8,414 research outputs found
Euclidean TSP with few inner points in linear space
Given a set of points in the Euclidean plane, such that just points
are strictly inside the convex hull of the whole set, we want to find the
shortest tour visiting every point. The fastest known algorithm for the version
when is significantly smaller than , i.e., when there are just few inner
points, works in time [Knauer and Spillner,
WG 2006], but also requires space of order . The best
linear space algorithm takes time [Deineko, Hoffmann, Okamoto,
Woeginer, Oper. Res. Lett. 34(1), 106-110]. We construct a linear space
time algorithm. The new insight is extending the
known divide-and-conquer method based on planar separators with a
matching-based argument to shrink the instance in every recursive call. This
argument also shows that the problem admits a quadratic bikernel.Comment: under submissio
Computing approximate PSD factorizations
We give an algorithm for computing approximate PSD factorizations of
nonnegative matrices. The running time of the algorithm is polynomial in the
dimensions of the input matrix, but exponential in the PSD rank and the
approximation error. The main ingredient is an exact factorization algorithm
when the rows and columns of the factors are constrained to lie in a general
polyhedron. This strictly generalizes nonnegative matrix factorizations which
can be captured by letting this polyhedron to be the nonnegative orthant.Comment: 10 page
Learning with Clustering Structure
We study supervised learning problems using clustering constraints to impose
structure on either features or samples, seeking to help both prediction and
interpretation. The problem of clustering features arises naturally in text
classification for instance, to reduce dimensionality by grouping words
together and identify synonyms. The sample clustering problem on the other
hand, applies to multiclass problems where we are allowed to make multiple
predictions and the performance of the best answer is recorded. We derive a
unified optimization formulation highlighting the common structure of these
problems and produce algorithms whose core iteration complexity amounts to a
k-means clustering step, which can be approximated efficiently. We extend these
results to combine sparsity and clustering constraints, and develop a new
projection algorithm on the set of clustered sparse vectors. We prove
convergence of our algorithms on random instances, based on a union of
subspaces interpretation of the clustering structure. Finally, we test the
robustness of our methods on artificial data sets as well as real data
extracted from movie reviews.Comment: Completely rewritten. New convergence proofs in the clustered and
sparse clustered case. New projection algorithm on sparse clustered vector
On k-Convex Polygons
We introduce a notion of -convexity and explore polygons in the plane that
have this property. Polygons which are \mbox{-convex} can be triangulated
with fast yet simple algorithms. However, recognizing them in general is a
3SUM-hard problem. We give a characterization of \mbox{-convex} polygons, a
particularly interesting class, and show how to recognize them in \mbox{} time. A description of their shape is given as well, which leads to
Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex
sets. Finally, we introduce the concept of generalized geometric permutations,
and show that their number can be exponential in the number of
\mbox{-convex} objects considered.Comment: 23 pages, 19 figure
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