1,037 research outputs found
Online Maximum k-Coverage
We study an online model for the maximum k-vertex-coverage problem, where given a graph G = (V,E) and an integer k, we ask for a subset A ⊆ V, such that |A | = k and the number of edges covered by A is maximized. In our model, at each step i, a new vertex vi is revealed, and we have to decide whether we will keep it or discard it. At any time of the process, only k vertices can be kept in memory; if at some point the current solution already contains k vertices, any inclusion of any new vertex in the solution must entail the irremediable deletion of one vertex of the current solution (a vertex not kept when revealed is irremediably deleted). We propose algorithms for several natural classes of graphs (mainly regular and bipartite), improving on an easy 1/2-competitive ratio. We next settle a set-version of the problem, called maximum k-(set)-coverage problem. For this problem we present an algorithm that improves upon former results for the same model for small and moderate values of k
From average case complexity to improper learning complexity
The basic problem in the PAC model of computational learning theory is to
determine which hypothesis classes are efficiently learnable. There is
presently a dearth of results showing hardness of learning problems. Moreover,
the existing lower bounds fall short of the best known algorithms.
The biggest challenge in proving complexity results is to establish hardness
of {\em improper learning} (a.k.a. representation independent learning).The
difficulty in proving lower bounds for improper learning is that the standard
reductions from -hard problems do not seem to apply in this
context. There is essentially only one known approach to proving lower bounds
on improper learning. It was initiated in (Kearns and Valiant 89) and relies on
cryptographic assumptions.
We introduce a new technique for proving hardness of improper learning, based
on reductions from problems that are hard on average. We put forward a (fairly
strong) generalization of Feige's assumption (Feige 02) about the complexity of
refuting random constraint satisfaction problems. Combining this assumption
with our new technique yields far reaching implications. In particular,
1. Learning 's is hard.
2. Agnostically learning halfspaces with a constant approximation ratio is
hard.
3. Learning an intersection of halfspaces is hard.Comment: 34 page
Computational Difficulty of Global Variations in the Density Matrix Renormalization Group
The density matrix renormalization group (DMRG) approach is arguably the most
successful method to numerically find ground states of quantum spin chains. It
amounts to iteratively locally optimizing matrix-product states, aiming at
better and better approximating the true ground state. To date, both a proof of
convergence to the globally best approximation and an assessment of its
complexity are lacking. Here we establish a result on the computational
complexity of an approximation with matrix-product states: The surprising
result is that when one globally optimizes over several sites of local
Hamiltonians, avoiding local optima, one encounters in the worst case a
computationally difficult NP-hard problem (hard even in approximation). The
proof exploits a novel way of relating it to binary quadratic programming. We
discuss intriguing ramifications on the difficulty of describing quantum
many-body systems.Comment: 5 pages, 1 figure, RevTeX, final versio
Finding Connected Dense -Subgraphs
Given a connected graph on vertices and a positive integer ,
a subgraph of on vertices is called a -subgraph in . We design
combinatorial approximation algorithms for finding a connected -subgraph in
such that its density is at least a factor
of the density of the densest -subgraph
in (which is not necessarily connected). These particularly provide the
first non-trivial approximations for the densest connected -subgraph problem
on general graphs
On Fast and Robust Information Spreading in the Vertex-Congest Model
This paper initiates the study of the impact of failures on the fundamental
problem of \emph{information spreading} in the Vertex-Congest model, in which
in every round, each of the nodes sends the same -bit message
to all of its neighbors.
Our contribution to coping with failures is twofold. First, we prove that the
randomized algorithm which chooses uniformly at random the next message to
forward is slow, requiring rounds on some graphs, which we
denote by , where is the vertex-connectivity.
Second, we design a randomized algorithm that makes dynamic message choices,
with probabilities that change over the execution. We prove that for
it requires only a near-optimal number of rounds, despite a
rate of failures per round. Our technique of choosing
probabilities that change according to the execution is of independent
interest.Comment: Appears in SIROCCO 2015 conferenc
Suggestive Annotation: A Deep Active Learning Framework for Biomedical Image Segmentation
Image segmentation is a fundamental problem in biomedical image analysis.
