6,324 research outputs found
Boolean Operations, Joins, and the Extended Low Hierarchy
We prove that the join of two sets may actually fall into a lower level of
the extended low hierarchy than either of the sets. In particular, there exist
sets that are not in the second level of the extended low hierarchy, EL_2, yet
their join is in EL_2. That is, in terms of extended lowness, the join operator
can lower complexity. Since in a strong intuitive sense the join does not lower
complexity, our result suggests that the extended low hierarchy is unnatural as
a complexity measure. We also study the closure properties of EL_ and prove
that EL_2 is not closed under certain Boolean operations. To this end, we
establish the first known (and optimal) EL_2 lower bounds for certain notions
generalizing Selman's P-selectivity, which may be regarded as an interesting
result in its own right.Comment: 12 page
Lower Bounds on Query Complexity for Testing Bounded-Degree CSPs
In this paper, we consider lower bounds on the query complexity for testing
CSPs in the bounded-degree model.
First, for any ``symmetric'' predicate except \equ
where , we show that every (randomized) algorithm that distinguishes
satisfiable instances of CSP(P) from instances -far
from satisfiability requires queries where is the
number of variables and is a constant that depends on and
. This breaks a natural lower bound , which is
obtained by the birthday paradox. We also show that every one-sided error
tester requires queries for such . These results are hereditary
in the sense that the same results hold for any predicate such that
. For EQU, we give a one-sided error tester
whose query complexity is . Also, for 2-XOR (or,
equivalently E2LIN2), we show an lower bound for
distinguishing instances between -close to and -far
from satisfiability.
Next, for the general k-CSP over the binary domain, we show that every
algorithm that distinguishes satisfiable instances from instances
-far from satisfiability requires queries. The
matching NP-hardness is not known, even assuming the Unique Games Conjecture or
the -to- Conjecture. As a corollary, for Maximum Independent Set on
graphs with vertices and a degree bound , we show that every
approximation algorithm within a factor d/\poly\log d and an additive error
of requires queries. Previously, only super-constant
lower bounds were known
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
Exponential Lower Bounds for Polytopes in Combinatorial Optimization
We solve a 20-year old problem posed by Yannakakis and prove that there
exists no polynomial-size linear program (LP) whose associated polytope
projects to the traveling salesman polytope, even if the LP is not required to
be symmetric. Moreover, we prove that this holds also for the cut polytope and
the stable set polytope. These results were discovered through a new connection
that we make between one-way quantum communication protocols and semidefinite
programming reformulations of LPs.Comment: 19 pages, 4 figures. This version of the paper will appear in the
Journal of the ACM. The earlier conference version in STOC'12 had the title
"Linear vs. Semidefinite Extended Formulations: Exponential Separation and
Strong Lower Bounds
Dura
The reactive event processing language, that is developed in the context of this project, has been called DEAL in previous documents. When we chose this name for our language it has not been used by other authors working in the same research area (complex event processing). However, in the meantime it appears in publications of other authors and because we have not used the name in publications yet we cannot claim that we were the first to use it. In order to avoid ambiguities and name conflicts in future publications we decided to rename our language to Dura which stands for “Declarative uniform reactive event processing language”. Therefore the title of this deliverable has been updated to “Dura – Concepts and Examples”
A real-time proximity querying algorithm for haptic-based molecular docking
Intermolecular binding underlies every metabolic and regulatory processes of the cell, and the therapeutic and pharmacological properties of drugs. Molecular docking systems model and simulate these interactions in silico and allow us to study the binding process. Haptic-based docking provides an immersive virtual docking environment where the user can interact with and guide the molecules to their binding pose. Moreover, it allows human perception, intuition and knowledge to assist and accelerate the docking process, and reduces incorrect binding poses. Crucial for interactive docking is the real-time calculation of interaction forces. For smooth and accurate haptic exploration and manipulation, force-feedback cues have to be updated at a rate of 1 kHz. Hence, force calculations must be performed within 1ms. To achieve this, modern haptic-based docking approaches often utilize pre-computed force grids and linear interpolation. However, such grids are time-consuming to pre-compute (especially for large molecules), memory hungry, can induce rough force transitions at cell boundaries and cannot be applied to flexible docking. Here we propose an efficient proximity querying method for computing intermolecular forces in real time. Our motivation is the eventual development of a haptic-based docking solution that can model molecular flexibility. Uniquely in a haptics application we use octrees to decompose the 3D search space in order to identify the set of interacting atoms within a cut-off distance. Force calculations are then performed on this set in real time. The implementation constructs the trees dynamically, and computes the interaction forces of large molecular structures (i.e. consisting of thousands of atoms) within haptic refresh rates. We have implemented this method in an immersive, haptic-based, rigid-body, molecular docking application called Haptimol_RD. The user can use the haptic device to orientate the molecules in space, sense the interaction forces on the device, and guide the molecules to their binding pose. Haptimol_RD is designed to run on consumer level hardware, i.e. there is no need for specialized/proprietary hardware
Neural Distributed Autoassociative Memories: A Survey
Introduction. Neural network models of autoassociative, distributed memory
allow storage and retrieval of many items (vectors) where the number of stored
items can exceed the vector dimension (the number of neurons in the network).
This opens the possibility of a sublinear time search (in the number of stored
items) for approximate nearest neighbors among vectors of high dimension. The
purpose of this paper is to review models of autoassociative, distributed
memory that can be naturally implemented by neural networks (mainly with local
learning rules and iterative dynamics based on information locally available to
neurons). Scope. The survey is focused mainly on the networks of Hopfield,
Willshaw and Potts, that have connections between pairs of neurons and operate
on sparse binary vectors. We discuss not only autoassociative memory, but also
the generalization properties of these networks. We also consider neural
networks with higher-order connections and networks with a bipartite graph
structure for non-binary data with linear constraints. Conclusions. In
conclusion we discuss the relations to similarity search, advantages and
drawbacks of these techniques, and topics for further research. An interesting
and still not completely resolved question is whether neural autoassociative
memories can search for approximate nearest neighbors faster than other index
structures for similarity search, in particular for the case of very high
dimensional vectors.Comment: 31 page
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