664 research outputs found
An Output-sensitive Algorithm for Computing Projections of Resultant Polytopes
We develop an incremental algorithm to compute the Newton polytope
of the resultant, aka resultant polytope, or its
projection along a given direction.
The resultant is fundamental in algebraic elimination and
in implicitization of parametric hypersurfaces.
Our algorithm exactly computes vertex- and halfspace-representations
of the desired polytope using an oracle producing resultant vertices in a
given direction.
It is output-sensitive as it uses one oracle call per vertex.
We overcome the bottleneck of determinantal predicates
by hashing, thus accelerating execution from to times.
We implement our algorithm using the experimental CGAL package {\tt
triangulation}.
A variant of the algorithm computes successively tighter inner and outer
approximations: when these polytopes have, respectively,
90\% and 105\% of the true volume, runtime is reduced up to times.
Our method computes instances of -, - or -dimensional polytopes
with K, K or vertices, resp., within hr.
Compared to tropical geometry software, ours is faster up to
dimension or , and competitive in higher dimensions
On nondegeneracy of curves
A curve is called nondegenerate if it can be modeled by a Laurent polynomial
that is nondegenerate with respect to its Newton polytope. We show that up to
genus 4, every curve is nondegenerate. We also prove that the locus of
nondegenerate curves inside the moduli space of curves of fixed genus g > 1 is
min(2g+1,3g-3)-dimensional, except in case g=7 where it is 16-dimensional
Recent progress on the combinatorial diameter of polytopes and simplicial complexes
The Hirsch conjecture, posed in 1957, stated that the graph of a
-dimensional polytope or polyhedron with facets cannot have diameter
greater than . The conjecture itself has been disproved, but what we
know about the underlying question is quite scarce. Most notably, no polynomial
upper bound is known for the diameters that were conjectured to be linear. In
contrast, no polyhedron violating the conjecture by more than 25% is known.
This paper reviews several recent attempts and progress on the question. Some
work in the world of polyhedra or (more often) bounded polytopes, but some try
to shed light on the question by generalizing it to simplicial complexes. In
particular, we include here our recent and previously unpublished proof that
the maximum diameter of arbitrary simplicial complexes is in and
we summarize the main ideas in the polymath 3 project, a web-based collective
effort trying to prove an upper bound of type nd for the diameters of polyhedra
and of more general objects (including, e. g., simplicial manifolds).Comment: 34 pages. This paper supersedes one cited as "On the maximum diameter
of simplicial complexes and abstractions of them, in preparation
Unwinding the Amplituhedron in Binary
We present new, fundamentally combinatorial and topological characterizations
of the amplituhedron. Upon projecting external data through the amplituhedron,
the resulting configuration of points has a specified (and maximal) generalized
'winding number'. Equivalently, the amplituhedron can be fully described in
binary: canonical projections of the geometry down to one dimension have a
specified (and maximal) number of 'sign flips' of the projected data. The
locality and unitarity of scattering amplitudes are easily derived as
elementary consequences of this binary code. Minimal winding defines a natural
'dual' of the amplituhedron. This picture gives us an avatar of the
amplituhedron purely in the configuration space of points in vector space
(momentum-twistor space in the physics), a new interpretation of the canonical
amplituhedron form, and a direct bosonic understanding of the scattering
super-amplitude in planar N = 4 SYM as a differential form on the space of
physical kinematical data.Comment: 42 pages, 13 figure
Efficient Algorithms for Privately Releasing Marginals via Convex Relaxations
Consider a database of people, each represented by a bit-string of length
corresponding to the setting of binary attributes. A -way marginal
query is specified by a subset of attributes, and a -dimensional
binary vector specifying their values. The result for this query is a
count of the number of people in the database whose attribute vector restricted
to agrees with .
Privately releasing approximate answers to a set of -way marginal queries
is one of the most important and well-motivated problems in differential
privacy. Information theoretically, the error complexity of marginal queries is
well-understood: the per-query additive error is known to be at least
and at most
. However, no polynomial
time algorithm with error complexity as low as the information theoretic upper
bound is known for small . In this work we present a polynomial time
algorithm that, for any distribution on marginal queries, achieves average
error at most . This error
bound is as good as the best known information theoretic upper bounds for
. This bound is an improvement over previous work on efficiently releasing
marginals when is small and when error is desirable. Using private
boosting we are also able to give nearly matching worst-case error bounds.
Our algorithms are based on the geometric techniques of Nikolov, Talwar, and
Zhang. The main new ingredients are convex relaxations and careful use of the
Frank-Wolfe algorithm for constrained convex minimization. To design our
relaxations, we rely on the Grothendieck inequality from functional analysis
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