146,179 research outputs found
On The Multiparty Communication Complexity of Testing Triangle-Freeness
In this paper we initiate the study of property testing in simultaneous and
non-simultaneous multi-party communication complexity, focusing on testing
triangle-freeness in graphs. We consider the model,
where we have players receiving private inputs, and a coordinator who
receives no input; the coordinator can communicate with all the players, but
the players cannot communicate with each other. In this model, we ask: if an
input graph is divided between the players, with each player receiving some of
the edges, how many bits do the players and the coordinator need to exchange to
determine if the graph is triangle-free, or from triangle-free?
For general communication protocols, we show that
bits are sufficient to test triangle-freeness in
graphs of size with average degree (the degree need not be known in
advance). For protocols, where there is only one
communication round, we give a protocol that uses bits
when and when ; here, again, the average degree does not need to be
known in advance. We show that for average degree , our simultaneous
protocol is asymptotically optimal up to logarithmic factors. For higher
degrees, we are not able to give lower bounds on testing triangle-freeness, but
we give evidence that the problem is hard by showing that finding an edge that
participates in a triangle is hard, even when promised that at least a constant
fraction of the edges must be removed in order to make the graph triangle-free.Comment: To Appear in PODC 201
Streaming Lower Bounds for Approximating MAX-CUT
We consider the problem of estimating the value of max cut in a graph in the
streaming model of computation. At one extreme, there is a trivial
-approximation for this problem that uses only space, namely,
count the number of edges and output half of this value as the estimate for max
cut value. On the other extreme, if one allows space, then a
near-optimal solution to the max cut value can be obtained by storing an
-size sparsifier that essentially preserves the max cut. An
intriguing question is if poly-logarithmic space suffices to obtain a
non-trivial approximation to the max-cut value (that is, beating the factor
). It was recently shown that the problem of estimating the size of a
maximum matching in a graph admits a non-trivial approximation in
poly-logarithmic space.
Our main result is that any streaming algorithm that breaks the
-approximation barrier requires space even if the
edges of the input graph are presented in random order. Our result is obtained
by exhibiting a distribution over graphs which are either bipartite or
-far from being bipartite, and establishing that
space is necessary to differentiate between these
two cases. Thus as a direct corollary we obtain that
space is also necessary to test if a graph is bipartite or -far
from being bipartite.
We also show that for any , any streaming algorithm that
obtains a -approximation to the max cut value when edges arrive
in adversarial order requires space, implying that
space is necessary to obtain an arbitrarily good approximation to
the max cut value
Local Conflict Coloring
Locally finding a solution to symmetry-breaking tasks such as
vertex-coloring, edge-coloring, maximal matching, maximal independent set,
etc., is a long-standing challenge in distributed network computing. More
recently, it has also become a challenge in the framework of centralized local
computation. We introduce conflict coloring as a general symmetry-breaking task
that includes all the aforementioned tasks as specific instantiations ---
conflict coloring includes all locally checkable labeling tasks from
[Naor\&Stockmeyer, STOC 1993]. Conflict coloring is characterized by two
parameters and , where the former measures the amount of freedom given
to the nodes for selecting their colors, and the latter measures the number of
constraints which colors of adjacent nodes are subject to.We show that, in the
standard LOCAL model for distributed network computing, if l/d \textgreater{}
\Delta, then conflict coloring can be solved in rounds in -node graphs with maximum degree
, where ignores the polylog factors in . The
dependency in~ is optimal, as a consequence of the lower
bound by [Linial, SIAM J. Comp. 1992] for -coloring. An important
special case of our result is a significant improvement over the best known
algorithm for distributed -coloring due to [Barenboim, PODC 2015],
which required rounds. Improvements for other
variants of coloring, including -list-coloring,
-edge-coloring, -coloring, etc., also follow from our general
result on conflict coloring. Likewise, in the framework of centralized local
computation algorithms (LCAs), our general result yields an LCA which requires
a smaller number of probes than the previously best known algorithm for
vertex-coloring, and works for a wide range of coloring problems
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