60 research outputs found
Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits
We deploy algebraic complexity theoretic techniques for constructing
symmetric determinantal representations of for00504925mulas and weakly skew
circuits. Our representations produce matrices of much smaller dimensions than
those given in the convex geometry literature when applied to polynomials
having a concise representation (as a sum of monomials, or more generally as an
arithmetic formula or a weakly skew circuit). These representations are valid
in any field of characteristic different from 2. In characteristic 2 we are led
to an almost complete solution to a question of B\"urgisser on the
VNP-completeness of the partial permanent. In particular, we show that the
partial permanent cannot be VNP-complete in a finite field of characteristic 2
unless the polynomial hierarchy collapses.Comment: To appear in the AMS Contemporary Mathematics volume on
Randomization, Relaxation, and Complexity in Polynomial Equation Solving,
edited by Gurvits, Pebay, Rojas and Thompso
The Limited Power of Powering: Polynomial Identity Testing and a Depth-four Lower Bound for the Permanent
Polynomial identity testing and arithmetic circuit lower bounds are two
central questions in algebraic complexity theory. It is an intriguing fact that
these questions are actually related. One of the authors of the present paper
has recently proposed a "real {\tau}-conjecture" which is inspired by this
connection. The real {\tau}-conjecture states that the number of real roots of
a sum of products of sparse univariate polynomials should be polynomially
bounded. It implies a superpolynomial lower bound on the size of arithmetic
circuits computing the permanent polynomial. In this paper we show that the
real {\tau}-conjecture holds true for a restricted class of sums of products of
sparse polynomials. This result yields lower bounds for a restricted class of
depth-4 circuits: we show that polynomial size circuits from this class cannot
compute the permanent, and we also give a deterministic polynomial identity
testing algorithm for the same class of circuits.Comment: 16 page
Factoring bivariate lacunary polynomials without heights
We present an algorithm which computes the multilinear factors of bivariate
lacunary polynomials. It is based on a new Gap Theorem which allows to test
whether a polynomial of the form P(X,X+1) is identically zero in time
polynomial in the number of terms of P(X,Y). The algorithm we obtain is more
elementary than the one by Kaltofen and Koiran (ISSAC'05) since it relies on
the valuation of polynomials of the previous form instead of the height of the
coefficients. As a result, it can be used to find some linear factors of
bivariate lacunary polynomials over a field of large finite characteristic in
probabilistic polynomial time.Comment: 25 pages, 1 appendi
On the Probabilistic Query Complexity of Transitively Symmetric Problems
We obtain optimal lower bounds on the nonadaptive probabilistic query complexity of a class of problems defined by a rather weak symmetry condition. In fact, for each problem in this class, given a number T of queries we compute exactly the performance (i.e., the probability of success on the worst instance) of the best nonadaptive probabilistic algorithm that makes T queries. We show that this optimal performance is given by a minimax formula involving certain probability distributions. Moreover, we identify two classes of problems for which adaptivity does not help. We illustrate these results on a few natural examples, including unordered search, Simon's problem, distinguishing one-to-one functions from two-to-one functions, and hidden translation. For these last three examples, which are of particular interest in quantum computing, the recent theorems of Aaronson, of Laplante and Magniez, and of Bar-Yossef, Kumar and Sivakumar on the probabilistic complexity of black-box problems do not yield any nonconstant lower bound
The Multivariate Resultant is NP-hard in any Characteristic
The multivariate resultant is a fundamental tool of computational algebraic
geometry. It can in particular be used to decide whether a system of n
homogeneous equations in n variables is satisfiable (the resultant is a
polynomial in the system's coefficients which vanishes if and only if the
system is satisfiable). In this paper we present several NP-hardness results
for testing whether a multivariate resultant vanishes, or equivalently for
deciding whether a square system of homogeneous equations is satisfiable. Our
main result is that testing the resultant for zero is NP-hard under
deterministic reductions in any characteristic, for systems of low-degree
polynomials with coefficients in the ground field (rather than in an
extension). We also observe that in characteristic zero, this problem is in the
Arthur-Merlin class AM if the generalized Riemann hypothesis holds true. In
positive characteristic, the best upper bound remains PSPACE.Comment: 13 page
Decidable and undecidable problems about quantum automata
We study the following decision problem: is the language recognized by a
quantum finite automaton empty or non-empty? We prove that this problem is
decidable or undecidable depending on whether recognition is defined by strict
or non-strict thresholds. This result is in contrast with the corresponding
situation for probabilistic finite automata for which it is known that strict
and non-strict thresholds both lead to undecidable problems.Comment: 10 page
Model counting for CNF formuals of bounded module treewidth
The modular treewidth of a graph is its treewidth after the contraction of modules. Modular treewidth properly generalizes treewidth and is itself properly generalized by clique-width. We show that the number of satisfying assignments of a CNF formula whose incidence graph has bounded modular treewidth can be computed in polynomial time. This provides new tractable classes of formulas for which #SAT is polynomial. In particular, our result generalizes known results for the treewidth of incidence graphs and is incomparable with known results for clique-width (or rank-width) of signed incidence graphs. The contraction of modules is an effective data reduction procedure. Our algorithm is the first one to harness this technique for #SAT. The order of the polynomial time bound of our algorithm depends on the modular treewidth. We show that this dependency cannot be avoided subject to an assumption from Parameterized Complexity
The set of realizations of a max-plus linear sequence is semi-polyhedral
We show that the set of realizations of a given dimension of a max-plus
linear sequence is a finite union of polyhedral sets, which can be computed
from any realization of the sequence. This yields an (expensive) algorithm to
solve the max-plus minimal realization problem. These results are derived from
general facts on rational expressions over idempotent commutative semirings: we
show more generally that the set of values of the coefficients of a commutative
rational expression in one letter that yield a given max-plus linear sequence
is a semi-algebraic set in the max-plus sense. In particular, it is a finite
union of polyhedral sets
R'esolutions universelles pour des probl`emes NP-complets
F33.43> n ) sur R tels que le polynome a 0 + a 1 X + ::: + a n X n admette une racine r'eelle. Si ÂŻa = (a 0 ; :::; a n ), se demander si ÂŻ a 2 X 0 , c'est se demander si le polynome a 0 + a 1 X + ::: + a n X n admet une racine r'eelle. D'efinition : Un probl`eme X sur M est PM , i.e. polynomial pour la structure M , si la question ÂŻ x 2 X peut etre d'ecid'ee en temps polynomial. Le temps repr'esente le nombre d'op'erations (calcul de fonction ou test d'appartenance `a une relation) `a effectuer pour obtenir la r'epons
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