and statistics. If the distribution is the
stationary distribution of the lazy random walk on a directed graph and the
graph is given as an input, then how many bits of the input graph do one need
to query in order to decide whether the distribution is uniform or "far" from
it? This is a problem of graph property testing and we consider this problem in
the orientation model (introduced by Halevy et al.). We reduce this problem to
test (in the orientation model) whether a directed graph is Eulerian. And using
result of Fischer et al. on query complexity of testing (in the orientation
model) whether a graph is Eulerian, we obtain bounds on the query complexity
for testing whether the stationary distribution is uniform
Testing formula satisfaction
We study the query complexity of testing for properties defined by read once formulae, as instances of massively parametrized properties, and prove several testability and non-testability results. First we prove the testability of any property accepted by a Boolean read-once formula involving any bounded arity gates, with a number of queries exponential in \epsilon and independent of all other parameters. When the gates are limited to being monotone, we prove that there is an estimation algorithm, that outputs an approximation of the distance of the input from
satisfying the property. For formulae only involving And/Or gates, we provide a more efficient test whose query complexity is only quasi-polynomial in \epsilon. On the other hand we show that such testability results do not hold in general for formulae over non-Boolean alphabets; specifically we construct a property defined by a read-once arity 2 (non-Boolean) formula over alphabets of size 4, such that any 1/4-test for it requires a number of queries depending on the formula size
On the Structure of Equilibria in Basic Network Formation
We study network connection games where the nodes of a network perform edge
swaps in order to improve their communication costs. For the model proposed by
Alon et al. (2010), in which the selfish cost of a node is the sum of all
shortest path distances to the other nodes, we use the probabilistic method to
provide a new, structural characterization of equilibrium graphs. We show how
to use this characterization in order to prove upper bounds on the diameter of
equilibrium graphs in terms of the size of the largest -vicinity (defined as
the the set of vertices within distance from a vertex), for any
and in terms of the number of edges, thus settling positively a conjecture of
Alon et al. in the cases of graphs of large -vicinity size (including graphs
of large maximum degree) and of graphs which are dense enough.
Next, we present a new swap-based network creation game, in which selfish
costs depend on the immediate neighborhood of each node; in particular, the
profit of a node is defined as the sum of the degrees of its neighbors. We
prove that, in contrast to the previous model, this network creation game
admits an exact potential, and also that any equilibrium graph contains an
induced star. The existence of the potential function is exploited in order to
show that an equilibrium can be reached in expected polynomial time even in the
case where nodes can only acquire limited knowledge concerning non-neighboring
nodes.Comment: 11 pages, 4 figure
Interconnection network with a shared whiteboard: Impact of (a)synchronicity on computing power
In this work we study the computational power of graph-based models of
distributed computing in which each node additionally has access to a global
whiteboard. A node can read the contents of the whiteboard and, when activated,
can write one message of O(log n) bits on it. When the protocol terminates,
each node computes the output based on the final contents of the whiteboard. We
consider several scheduling schemes for nodes, providing a strict ordering of
their power in terms of the problems which can be solved with exactly one
activation per node. The problems used to separate the models are related to
Maximal Independent Set, detection of cycles of length 4, and BFS spanning tree
constructions
Testing Odd Direct Sums Using High Dimensional Expanders
In this work, using methods from high dimensional expansion, we show that the property of k-direct-sum is testable for odd values of k . Previous work of [Kaufman and Lubotzky, 2014] could inherently deal only with the case that k is even, using a reduction to linearity testing. Interestingly, our work is the first to combine the topological notion of high dimensional expansion (called co-systolic expansion) with the combinatorial/spectral notion of high dimensional expansion (called colorful expansion) to obtain the result.
The classical k-direct-sum problem applies to the complete complex; Namely it considers a function defined over all k-subsets of some n sized universe. Our result here applies to any collection of k-subsets of an n-universe, assuming this collection of subsets forms a high dimensional expander
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