536 research outputs found
Buyback Problem - Approximate matroid intersection with cancellation costs
In the buyback problem, an algorithm observes a sequence of bids and must
decide whether to accept each bid at the moment it arrives, subject to some
constraints on the set of accepted bids. Decisions to reject bids are
irrevocable, whereas decisions to accept bids may be canceled at a cost that is
a fixed fraction of the bid value. Previous to our work, deterministic and
randomized algorithms were known when the constraint is a matroid constraint.
We extend this and give a deterministic algorithm for the case when the
constraint is an intersection of matroid constraints. We further prove a
matching lower bound on the competitive ratio for this problem and extend our
results to arbitrary downward closed set systems. This problem has applications
to banner advertisement, semi-streaming, routing, load balancing and other
problems where preemption or cancellation of previous allocations is allowed
Exact bounds for distributed graph colouring
We prove exact bounds on the time complexity of distributed graph colouring.
If we are given a directed path that is properly coloured with colours, by
prior work it is known that we can find a proper 3-colouring in communication rounds. We close the gap between upper and
lower bounds: we show that for infinitely many the time complexity is
precisely communication rounds.Comment: 16 pages, 3 figure
Almost-Tight Distributed Minimum Cut Algorithms
We study the problem of computing the minimum cut in a weighted distributed
message-passing networks (the CONGEST model). Let be the minimum cut,
be the number of nodes in the network, and be the network diameter. Our
algorithm can compute exactly in time. To the best of our knowledge, this is the first paper that
explicitly studies computing the exact minimum cut in the distributed setting.
Previously, non-trivial sublinear time algorithms for this problem are known
only for unweighted graphs when due to Pritchard and
Thurimella's -time and -time algorithms for
computing -edge-connected and -edge-connected components.
By using the edge sampling technique of Karger's, we can convert this
algorithm into a -approximation -time algorithm for any . This improves
over the previous -approximation -time algorithm and
-approximation -time algorithm of Ghaffari and Kuhn. Due to the lower
bound of by Das Sarma et al. which holds for any
approximation algorithm, this running time is tight up to a factor.
To get the stated running time, we developed an approximation algorithm which
combines the ideas of Thorup's algorithm and Matula's contraction algorithm. It
saves an factor as compared to applying Thorup's tree
packing theorem directly. Then, we combine Kutten and Peleg's tree partitioning
algorithm and Karger's dynamic programming to achieve an efficient distributed
algorithm that finds the minimum cut when we are given a spanning tree that
crosses the minimum cut exactly once
Distributed Symmetry Breaking in Hypergraphs
Fundamental local symmetry breaking problems such as Maximal Independent Set
(MIS) and coloring have been recognized as important by the community, and
studied extensively in (standard) graphs. In particular, fast (i.e.,
logarithmic run time) randomized algorithms are well-established for MIS and
-coloring in both the LOCAL and CONGEST distributed computing
models. On the other hand, comparatively much less is known on the complexity
of distributed symmetry breaking in {\em hypergraphs}. In particular, a key
question is whether a fast (randomized) algorithm for MIS exists for
hypergraphs.
In this paper, we study the distributed complexity of symmetry breaking in
hypergraphs by presenting distributed randomized algorithms for a variety of
fundamental problems under a natural distributed computing model for
hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can
be solved in rounds ( is the number of nodes of the
hypergraph) in the LOCAL model. We then present a key result of this paper ---
an -round hypergraph MIS algorithm in
the CONGEST model where is the maximum node degree of the hypergraph
and is any arbitrarily small constant.
To demonstrate the usefulness of hypergraph MIS, we present applications of
our hypergraph algorithm to solving problems in (standard) graphs. In
particular, the hypergraph MIS yields fast distributed algorithms for the {\em
balanced minimal dominating set} problem (left open in Harris et al. [ICALP
2013]) and the {\em minimal connected dominating set problem}. We also present
distributed algorithms for coloring, maximal matching, and maximal clique in
hypergraphs.Comment: Changes from the previous version: More references adde
General Relativity as an Attractor in Scalar-Tensor Stochastic Inflation
Quantum fluctuations of scalar fields during inflation could determine the
very large-scale structure of the universe. In the case of general
scalar-tensor gravity theories these fluctuations lead to the diffusion of
fundamental constants like the Planck mass and the effective Brans--Dicke
parameter, . In the particular case of Brans--Dicke gravity, where
is constant, this leads to runaway solutions with infinitely large
values of the Planck mass. However, in a theory with variable we find
stationary probability distributions with a finite value of the Planck mass
peaked at exponentially large values of after inflation. We conclude
that general relativity is an attractor during the quantum diffusion of the
fields.Comment: LaTeX (with RevTex) 11 pages, 2 uuencoded figures appended, also
available on WWW via http://star.maps.susx.ac.uk/index.htm
Implications of Space-Time foam for Entanglement Correlations of Neutral Kaons
The role of invariance and consequences for bipartite entanglement of
neutral (K) mesons are discussed. A relaxation of leads to a modification
of the entanglement which is known as the effect. The relaxation of
assumptions required to prove the theorem are examined within the context
of models of space-time foam. It is shown that the evasion of the EPR type
entanglement implied by (which is connected with spin statistics) is
rather elusive. Relaxation of locality (through non-commutative geometry) or
the introduction of decoherence by themselves do not lead to a destruction of
the entanglement. So far we find only one model which is based on non-critical
strings and D-particle capture and recoil that leads to a stochastic
contribution to the space-time metric and consequent change in the neutral
meson bipartite entanglement. The lack of an omega effect is demonstrated for a
class of models based on thermal like baths which are generally considered as
generic models of decoherence
STATIONARY SOLUTIONS IN BRANS-DICKE STOCHASTIC INFLATIONARY COSMOLOGY
In Brans-Dicke theory the Universe becomes divided after inflation into many
exponentially large domains with different values of the effective
gravitational constant. Such a process can be described by diffusion equations
for the probability of finding a certain value of the inflaton and dilaton
fields in a physical volume of the Universe. For a typical chaotic inflation
potential, the solutions for the probability distribution never become
stationary but grow forever towards larger values of the fields. We show here
that a non-minimal conformal coupling of the inflaton to the curvature scalar,
as well as radiative corrections to the effective potential, may provide a
dynamical cutoff and generate stationary solutions. We also analyze the
possibility of large nonperturbative jumps of the fluctuating inflaton scalar
field, which was recently revealed in the context of the Einstein theory. We
find that in the Brans--Dicke theory the amplitude of such jumps is strongly
suppressed.Comment: 19 pages, LaTe
Constraints from Inflation on Scalar-Tensor Gravity Theories
We show how observations of the perturbation spectra produced during
inflation may be used to constrain the parameters of general scalar-tensor
theories of gravity, which include both an inflaton and dilaton field. An
interesting feature of these models is the possibility that the curvature
perturbations on super-horizon scales may not be constant due to non-adiabatic
perturbations of the two fields. Within a given model, the tilt and relative
amplitude of the scalar and tensor perturbation spectra gives constraints on
the parameters of the gravity theory, which may be comparable with those from
primordial nucleosynthesis and post-Newtonian experiments.Comment: LaTeX (with RevTex) 19 pages, 8 uuencoded figures appended, also
available on WWW via http://star.maps.susx.ac.uk/index.htm
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