Recent advances in deep learning have achieved promising results on many
biomedical image segmentation benchmarks. However, due to large variations in
biomedical images (different modalities, image settings, objects, noise, etc),
to utilize deep learning on a new application, it usually needs a new set of
training data. This can incur a great deal of annotation effort and cost,
because only biomedical experts can annotate effectively, and often there are
too many instances in images (e.g., cells) to annotate. In this paper, we aim
to address the following question: With limited effort (e.g., time) for
annotation, what instances should be annotated in order to attain the best
performance? We present a deep active learning framework that combines fully
convolutional network (FCN) and active learning to significantly reduce
annotation effort by making judicious suggestions on the most effective
annotation areas. We utilize uncertainty and similarity information provided by
FCN and formulate a generalized version of the maximum set cover problem to
determine the most representative and uncertain areas for annotation. Extensive
experiments using the 2015 MICCAI Gland Challenge dataset and a lymph node
ultrasound image segmentation dataset show that, using annotation suggestions
by our method, state-of-the-art segmentation performance can be achieved by
using only 50% of training data.Comment: Accepted at MICCAI 201
Quantum Interactive Proofs with Competing Provers
This paper studies quantum refereed games, which are quantum interactive
proof systems with two competing provers: one that tries to convince the
verifier to accept and the other that tries to convince the verifier to reject.
We prove that every language having an ordinary quantum interactive proof
system also has a quantum refereed game in which the verifier exchanges just
one round of messages with each prover. A key part of our proof is the fact
that there exists a single quantum measurement that reliably distinguishes
between mixed states chosen arbitrarily from disjoint convex sets having large
minimal trace distance from one another. We also show how to reduce the
probability of error for some classes of quantum refereed games.Comment: 13 pages, to appear in STACS 200
Sampling and Representation Complexity of Revenue Maximization
We consider (approximate) revenue maximization in auctions where the
distribution on input valuations is given via "black box" access to samples
from the distribution. We observe that the number of samples required -- the
sample complexity -- is tightly related to the representation complexity of an
approximately revenue-maximizing auction. Our main results are upper bounds and
an exponential lower bound on these complexities
Parallel Repetition of Entangled Games with Exponential Decay via the Superposed Information Cost
In a two-player game, two cooperating but non communicating players, Alice
and Bob, receive inputs taken from a probability distribution. Each of them
produces an output and they win the game if they satisfy some predicate on
their inputs/outputs. The entangled value of a game is the
maximum probability that Alice and Bob can win the game if they are allowed to
share an entangled state prior to receiving their inputs.
The -fold parallel repetition of consists of instances of
where the players receive all the inputs at the same time and produce all
the outputs at the same time. They win if they win each instance of .
In this paper we show that for any game such that , decreases exponentially in . First, for
any game on the uniform distribution, we show that , where and are the sizes of the input
and output sets. From this result, we show that for any entangled game ,
where is the input distribution of and
. This implies parallel
repetition with exponential decay as long as for
general games. To prove this parallel repetition, we introduce the concept of
\emph{Superposed Information Cost} for entangled games which is inspired from
the information cost used in communication complexity.Comment: In the first version of this paper we presented a different, stronger
Corollary 1 but due to an error in the proof we had to modify it in the
second version. This third version is a minor update. We correct some typos
and re-introduce a proof accidentally commented out in the second versio
Algorithmic and Hardness Results for the Colorful Components Problems
In this paper we investigate the colorful components framework, motivated by
applications emerging from comparative genomics. The general goal is to remove
a collection of edges from an undirected vertex-colored graph such that in
the resulting graph all the connected components are colorful (i.e., any
two vertices of the same color belong to different connected components). We
want to optimize an objective function, the selection of this function
being specific to each problem in the framework.
We analyze three objective functions, and thus, three different problems,
which are believed to be relevant for the biological applications: minimizing
the number of singleton vertices, maximizing the number of edges in the
transitive closure, and minimizing the number of connected components.
Our main result is a polynomial time algorithm for the first problem. This
result disproves the conjecture of Zheng et al. that the problem is -hard
(assuming ). Then, we show that the second problem is -hard,
thus proving and strengthening the conjecture of Zheng et al. that the problem
is -hard. Finally, we show that the third problem does not admit
polynomial time approximation within a factor of for
any , assuming (or within a factor of , assuming ).Comment: 18 pages, 3 figure
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