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#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\begin_document
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January 30th, 2004
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Outline
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\begin_layout Standard
\begin_inset CommandInset toc
LatexCommand tableofcontents
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-\begin_layout EndFrame
-
-\end_layout
-
+\end_deeper
\begin_layout Standard
\begin_inset ERT
-status collapsed
+status open
\begin_layout Plain Layout
What are Tournaments?
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-\begin_layout BeginFrame
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Tournaments Consist of Jousts Between Knights
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+
+
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+
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\begin_layout Columns
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\end_deeper
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Tournaments are Complete Directed Graphs
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+
+
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+
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\end_deeper
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-
-[<+>]
++
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+\begin_inset Argument 4
+status open
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+\begin_layout Plain Layout
Tournaments Arise Naturally in Different Situations
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout ExampleBlock
\begin_inset Argument 2
status collapsed
.
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+\end_deeper
\end_deeper
\begin_layout Subsection
What Does ``Finding Paths'' Mean?
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+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
``Finding Paths'' is Ambiguous
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+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Block
\begin_inset Argument 2
status open
approximately their distance.
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+\end_deeper
\end_deeper
\end_deeper
\begin_layout Section
\end_layout
-\begin_layout BeginFrame
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+\begin_inset Argument 4
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+
+\begin_layout Plain Layout
The Classes L and NL are Defined via
\begin_inset Newline newline
\end_inset
Logspace Turing Machines
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+\end_inset
+
+
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+
+\begin_deeper
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status collapsed
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Logspace Turing Machines Are Quite Powerful
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+
+
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+
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\begin_layout Block
\begin_inset Argument 2
status collapsed
\end_layout
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status collapsed
\begin_layout Plain Layout
+1
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+
+\end_inset
+
+
+\begin_inset Argument 3
+status collapsed
-<1>[label=hierarchy]
+\begin_layout Plain Layout
+label=hierarchy
\end_layout
\end_inset
+
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
The Complexity Class Hierarchy
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+
+
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+
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\begin_inset ERT
status collapsed
\end_layout
-\begin_layout BeginFrame
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The Circuit Complexity Classes AC
\begin_inset Formula $^{0}$
\end_inset
Limit the Circuit Depth
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+\end_inset
+
+
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+
+\begin_deeper
\begin_layout Standard
\begin_inset ERT
status collapsed
.
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+\end_deeper
\end_deeper
\end_deeper
\begin_layout AgainFrame
Standard Complexity Results on Finding Paths
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-\begin_layout BeginFrame
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All Variants of Finding Paths in Directed Graphs
\begin_inset Newline newline
\end_inset
Are Equally Difficult
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Fact
\begin_inset Formula $\Lang{reach}$
\end_inset
.
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+\end_deeper
\end_deeper
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\begin_inset Argument 1
hierarchy
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-\begin_layout BeginFrame
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Finding Paths in Forests and Directed Paths is Easy,
\begin_inset Newline newline
\end_inset
But Not Trivial
\end_layout
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+
+
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+
+\begin_deeper
\begin_layout Fact
\begin_inset Formula $\Lang{reach}_{\operatorname{forest}}$
\end_inset
-complete.
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\begin_inset Argument 1
status collapsed
Complexity of: Does a Path Exist?
\end_layout
-\begin_layout BeginFrame
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+
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Definition of the Tournament Reachability Problem
\end_layout
+\end_inset
+
+
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+
+\begin_deeper
\begin_layout Definition
Let
\color none
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\end_deeper
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The Tournament Reachability Problem is Very Easy
\end_layout
+\end_inset
+
+
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+
+\begin_deeper
\begin_layout Theorem
\begin_inset Formula $\Lang{reach}_{\operatorname{tourn}}\in\Class{AC}^{0}$
\end_inset
.
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+\end_deeper
\end_deeper
\begin_layout AgainFrame
\begin_inset Argument 1
Complexity of: Construct a Shortest Path
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Finding a Shortest Path Is as Difficult as
\begin_inset Newline newline
\end_inset
the Distance Problem
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+
+
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+
+\begin_deeper
\begin_layout Definition
Let
\color none
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The Tournament Distance Problem is Hard
\end_layout
+\end_inset
+
+
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+
+\begin_deeper
\begin_layout Theorem
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
\end_inset
.
