2,368 research outputs found
The formation of black holes in spherically symmetric gravitational collapse
We consider the spherically symmetric, asymptotically flat Einstein-Vlasov
system. We find explicit conditions on the initial data, with ADM mass M, such
that the resulting spacetime has the following properties: there is a family of
radially outgoing null geodesics where the area radius r along each geodesic is
bounded by 2M, the timelike lines are incomplete, and for r>2M
the metric converges asymptotically to the Schwarzschild metric with mass M.
The initial data that we construct guarantee the formation of a black hole in
the evolution. We also give examples of such initial data with the additional
property that the solutions exist for all and all Schwarzschild time,
i.e., we obtain global existence in Schwarzschild coordinates in situations
where the initial data are not small. Some of our results are also established
for the Einstein equations coupled to a general matter model characterized by
conditions on the matter quantities.Comment: 36 pages. A corollary on global existence in Schwarzschild
coordinates for data which are not small is added together with minor
modification
Improving the Price of Anarchy for Selfish Routing via Coordination Mechanisms
We reconsider the well-studied Selfish Routing game with affine latency
functions. The Price of Anarchy for this class of games takes maximum value
4/3; this maximum is attained already for a simple network of two parallel
links, known as Pigou's network. We improve upon the value 4/3 by means of
Coordination Mechanisms.
We increase the latency functions of the edges in the network, i.e., if
is the latency function of an edge , we replace it by
with for all . Then an
adversary fixes a demand rate as input. The engineered Price of Anarchy of the
mechanism is defined as the worst-case ratio of the Nash social cost in the
modified network over the optimal social cost in the original network.
Formally, if \CM(r) denotes the cost of the worst Nash flow in the modified
network for rate and \Copt(r) denotes the cost of the optimal flow in the
original network for the same rate then [\ePoA = \max_{r \ge 0}
\frac{\CM(r)}{\Copt(r)}.]
We first exhibit a simple coordination mechanism that achieves for any
network of parallel links an engineered Price of Anarchy strictly less than
4/3. For the case of two parallel links our basic mechanism gives 5/4 = 1.25.
Then, for the case of two parallel links, we describe an optimal mechanism; its
engineered Price of Anarchy lies between 1.191 and 1.192.Comment: 17 pages, 2 figures, preliminary version appeared at ESA 201
How to detect an anti-spacetime
Is it possible, in principle, to measure the sign of the Lapse? We show that
fermion dynamics distinguishes spacetimes having the same metric but different
tetrads, for instance a Lapse with opposite sign. This sign might be a physical
quantity not captured by the metric. We discuss its possible role in quantum
gravity.Comment: Article awarded with an "Honorable Mention" from the 2012 Gravity
Foundation Award. 6 pages, 8 (pretty) figure
A New Lower Bound for Deterministic Truthful Scheduling
We study the problem of truthfully scheduling tasks to selfish
unrelated machines, under the objective of makespan minimization, as was
introduced in the seminal work of Nisan and Ronen [STOC'99]. Closing the
current gap of on the approximation ratio of deterministic truthful
mechanisms is a notorious open problem in the field of algorithmic mechanism
design. We provide the first such improvement in more than a decade, since the
lower bounds of (for ) and (for ) by
Christodoulou et al. [SODA'07] and Koutsoupias and Vidali [MFCS'07],
respectively. More specifically, we show that the currently best lower bound of
can be achieved even for just machines; for we already get
the first improvement, namely ; and allowing the number of machines to
grow arbitrarily large we can get a lower bound of .Comment: 15 page
Improving the Price of Anarchy for Selfish Routing via Coordination Mechanisms
We reconsider the well-studied Selfish Routing game with affine latency functions. The Price of Anarchy for this class of games takes maximum value 4/3; this maximum is attained already for a simple network of two parallel links, known as Pigou's network. We improve upon the value 4/3 by means of Coordination Mechanisms. We increase the latency functions of the edges in the network, i.e., if is the latency function of an edge , we replace it by with for all . Then an adversary fixes a demand rate as input. The engineered Price of Anarchy of the mechanism is defined as the worst-case ratio of the Nash social cost in the modified network over the optimal social cost in the original network. Formally, if \CM(r) denotes the cost of the worst Nash flow in the modified network for rate and \Copt(r) denotes the cost of the optimal flow in the original network for the same rate then [\ePoA = \max_{r \ge 0} \frac{\CM(r)}{\Copt(r)}.] We first exhibit a simple coordination mechanism that achieves for any network of parallel links an engineered Price of Anarchy strictly less than 4/3. For the case of two parallel links our basic mechanism gives 5/4 = 1.25. Then, for the case of two parallel links, we describe an optimal mechanism; its engineered Price of Anarchy lies between 1.191 and 1.192
Designing cost-sharing methods for Bayesian games
