294 research outputs found
Univariate Ideal Membership Parameterized by Rank, Degree, and Number of Generators
Let F[X] be the polynomial ring over the variables X={x_1,x_2, ..., x_n}. An ideal I= generated by univariate polynomials {p_i(x_i)}_{i=1}^n is a univariate ideal. We study the ideal membership problem for the univariate ideals and show the following results.
- Let f(X) in F[l_1, ..., l_r] be a (low rank) polynomial given by an arithmetic circuit where l_i : 1 be a univariate ideal. Given alpha in F^n, the (unique) remainder f(X) mod I can be evaluated at alpha in deterministic time d^{O(r)} * poly(n), where d=max {deg(f),deg(p_1)...,deg(p_n)}. This yields a randomized n^{O(r)} algorithm for minimum vertex cover in graphs with rank-r adjacency matrices. It also yields an n^{O(r)} algorithm for evaluating the permanent of a n x n matrix of rank r, over any field F. Over Q, an algorithm of similar run time for low rank permanent is due to Barvinok [Barvinok, 1996] via a different technique.
- Let f(X)in F[X] be given by an arithmetic circuit of degree k (k treated as fixed parameter) and I=. We show that in the special case when I=, we obtain a randomized O^*(4.08^k) algorithm that uses poly(n,k) space.
- Given f(X)in F[X] by an arithmetic circuit and I=, membership testing is W[1]-hard, parameterized by k. The problem is MINI[1]-hard in the special case when I=
Polynomial Identity Testing via Evaluation of Rational Functions
We introduce a hitting set generator for Polynomial Identity Testing based on evaluations of low-degree univariate rational functions at abscissas associated with the variables. In spite of the univariate nature, we establish an equivalence up to rescaling with a generator introduced by Shpilka and Volkovich, which has a similar structure but uses multivariate polynomials in the abscissas.
We study the power of the generator by characterizing its vanishing ideal, i.e., the set of polynomials that it fails to hit. Capitalizing on the univariate nature, we develop a small collection of polynomials that jointly produce the vanishing ideal. As corollaries, we obtain tight bounds on the minimum degree, sparseness, and partition size of set-multi-linearity in the vanishing ideal. Inspired by an alternating algebra representation, we develop a structured deterministic membership test for the vanishing ideal. As a proof of concept we rederive known derandomization results based on the generator by Shpilka and Volkovich, and present a new application for read-once oblivious arithmetic branching programs that provably transcends the usual combinatorial techniques
Efficient Quantum Algorithm for Identifying Hidden Polynomials
We consider a natural generalization of an abelian Hidden Subgroup Problem
where the subgroups and their cosets correspond to graphs of linear functions
over a finite field F with d elements. The hidden functions of the generalized
problem are not restricted to be linear but can also be m-variate polynomial
functions of total degree n>=2.
The problem of identifying hidden m-variate polynomials of degree less or
equal to n for fixed n and m is hard on a classical computer since
Omega(sqrt{d}) black-box queries are required to guarantee a constant success
probability. In contrast, we present a quantum algorithm that correctly
identifies such hidden polynomials for all but a finite number of values of d
with constant probability and that has a running time that is only
polylogarithmic in d.Comment: 17 page
On Learning Polynomial Recursive Programs
We introduce the class of P-finite automata. These are a generalisation of
weighted automata, in which the weights of transitions can depend polynomially
on the length of the input word. P-finite automata can also be viewed as simple
tail-recursive programs in which the arguments of recursive calls can
non-linearly refer to a variable that counts the number of recursive calls. The
nomenclature is motivated by the fact that over a unary alphabet P-finite
automata compute so-called P-finite sequences, that is, sequences that satisfy
a linear recurrence with polynomial coefficients. Our main result shows that
P-finite automata can be learned in polynomial time in Angluin's MAT exact
learning model. This generalises the classical results that deterministic
finite automata and weighted automata over a field are respectively
polynomial-time learnable in the MAT model
Moment Varieties for Mixtures of Products
The setting of this article is nonparametric algebraic statistics. We study
moment varieties of conditionally independent mixture distributions on
. These are the secant varieties of toric varieties that express
independence in terms of univariate moments. Our results revolve around the
dimensions and defining polynomials of these varieties.Comment: 14 page
Numerical Algebraic Geometry: A New Perspective on String and Gauge Theories
The interplay rich between algebraic geometry and string and gauge theories
has recently been immensely aided by advances in computational algebra.
However, these symbolic (Gr\"{o}bner) methods are severely limited by
algorithmic issues such as exponential space complexity and being highly
sequential. In this paper, we introduce a novel paradigm of numerical algebraic
geometry which in a plethora of situations overcomes these short-comings. Its
so-called 'embarrassing parallelizability' allows us to solve many problems and
extract physical information which elude the symbolic methods. We describe the
method and then use it to solve various problems arising from physics which
could not be otherwise solved.Comment: 36 page
Paths and walks, forests and planes : arcadian algorithms and complexity
This dissertation is concerned with new results in the area of parameterized algorithms and complexity. We develop a new technique for hard graph problems that generalizes and unifies established methods such as Color-Coding, representative families, labelled walks and algebraic fingerprinting. At the heart of the approach lies an algebraic formulation of the problems, which is effected by means of a suitable exterior algebra. This allows us to estimate the number of simple paths of given length in directed graphs faster than before. Additionally, we give fast deterministic algorithms for finding paths of given length if the input graph contains only few of such paths. Moreover, we develop faster deterministic algorithms to find spanning trees with few leaves. We also consider the algebraic foundations of our new method. Additionally, we investigate the fine-grained complexity of determining the precise number of forests with a given number of edges in a given undirected graph. To wit, this happens in two ways. Firstly, we complete the complexity classification of the Tutte plane, assuming the exponential time hypothesis. Secondly, we prove that counting forests with a given number of edges is at least as hard as counting cliques of a given size.Diese Dissertation befasst sich mit neuen Ergebnissen auf dem Gebiet parametrisierter Algorithmen und Komplexitätstheorie. Wir entwickeln eine neue Technik für schwere Graphprobleme, die etablierte Methoden wie Color-Coding, representative families, labelled walks oder algebraic fingerprinting verallgemeinert und vereinheitlicht. Kern der Herangehensweise ist eine algebraische Formulierung der Probleme, die vermittels passender Graßmannalgebren geschieht. Das erlaubt uns, die Anzahl einfacher Pfade gegebener Länge in gerichteten Graphen schneller als bisher zu schätzen. Außerdem geben wir schnelle deterministische Verfahren an, Pfade gegebener Länge zu finden, falls der Eingabegraph nur wenige solche Pfade enthält. Übrigens entwickeln wir schnellere deterministische Algorithmen, um Spannbäume mit wenigen Blättern zu finden. Wir studieren außerdem die algebraischen Grundlagen unserer neuen Methode. Weiters untersuchen wir die fine-grained-Komplexität davon, die genaue Anzahl von Wäldern einer gegebenen Kantenzahl in einem gegebenen ungerichteten Graphen zu bestimmen. Und zwar erfolgt das auf zwei verschiedene Arten. Erstens vervollständigen wir die Komplexitätsklassifizierung der Tutte-Ebene unter Annahme der Expo- nentialzeithypothese. Zweitens beweisen wir, dass Wälder mit gegebener Kantenzahl zu zählen, wenigstens so schwer ist, wie Cliquen gegebener Größe zu zählen.Cluster of Excellence (Multimodal Computing and Interaction
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