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-\begin_layout BeginFrame
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+
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Proof That
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
\end_inset
is NL-complete
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+
+
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+
+\begin_deeper
\begin_layout Standard
\begin_inset ERT
status collapsed
\end_layout
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\end_deeper
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\begin_inset Argument 1
Complexity of: Approximating the Shortest Path
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-\begin_layout BeginFrame
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+
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Approximators Compute Paths that Are Nearly As Short As a Shortest Path
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+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Definition
An
\color none
\end_layout
\end_deeper
-\begin_layout BeginFrame
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+
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There Exists a Logspace Approximation Scheme for
\begin_inset Newline newline
\end_inset
the Tournament Shortest Path Problem
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Theorem
There exists an approximation scheme for
\begin_inset Formula $\Lang{tournament-shortest-path}$
\end_layout
+\end_deeper
\begin_layout AgainFrame
\begin_inset Argument 1
status collapsed
hierarchy
\end_layout
-\begin_layout EndFrame
-
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-
\begin_layout Standard
\begin_inset Note Note
status open
Summary
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+
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Summary
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Block
\begin_inset Argument 2
status collapsed
\end_layout
+\end_deeper
\end_deeper
\begin_layout Subsection*
For Further Reading
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+
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For Further Reading
\end_layout
+\end_inset
+
+
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+
+\begin_deeper
\begin_layout Standard
\begin_inset ERT
status open
In press.
\end_layout
-\begin_layout EndFrame
-
-\end_layout
-
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\begin_layout Standard
\start_of_appendix
\begin_inset ERT
Graphs With Bounded Independence Number
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-
-[label=independence]
+label=independence
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\end_inset
+
+\begin_inset Argument 4
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+
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Definition of Independence Number of a Graph
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Definition
The
\color none
\end_layout
-\begin_layout BeginFrame
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+
+\end_layout
+
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+\begin_inset Argument 4
+status open
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The Results for Tournaments also Apply to
\begin_inset Newline newline
\end_inset
Graphs With Bounded Independence Number
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Theorem
For each
\begin_inset space ~
.
\end_layout
+\end_deeper
\begin_layout Subsection
Finding Paths in Undirected Graphs
\end_layout
-\begin_layout BeginFrame
-\begin_inset ERT
+\begin_layout Frame
+\begin_inset Argument 1
status collapsed
\begin_layout Plain Layout
+1-2
+\end_layout
+
+\end_inset
+
-<1-2>[label=undirected]
+\begin_inset Argument 3
+status collapsed
+
+\begin_layout Plain Layout
+label=undirected
\end_layout
\end_inset
+
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
The Complexity of Finding Paths in Undirected Graphs
\begin_inset Newline newline
\end_inset
Is Party Unknown.
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Fact
\begin_inset Formula $\Lang{reach}_{\operatorname{undirected}}$
\end_inset
\end_layout
+\end_deeper
\end_deeper
\begin_layout Subsection
The Approximation Scheme is Optimal
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-\begin_layout BeginFrame
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status collapsed
\begin_layout Plain Layout
-
-[label=optimality]
+label=optimality
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\end_inset
+
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
The Approximation Scheme is Optimal
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Theorem
Suppose there exists an approximation scheme for
\begin_inset Formula $\Lang{tournament-shortest-path}$
\end_layout
\end_deeper
-\begin_layout EndFrame
-
-\end_layout
-
+\end_deeper
\end_body
\end_document
#LyX 2.1 created this file. For more info see http://www.lyx.org/
-\lyxformat 452
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\begin_document
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Outline
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+
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\begin_layout Standard
\begin_inset CommandInset toc
LatexCommand tableofcontents
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\begin_layout Section
Motivation
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Make Titles Informative.
Use Uppercase Letters.
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+
+
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+
+\begin_deeper
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Frame subtitles are optional.
Use upper- or lowercase letters.
These overlays are created using the Pause style.
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Make Titles Informative.
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status open
This is shown second.
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Make Titles Informative.
