59,832 research outputs found
On the extension complexity of combinatorial polytopes
In this paper we extend recent results of Fiorini et al. on the extension
complexity of the cut polytope and related polyhedra. We first describe a
lifting argument to show exponential extension complexity for a number of
NP-complete problems including subset-sum and three dimensional matching. We
then obtain a relationship between the extension complexity of the cut polytope
of a graph and that of its graph minors. Using this we are able to show
exponential extension complexity for the cut polytope of a large number of
graphs, including those used in quantum information and suspensions of cubic
planar graphs.Comment: 15 pages, 3 figures, 2 table
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
JigsawNet: Shredded Image Reassembly using Convolutional Neural Network and Loop-based Composition
This paper proposes a novel algorithm to reassemble an arbitrarily shredded
image to its original status. Existing reassembly pipelines commonly consist of
a local matching stage and a global compositions stage. In the local stage, a
key challenge in fragment reassembly is to reliably compute and identify
correct pairwise matching, for which most existing algorithms use handcrafted
features, and hence, cannot reliably handle complicated puzzles. We build a
deep convolutional neural network to detect the compatibility of a pairwise
stitching, and use it to prune computed pairwise matches. To improve the
network efficiency and accuracy, we transfer the calculation of CNN to the
stitching region and apply a boost training strategy. In the global composition
stage, we modify the commonly adopted greedy edge selection strategies to two
new loop closure based searching algorithms. Extensive experiments show that
our algorithm significantly outperforms existing methods on solving various
puzzles, especially those challenging ones with many fragment pieces
Space Complexity of Perfect Matching in Bounded Genus Bipartite Graphs
We investigate the space complexity of certain perfect matching problems over
bipartite graphs embedded on surfaces of constant genus (orientable or
non-orientable). We show that the problems of deciding whether such graphs have
(1) a perfect matching or not and (2) a unique perfect matching or not, are in
the logspace complexity class \SPL. Since \SPL\ is contained in the logspace
counting classes \oplus\L (in fact in \modk\ for all ), \CeqL, and
\PL, our upper bound places the above-mentioned matching problems in these
counting classes as well. We also show that the search version, computing a
perfect matching, for this class of graphs is in \FL^{\SPL}. Our results
extend the same upper bounds for these problems over bipartite planar graphs
known earlier. As our main technical result, we design a logspace computable
and polynomially bounded weight function which isolates a minimum weight
perfect matching in bipartite graphs embedded on surfaces of constant genus. We
use results from algebraic topology for proving the correctness of the weight
function.Comment: 23 pages, 13 figure
Spectral Clustering with Imbalanced Data
Spectral clustering is sensitive to how graphs are constructed from data
particularly when proximal and imbalanced clusters are present. We show that
Ratio-Cut (RCut) or normalized cut (NCut) objectives are not tailored to
imbalanced data since they tend to emphasize cut sizes over cut values. We
propose a graph partitioning problem that seeks minimum cut partitions under
minimum size constraints on partitions to deal with imbalanced data. Our
approach parameterizes a family of graphs, by adaptively modulating node
degrees on a fixed node set, to yield a set of parameter dependent cuts
reflecting varying levels of imbalance. The solution to our problem is then
obtained by optimizing over these parameters. We present rigorous limit cut
analysis results to justify our approach. We demonstrate the superiority of our
method through unsupervised and semi-supervised experiments on synthetic and
real data sets.Comment: 24 pages, 7 figures. arXiv admin note: substantial text overlap with
arXiv:1302.513
Logic Programming and Logarithmic Space
We present an algebraic view on logic programming, related to proof theory
and more specifically linear logic and geometry of interaction. Within this
construction, a characterization of logspace (deterministic and
non-deterministic) computation is given via a synctactic restriction, using an
encoding of words that derives from proof theory.
We show that the acceptance of a word by an observation (the counterpart of a
program in the encoding) can be decided within logarithmic space, by reducing
this problem to the acyclicity of a graph. We show moreover that observations
are as expressive as two-ways multi-heads finite automata, a kind of pointer
machines that is a standard model of logarithmic space computation
Clustering and Community Detection with Imbalanced Clusters
Spectral clustering methods which are frequently used in clustering and
community detection applications are sensitive to the specific graph
constructions particularly when imbalanced clusters are present. We show that
ratio cut (RCut) or normalized cut (NCut) objectives are not tailored to
imbalanced cluster sizes since they tend to emphasize cut sizes over cut
values. We propose a graph partitioning problem that seeks minimum cut
partitions under minimum size constraints on partitions to deal with imbalanced
cluster sizes. Our approach parameterizes a family of graphs by adaptively
modulating node degrees on a fixed node set, yielding a set of parameter
dependent cuts reflecting varying levels of imbalance. The solution to our
problem is then obtained by optimizing over these parameters. We present
rigorous limit cut analysis results to justify our approach and demonstrate the
superiority of our method through experiments on synthetic and real datasets
for data clustering, semi-supervised learning and community detection.Comment: Extended version of arXiv:1309.2303 with new applications. Accepted
to IEEE TSIP
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