9,523 research outputs found
Random Sampling in Computational Algebra: Helly Numbers and Violator Spaces
This paper transfers a randomized algorithm, originally used in geometric
optimization, to computational problems in commutative algebra. We show that
Clarkson's sampling algorithm can be applied to two problems in computational
algebra: solving large-scale polynomial systems and finding small generating
sets of graded ideals. The cornerstone of our work is showing that the theory
of violator spaces of G\"artner et al.\ applies to polynomial ideal problems.
To show this, one utilizes a Helly-type result for algebraic varieties. The
resulting algorithms have expected runtime linear in the number of input
polynomials, making the ideas interesting for handling systems with very large
numbers of polynomials, but whose rank in the vector space of polynomials is
small (e.g., when the number of variables and degree is constant).Comment: Minor edits, added two references; results unchange
Understanding the Complexity of Lifted Inference and Asymmetric Weighted Model Counting
In this paper we study lifted inference for the Weighted First-Order Model
Counting problem (WFOMC), which counts the assignments that satisfy a given
sentence in first-order logic (FOL); it has applications in Statistical
Relational Learning (SRL) and Probabilistic Databases (PDB). We present several
results. First, we describe a lifted inference algorithm that generalizes prior
approaches in SRL and PDB. Second, we provide a novel dichotomy result for a
non-trivial fragment of FO CNF sentences, showing that for each sentence the
WFOMC problem is either in PTIME or #P-hard in the size of the input domain; we
prove that, in the first case our algorithm solves the WFOMC problem in PTIME,
and in the second case it fails. Third, we present several properties of the
algorithm. Finally, we discuss limitations of lifted inference for symmetric
probabilistic databases (where the weights of ground literals depend only on
the relation name, and not on the constants of the domain), and prove the
impossibility of a dichotomy result for the complexity of probabilistic
inference for the entire language FOL
An Inflationary Fixed Point Operator in XQuery
We introduce a controlled form of recursion in XQuery, inflationary fixed
points, familiar in the context of relational databases. This imposes
restrictions on the expressible types of recursion, but we show that
inflationary fixed points nevertheless are sufficiently versatile to capture a
wide range of interesting use cases, including the semantics of Regular XPath
and its core transitive closure construct.
While the optimization of general user-defined recursive functions in XQuery
appears elusive, we will describe how inflationary fixed points can be
efficiently evaluated, provided that the recursive XQuery expressions exhibit a
distributivity property. We show how distributivity can be assessed both,
syntactically and algebraically, and provide experimental evidence that XQuery
processors can substantially benefit during inflationary fixed point
evaluation.Comment: 11 pages, 10 figures, 2 table
Algorithmic Algebraic Geometry and Flux Vacua
We develop a new and efficient method to systematically analyse four
dimensional effective supergravities which descend from flux compactifications.
The issue of finding vacua of such systems, both supersymmetric and
non-supersymmetric, is mapped into a problem in computational algebraic
geometry. Using recent developments in computer algebra, the problem can then
be rapidly dealt with in a completely algorithmic fashion. Two main results are
(1) a procedure for calculating constraints which the flux parameters must
satisfy in these models if any given type of vacuum is to exist; (2) a stepwise
process for finding all of the isolated vacua of such systems and their
physical properties. We illustrate our discussion with several concrete
examples, some of which have eluded conventional methods so far.Comment: 41 pages, 4 figure
Fast Witness Extraction Using a Decision Oracle
The gist of many (NP-)hard combinatorial problems is to decide whether a
universe of elements contains a witness consisting of elements that
match some prescribed pattern. For some of these problems there are known
advanced algebra-based FPT algorithms which solve the decision problem but do
not return the witness. We investigate techniques for turning such a
YES/NO-decision oracle into an algorithm for extracting a single witness, with
an objective to obtain practical scalability for large values of . By
relying on techniques from combinatorial group testing, we demonstrate that a
witness may be extracted with queries to either a deterministic or
a randomized set inclusion oracle with one-sided probability of error.
Furthermore, we demonstrate through implementation and experiments that the
algebra-based FPT algorithms are practical, in particular in the setting of the
-path problem. Also discussed are engineering issues such as optimizing
finite field arithmetic.Comment: Journal version, 16 pages. Extended abstract presented at ESA'1
The Dual Polynomial of Bipartite Perfect Matching
We obtain a description of the Boolean dual function of the Bipartite Perfect
Matching decision problem, as a multilinear polynomial over the Reals. We show
that in this polynomial, both the number of monomials and the magnitude of
their coefficients are at most exponential in . As an
application, we obtain a new upper bound of on the approximate degree of the bipartite perfect matching function,
improving the previous best known bound of . We deduce
that, beyond a factor, the polynomial method
cannot be used to improve the lower bound on the bounded-error quantum query
complexity of bipartite perfect matching
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