January 11, 2010
Humboldt-Universität zu Berlin
room 4.112
Institut für Informatik
Johann von Neumann-Haus,
Rudower Chaussee 25
Lecture - 14:15
Abstract:
The 17th of the problems proposed by Steve Smale for the 21st century
asks for the existence of a deterministic algorithm computing an
approximate solution of a system of $n$ complex polynomials in $n$
unknowns in time polynomial, on the average, in the size $N$ of the
input system. A partial solution to this problem was given by Carlos
Beltran and Luis Miguel Pardo who exhibited a randomized algorithm
doing so. In this paper we further extend this result in several
directions. Firstly, we exhibit a linear homotopy algorithm that
efficiently implements a non-constructive idea of Mike Shub. This
algorithm is then used in a randomized algorithm, call it LV, a la
Beltran-Pardo. Secondly, we perform a smoothed analysis (in the sense
of Spielman and Teng) of algorithm LV and prove that its smoothed
complexity is polynomial in the input size and $\s^{-1}$, where $\s$
controls the size of of the random perturbation of the input systems.
Thirdly, we perform a condition-based analysis of LV. That is, we give
a bound, for each system $f$, of the expected running time of LV with
input $f$. In addition to its dependence on $N$ this bound also
depends on the condition of $f$. Fourthly, and to conclude, we return
to Smale's 17th problem as originally formulated for deterministic
algorithms. We exhibit such an algorithm and show that its average
complexity is $N^{O(\log\log N)}$. This is nearly a solution to
Smale's 17th problem. This is joint work with Felipe Cucker.
Colloquium - 16:00
Abstract:
The main topic of this talk is the characterization of all
polynomial-time computable queries on the class of interval graphs by
sentences of fixed-point logic with counting. The result is one of the
first establishing the capturing of polynomial time on a graph class
which is defined by forbidden induced subgraphs. Furthermore, it is
shown that fixed-point logic with counting is not expressive enough to
capture polynomial time on the classes of chordal graphs and
incomparability graphs.
The capturing is done by exhibiting an interval graph canonization
procedure which is definable in fixed-point logic with counting. This
procedure is not based on known methods for interval graphs and might be
interesting in its own right for its conceptual simplicity. Apart from
the canonization, the talk will also introduce background on the
employed logic and other capturing results.