10,613 research outputs found
Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy
We consider quantum computations comprising only commuting gates, known as
IQP computations, and provide compelling evidence that the task of sampling
their output probability distributions is unlikely to be achievable by any
efficient classical means. More specifically we introduce the class post-IQP of
languages decided with bounded error by uniform families of IQP circuits with
post-selection, and prove first that post-IQP equals the classical class PP.
Using this result we show that if the output distributions of uniform IQP
circuit families could be classically efficiently sampled, even up to 41%
multiplicative error in the probabilities, then the infinite tower of classical
complexity classes known as the polynomial hierarchy, would collapse to its
third level. We mention some further results on the classical simulation
properties of IQP circuit families, in particular showing that if the output
distribution results from measurements on only O(log n) lines then it may in
fact be classically efficiently sampled.Comment: 13 page
Structure-Aware Sampling: Flexible and Accurate Summarization
In processing large quantities of data, a fundamental problem is to obtain a
summary which supports approximate query answering. Random sampling yields
flexible summaries which naturally support subset-sum queries with unbiased
estimators and well-understood confidence bounds.
Classic sample-based summaries, however, are designed for arbitrary subset
queries and are oblivious to the structure in the set of keys. The particular
structure, such as hierarchy, order, or product space (multi-dimensional),
makes range queries much more relevant for most analysis of the data.
Dedicated summarization algorithms for range-sum queries have also been
extensively studied. They can outperform existing sampling schemes in terms of
accuracy on range queries per summary size. Their accuracy, however, rapidly
degrades when, as is often the case, the query spans multiple ranges. They are
also less flexible - being targeted for range sum queries alone - and are often
quite costly to build and use.
In this paper we propose and evaluate variance optimal sampling schemes that
are structure-aware. These summaries improve over the accuracy of existing
structure-oblivious sampling schemes on range queries while retaining the
benefits of sample-based summaries: flexible summaries, with high accuracy on
both range queries and arbitrary subset queries
Sherali-Adams gaps, flow-cover inequalities and generalized configurations for capacity-constrained Facility Location
Metric facility location is a well-studied problem for which linear
programming methods have been used with great success in deriving approximation
algorithms. The capacity-constrained generalizations, such as capacitated
facility location (CFL) and lower-bounded facility location (LBFL), have proved
notorious as far as LP-based approximation is concerned: while there are
local-search-based constant-factor approximations, there is no known linear
relaxation with constant integrality gap. According to Williamson and Shmoys
devising a relaxation-based approximation for \cfl\ is among the top 10 open
problems in approximation algorithms.
This paper advances significantly the state-of-the-art on the effectiveness
of linear programming for capacity-constrained facility location through a host
of impossibility results for both CFL and LBFL. We show that the relaxations
obtained from the natural LP at levels of the Sherali-Adams
hierarchy have an unbounded gap, partially answering an open question of
\cite{LiS13, AnBS13}. Here, denotes the number of facilities in the
instance. Building on the ideas for this result, we prove that the standard CFL
relaxation enriched with the generalized flow-cover valid inequalities
\cite{AardalPW95} has also an unbounded gap. This disproves a long-standing
conjecture of \cite{LeviSS12}. We finally introduce the family of proper
relaxations which generalizes to its logical extreme the classic star
relaxation and captures general configuration-style LPs. We characterize the
behavior of proper relaxations for CFL and LBFL through a sharp threshold
phenomenon.Comment: arXiv admin note: substantial text overlap with arXiv:1305.599
Probabilistic Computability and Choice
We study the computational power of randomized computations on infinite
objects, such as real numbers. In particular, we introduce the concept of a Las
Vegas computable multi-valued function, which is a function that can be
computed on a probabilistic Turing machine that receives a random binary
sequence as auxiliary input. The machine can take advantage of this random
sequence, but it always has to produce a correct result or to stop the
computation after finite time if the random advice is not successful. With
positive probability the random advice has to be successful. We characterize
the class of Las Vegas computable functions in the Weihrauch lattice with the
help of probabilistic choice principles and Weak Weak K\H{o}nig's Lemma. Among
other things we prove an Independent Choice Theorem that implies that Las Vegas
computable functions are closed under composition. In a case study we show that
Nash equilibria are Las Vegas computable, while zeros of continuous functions
with sign changes cannot be computed on Las Vegas machines. However, we show
that the latter problem admits randomized algorithms with weaker failure
recognition mechanisms. The last mentioned results can be interpreted such that
the Intermediate Value Theorem is reducible to the jump of Weak Weak
K\H{o}nig's Lemma, but not to Weak Weak K\H{o}nig's Lemma itself. These
examples also demonstrate that Las Vegas computable functions form a proper
superclass of the class of computable functions and a proper subclass of the
class of non-deterministically computable functions. We also study the impact
of specific lower bounds on the success probabilities, which leads to a strict
hierarchy of classes. In particular, the classical technique of probability
amplification fails for computations on infinite objects. We also investigate
the dependency on the underlying probability space.Comment: Information and Computation (accepted for publication
A Casual Tour Around a Circuit Complexity Bound
I will discuss the recent proof that the complexity class NEXP
(nondeterministic exponential time) lacks nonuniform ACC circuits of polynomial
size. The proof will be described from the perspective of someone trying to
discover it.Comment: 21 pages, 2 figures. An earlier version appeared in SIGACT News,
September 201
Black Box White Arrow
The present paper proposes a new and systematic approach to the so-called
black box group methods in computational group theory. Instead of a single
black box, we consider categories of black boxes and their morphisms. This
makes new classes of black box problems accessible. For example, we can enrich
black box groups by actions of outer automorphisms.
As an example of application of this technique, we construct Frobenius maps
on black box groups of untwisted Lie type in odd characteristic (Section 6) and
inverse-transpose automorphisms on black box groups encrypting .
One of the advantages of our approach is that it allows us to work in black
box groups over finite fields of big characteristic. Another advantage is
explanatory power of our methods; as an example, we explain Kantor's and
Kassabov's construction of an involution in black box groups encrypting .
Due to the nature of our work we also have to discuss a few methodological
issues of the black box group theory.
The paper is further development of our text "Fifty shades of black"
[arXiv:1308.2487], and repeats parts of it, but under a weaker axioms for black
box groups.Comment: arXiv admin note: substantial text overlap with arXiv:1308.248
Pseudorandomness for Approximate Counting and Sampling
We study computational procedures that use both randomness and nondeterminism. The goal of this paper is to derandomize such procedures under the weakest possible assumptions.
Our main technical contribution allows one to âboostâ a given hardness assumption: We show that if there is a problem in EXP that cannot be computed by poly-size nondeterministic circuits then there is one which cannot be computed by poly-size circuits that make non-adaptive NP oracle queries. This in particular shows that the various assumptions used over the last few years by several authors to derandomize Arthur-Merlin games (i.e., show AM = NP) are in fact all equivalent.
We also define two new primitives that we regard as the natural pseudorandom objects associated with approximate counting and sampling of NP-witnesses. We use the âboostingâ theorem and hashing techniques to construct these primitives using an assumption that is no stronger than that used to derandomize AM.
We observe that Cai's proof that S_2^P â PPâ(NP) and the learning algorithm of Bshouty et al. can be seen as reductions to sampling that are not probabilistic. As a consequence they can be derandomized under an assumption which is weaker than the assumption that was previously known to suffice
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