We study the design of cost-sharing protocols for two fundamental resource allocation problems, the Set Cover and the Steiner Tree Problem, under environments of incomplete information (Bayesian model). Our objective is to design protocols where the worst-case Bayesian Nash equilibria, have low cost, i.e. the Bayesian Price of Anarchy (PoA) is minimized. Although budget balance is a very natural requirement, it puts considerable restrictions on the design space, resulting in high PoA. We propose an alternative, relaxed requirement called budget balance in the equilibrium (BBiE).We show an interesting connection between algorithms for Oblivious Stochastic optimization problems and cost-sharing design with low PoA. We exploit this connection for both problems and we enforce approximate solutions of the stochastic problem, as Bayesian Nash equilibria, with the same guarantees on the PoA. More interestingly, we show how to obtain the same bounds on the PoA, by using anonymous posted prices which are desirable because they are easy to implement and, as we show, induce dominant strategies for the players
Welfare guarantees for proportional allocations
According to the proportional allocation mechanism from the network
optimization literature, users compete for a divisible resource -- such as
bandwidth -- by submitting bids. The mechanism allocates to each user a
fraction of the resource that is proportional to her bid and collects an amount
equal to her bid as payment. Since users act as utility-maximizers, this
naturally defines a proportional allocation game. Recently, Syrgkanis and
Tardos (STOC 2013) quantified the inefficiency of equilibria in this game with
respect to the social welfare and presented a lower bound of 26.8% on the price
of anarchy over coarse-correlated and Bayes-Nash equilibria in the full and
incomplete information settings, respectively. In this paper, we improve this
bound to 50% over both equilibrium concepts. Our analysis is simpler and,
furthermore, we argue that it cannot be improved by arguments that do not take
the equilibrium structure into account. We also extend it to settings with
budget constraints where we show the first constant bound (between 36% and 50%)
on the price of anarchy of the corresponding game with respect to an effective
welfare benchmark that takes budgets into account.Comment: 15 page
On Linear Congestion Games with Altruistic Social Context
We study the issues of existence and inefficiency of pure Nash equilibria in
linear congestion games with altruistic social context, in the spirit of the
model recently proposed by de Keijzer {\em et al.} \cite{DSAB13}. In such a
framework, given a real matrix specifying a particular
social context, each player aims at optimizing a linear combination of the
payoffs of all the players in the game, where, for each player , the
multiplicative coefficient is given by the value . We give a broad
characterization of the social contexts for which pure Nash equilibria are
always guaranteed to exist and provide tight or almost tight bounds on their
prices of anarchy and stability. In some of the considered cases, our
achievements either improve or extend results previously known in the
literature
On the Impact of Fair Best Response Dynamics
In this work we completely characterize how the frequency with which each
player participates in the game dynamics affects the possibility of reaching
efficient states, i.e., states with an approximation ratio within a constant
factor from the price of anarchy, within a polynomially bounded number of best
responses. We focus on the well known class of congestion games and we show
that, if each player is allowed to play at least once and at most times
any best responses, states with approximation ratio times the
price of anarchy are reached after best
responses, and that such a bound is essentially tight also after exponentially
many ones. One important consequence of our result is that the fairness among
players is a necessary and sufficient condition for guaranteeing a fast
convergence to efficient states. This answers the important question of the
maximum order of needed to fast obtain efficient states, left open by
[9,10] and [3], in which fast convergence for constant and very slow
convergence for have been shown, respectively. Finally, we show
that the structure of the game implicitly affects its performances. In
particular, we show that in the symmetric setting, in which all players share
the same set of strategies, the game always converges to an efficient state
after a polynomial number of best responses, regardless of the frequency each
player moves with
Self-Similar Collapse of Conformally Coupled Scalar Fields
A massless scalar field minimally coupled to the gravitational field in a
simplified spherical symmetry is discussed. It is shown that, in this case, the
solution found by Roberts, describing a scalar field collapse, is in fact the
most general one. Taking that solution as departure point, a study of the
gravitational collapse for the self-similar conformal case is presented.Comment: 9 pages, accepted for publication, Classical and Quantum Gravity.
Available at http://dft.if.uerj.br/preprint/e-17.tex or at
ftp://dft.if.uerj.br/preprint/e-17.tex . Figures can be obtained on request
at [email protected]
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