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+
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.
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\begin_layout Subsection
Previous Work
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Make Titles Informative.
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+
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On second slide.
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\begin_layout Section
Our Results/Contribution
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Main Results
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Make Titles Informative.
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+
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+
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\begin_layout Theorem
On first slide.
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On second slide.
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+
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+
+\begin_layout Frame
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Make Titles Informative.
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+
+
+\end_layout
+
+\begin_deeper
\begin_layout ColumnsTopAligned
\end_layout
New line
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+\end_deeper
\end_deeper
\begin_layout Subsection
Basic Ideas for Proofs/Implementations
Summary
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Summary
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+
+
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+
+\begin_deeper
\begin_layout Itemize
The
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first main message
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+
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+
+
\color none
of your talk in one or two lines.
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The
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second main message
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+
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+
+
\color none
of your talk in one or two lines.
\end_layout
Perhaps a
\color none
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-third message
-\color none
-, but not more than that.
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-\begin_layout Standard
-\begin_inset Note Note
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status open
\begin_layout Plain Layout
-An outlook is always optional.
+third message
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+\color none
+, but not more than that.
\end_layout
\begin_layout Standard
-\begin_inset ERT
+\begin_inset Note Note
status open
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+An outlook is always optional.
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-\backslash
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\end_deeper
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-
-\end_layout
-
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\begin_layout Section*
\start_of_appendix
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For Further Reading
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-
-[allowframebreaks]
+allowframebreaks
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For Further Reading
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+\end_inset
+
+
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+
+\begin_deeper
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\begin_layout Plain Layout
2(1):50--100, 2000.
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-\begin_layout EndFrame
-
-\end_layout
-
+\end_deeper
\end_body
\end_document
#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\begin_document
\begin_header
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status open
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-Author, Another
+Autor, Anders
\begin_inset Note Note
status collapsed
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Gliederung
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+
+
+\end_layout
+
+\begin_deeper
\begin_layout Standard
\begin_inset CommandInset toc
LatexCommand tableofcontents
\end_layout
+\end_deeper
\begin_layout Section
Motivation
\end_layout
Das behandelte Problem
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-\begin_layout BeginFrame
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+
+\begin_layout Plain Layout
Überschriften sollten informativ sein
-\begin_inset Newline newline
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+
\end_inset
-Korrekte Groß-/Kleinschreibung beachten
+
\end_layout
+\begin_deeper
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Untertitel sind optional
\begin_inset Note Note
erzeugt.
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+
+\end_layout
+
+\begin_layout Frame
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Überschriften müssen informativ sein
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+
+
+\end_layout
+
+\begin_deeper
\begin_layout Itemize
\begin_inset Argument item:2
status open
Dies wird als Zweites gezeigt.
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-\begin_layout BeginFrame
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+
+\end_layout
+
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
Überschriften müssen informativ sein
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Block
\begin_inset Argument 1
status open
.
\end_layout
+\end_deeper
\end_deeper
\begin_layout Subsection
Frühere Arbeiten
\end_layout
-\begin_layout BeginFrame
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
Überschriften müssen informativ sein
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Example
\begin_inset Argument 1
status open
Auf dem zweiten Overlay.
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\begin_layout Section
Unsere Resultate/Beitrag
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Hauptresultate
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-\begin_layout BeginFrame
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Überschriften müssen informativ sein
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+
+
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+
+\begin_deeper
\begin_layout Theorem
Auf dem ersten Overlay.
\end_layout
Auf dem zweiten Overlay.
\end_layout
-\begin_layout BeginFrame
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+
+\end_layout
+
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
Überschriften müssen informativ sein
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout ColumnsTopAligned
\end_layout
Neue Zeile.
\end_layout
+\end_deeper
\end_deeper
\begin_layout Subsection
Ideen für Beweise/Umsetzung
Zusammenfassung
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-\begin_layout BeginFrame
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+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
Zusammenfassung
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
\begin_layout Standard
\begin_inset Note Note
status open
Die
\color none
-\color red
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+
+\begin_inset Flex Alert
+status open
+
+\begin_layout Plain Layout
erste Hauptbotschaft
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