17,450 research outputs found
The Minimum Expectation Selection Problem
We define the min-min expectation selection problem (resp. max-min
expectation selection problem) to be that of selecting k out of n given
discrete probability distributions, to minimize (resp. maximize) the
expectation of the minimum value resulting when independent random variables
are drawn from the selected distributions. We assume each distribution has
finitely many atoms. Let d be the number of distinct values in the support of
the distributions. We show that if d is a constant greater than 2, the min-min
expectation problem is NP-complete but admits a fully polynomial time
approximation scheme. For d an arbitrary integer, it is NP-hard to approximate
the min-min expectation problem with any constant approximation factor. The
max-min expectation problem is polynomially solvable for constant d; we leave
open its complexity for variable d. We also show similar results for binary
selection problems in which we must choose one distribution from each of n
pairs of distributions.Comment: 13 pages, 1 figure. Full version of paper presented at 10th Int.
Conf. Random Structures and Algorithms, Poznan, Poland, August 200
A quantum-inspired algorithm for estimating the permanent of positive semidefinite matrices
We construct a quantum-inspired classical algorithm for computing the
permanent of Hermitian positive semidefinite matrices, by exploiting a
connection between these mathematical structures and the boson sampling model.
Specifically, the permanent of a Hermitian positive semidefinite matrix can be
expressed in terms of the expected value of a random variable, which stands for
a specific photon-counting probability when measuring a linear-optically
evolved random multimode coherent state. Our algorithm then approximates the
matrix permanent from the corresponding sample mean and is shown to run in
polynomial time for various sets of Hermitian positive semidefinite matrices,
achieving a precision that improves over known techniques. This work
illustrates how quantum optics may benefit algorithms development.Comment: 9 pages, 1 figure. Updated version for publicatio
New Results for the MAP Problem in Bayesian Networks
This paper presents new results for the (partial) maximum a posteriori (MAP)
problem in Bayesian networks, which is the problem of querying the most
probable state configuration of some of the network variables given evidence.
First, it is demonstrated that the problem remains hard even in networks with
very simple topology, such as binary polytrees and simple trees (including the
Naive Bayes structure). Such proofs extend previous complexity results for the
problem. Inapproximability results are also derived in the case of trees if the
number of states per variable is not bounded. Although the problem is shown to
be hard and inapproximable even in very simple scenarios, a new exact algorithm
is described that is empirically fast in networks of bounded treewidth and
bounded number of states per variable. The same algorithm is used as basis of a
Fully Polynomial Time Approximation Scheme for MAP under such assumptions.
Approximation schemes were generally thought to be impossible for this problem,
but we show otherwise for classes of networks that are important in practice.
The algorithms are extensively tested using some well-known networks as well as
random generated cases to show their effectiveness.Comment: A couple of typos were fixed, as well as the notation in part of
section 4, which was misleading. Theoretical and empirical results have not
change
A Birthday Repetition Theorem and Complexity of Approximating Dense CSPs
A -birthday repetition of a
two-prover game is a game in which the two provers are sent
random sets of questions from of sizes and respectively.
These two sets are sampled independently uniformly among all sets of questions
of those particular sizes. We prove the following birthday repetition theorem:
when satisfies some mild conditions, decreases exponentially in where is the total number of
questions. Our result positively resolves an open question posted by Aaronson,
Impagliazzo and Moshkovitz (CCC 2014).
As an application of our birthday repetition theorem, we obtain new
fine-grained hardness of approximation results for dense CSPs. Specifically, we
establish a tight trade-off between running time and approximation ratio for
dense CSPs by showing conditional lower bounds, integrality gaps and
approximation algorithms. In particular, for any sufficiently large and for
every , we show the following results:
- We exhibit an -approximation algorithm for dense Max -CSPs
with alphabet size via -level of Sherali-Adams relaxation.
- Through our birthday repetition theorem, we obtain an integrality gap of
for -level Lasserre relaxation for fully-dense Max
-CSP.
- Assuming that there is a constant such that Max 3SAT cannot
be approximated to within of the optimal in sub-exponential
time, our birthday repetition theorem implies that any algorithm that
approximates fully-dense Max -CSP to within a factor takes
time, almost tightly matching the algorithmic
result based on Sherali-Adams relaxation.Comment: 45 page
The matching polytope does not admit fully-polynomial size relaxation schemes
The groundbreaking work of Rothvo{\ss} [arxiv:1311.2369] established that
every linear program expressing the matching polytope has an exponential number
of inequalities (formally, the matching polytope has exponential extension
complexity). We generalize this result by deriving strong bounds on the
polyhedral inapproximability of the matching polytope: for fixed , every polyhedral -approximation
requires an exponential number of inequalities, where is the number of
vertices. This is sharp given the well-known -approximation of size
provided by the odd-sets of size up to
. Thus matching is the first problem in , whose natural
linear encoding does not admit a fully polynomial-size relaxation scheme (the
polyhedral equivalent of an FPTAS), which provides a sharp separation from the
polynomial-size relaxation scheme obtained e.g., via constant-sized odd-sets
mentioned above.
Our approach reuses ideas from Rothvo{\ss} [arxiv:1311.2369], however the
main lower bounding technique is different. While the original proof is based
on the hyperplane separation bound (also called the rectangle corruption
bound), we employ the information-theoretic notion of common information as
introduced in Braun and Pokutta [http://eccc.hpi-web.de/report/2013/056/],
which allows to analyze perturbations of slack matrices. It turns out that the
high extension complexity for the matching polytope stem from the same source
of hardness as for the correlation polytope: a direct sum structure.Comment: 21 pages, 3 figure
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