904 research outputs found
Exponential Separation of Quantum Communication and Classical Information
We exhibit a Boolean function for which the quantum communication complexity
is exponentially larger than the classical information complexity. An
exponential separation in the other direction was already known from the work
of Kerenidis et. al. [SICOMP 44, pp. 1550-1572], hence our work implies that
these two complexity measures are incomparable. As classical information
complexity is an upper bound on quantum information complexity, which in turn
is equal to amortized quantum communication complexity, our work implies that a
tight direct sum result for distributional quantum communication complexity
cannot hold. The function we use to present such a separation is the Symmetric
k-ary Pointer Jumping function introduced by Rao and Sinha [ECCC TR15-057],
whose classical communication complexity is exponentially larger than its
classical information complexity. In this paper, we show that the quantum
communication complexity of this function is polynomially equivalent to its
classical communication complexity. The high-level idea behind our proof is
arguably the simplest so far for such an exponential separation between
information and communication, driven by a sequence of round-elimination
arguments, allowing us to simplify further the approach of Rao and Sinha.
As another application of the techniques that we develop, we give a simple
proof for an optimal trade-off between Alice's and Bob's communication while
computing the related Greater-Than function on n bits: say Bob communicates at
most b bits, then Alice must send n/exp(O(b)) bits to Bob. This holds even when
allowing pre-shared entanglement. We also present a classical protocol
achieving this bound.Comment: v1, 36 pages, 3 figure
An Improved Private Mechanism for Small Databases
We study the problem of answering a workload of linear queries ,
on a database of size at most drawn from a universe
under the constraint of (approximate) differential privacy.
Nikolov, Talwar, and Zhang~\cite{NTZ} proposed an efficient mechanism that, for
any given and , answers the queries with average error that is
at most a factor polynomial in and
worse than the best possible. Here we improve on this guarantee and give a
mechanism whose competitiveness ratio is at most polynomial in and
, and has no dependence on . Our mechanism
is based on the projection mechanism of Nikolov, Talwar, and Zhang, but in
place of an ad-hoc noise distribution, we use a distribution which is in a
sense optimal for the projection mechanism, and analyze it using convex duality
and the restricted invertibility principle.Comment: To appear in ICALP 2015, Track
Optimal Single-Choice Prophet Inequalities from Samples
We study the single-choice Prophet Inequality problem when the gambler is
given access to samples. We show that the optimal competitive ratio of
can be achieved with a single sample from each distribution. When the
distributions are identical, we show that for any constant ,
samples from the distribution suffice to achieve the optimal competitive
ratio () within , resolving an open problem of
Correa, D\"utting, Fischer, and Schewior.Comment: Appears in Innovations in Theoretical Computer Science (ITCS) 202
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