934 research outputs found
Enumeration of the Monomials of a Polynomial and Related Complexity Classes
We study the problem of generating monomials of a polynomial in the context
of enumeration complexity. In this setting, the complexity measure is the delay
between two solutions and the total time. We present two new algorithms for
restricted classes of polynomials, which have a good delay and the same global
running time as the classical ones. Moreover they are simple to describe, use
little evaluation points and one of them is parallelizable. We introduce three
new complexity classes, TotalPP, IncPP and DelayPP, which are probabilistic
counterparts of the most common classes for enumeration problems, hoping that
randomization will be a tool as strong for enumeration as it is for decision.
Our interpolation algorithms proves that a lot of interesting problems are in
these classes like the enumeration of the spanning hypertrees of a 3-uniform
hypergraph.
Finally we give a method to interpolate a degree 2 polynomials with an
acceptable (incremental) delay. We also prove that finding a specified monomial
in a degree 2 polynomial is hard unless RP = NP. It suggests that there is no
algorithm with a delay as good (polynomial) as the one we achieve for
multilinear polynomials
A Dichotomy Theorem for Homomorphism Polynomials
In the present paper we show a dichotomy theorem for the complexity of
polynomial evaluation. We associate to each graph H a polynomial that encodes
all graphs of a fixed size homomorphic to H. We show that this family is
computable by arithmetic circuits in constant depth if H has a loop or no edge
and that it is hard otherwise (i.e., complete for VNP, the arithmetic class
related to #P). We also demonstrate the hardness over the rational field of cut
eliminator, a polynomial defined by B\"urgisser which is known to be neither VP
nor VNP-complete in the field of two elements, if VP is not equal to VNP (VP is
the class of polynomials computable by arithmetic circuit of polynomial size)
Calabi-Yau threefolds with large h^{2, 1}
We carry out a systematic analysis of Calabi-Yau threefolds that are
elliptically fibered with section ("EFS") and have a large Hodge number h^{2,
1}. EFS Calabi-Yau threefolds live in a single connected space, with regions of
moduli space associated with different topologies connected through transitions
that can be understood in terms of singular Weierstrass models. We determine
the complete set of such threefolds that have h^{2, 1} >= 350 by tuning
coefficients in Weierstrass models over Hirzebruch surfaces. The resulting set
of Hodge numbers includes those of all known Calabi-Yau threefolds with h^{2,
1} >= 350, as well as three apparently new Calabi-Yau threefolds. We speculate
that there are no other Calabi-Yau threefolds (elliptically fibered or not)
with Hodge numbers that exceed this bound. We summarize the theoretical and
practical obstacles to a complete enumeration of all possible EFS Calabi-Yau
threefolds and fourfolds, including those with small Hodge numbers, using this
approach.Comment: 44 pages, 5 tables, 5 figures; v2: minor corrections; v3: minor
corrections, moved figure; v4: typo in Table 2 correcte
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
- …