# Graduiertenkolleg: Methods for Discrete Structures

Deutsche Forschungsgemeinschaft
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# Monday Lecture and Colloquium

January 11, 2010

Humboldt-Universität zu Berlin
room 4.112
Institut für Informatik
Johann von Neumann-Haus,
Rudower Chaussee 25

Lecture - 14:15

### On a Problem Posed by Steve Smale

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

### Capturing Polynomial Time on Interval Graphs

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.

Letzte Aktualisierung: 14.12.2009