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The Complexity of
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Finding Paths in Tournaments
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Till Tantau
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January 30th, 2004
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Introduction
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What are Tournaments?
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Tournaments Consist of Jousts Between Knights
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Standard Complexity Results on Finding Paths
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All Variants of Finding Paths in Directed Graphs
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Are Equally Difficult
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For directed graphs, we can solve
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Finding Paths in Tournaments
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Complexity of: Does a Path Exist?
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Definition of the Tournament Reachability Problem
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The Tournament Reachability Problem is Very Easy
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The problem is
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-\layout Itemize
-
+\end_layout
+\begin_layout Itemize
\begin_inset Formula $\Lang{reach}\not\le_{\operatorname{m}}^{\Class{AC}^{0}}\Lang{reach}_{\operatorname{tourn}}$
-\end_inset
+\end_inset
.
-\end_deeper
-\layout AgainFrame
+\end_layout
+\end_deeper
+\end_deeper
+\begin_layout AgainFrame
+\begin_inset Argument 1
+status open
-\begin_inset ERT
-status Collapsed
+\begin_layout Plain Layout
+5
+\end_layout
-\layout Standard
-<5>
-\end_inset
+\end_inset
hierarchy
-\layout Subsection
+\end_layout
+\begin_layout Subsection
Complexity of: Construct a Shortest Path
-\layout BeginFrame
+\end_layout
+
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+\begin_layout Plain Layout
Finding a Shortest Path Is as Difficult as
-\newline
+\begin_inset Newline newline
+\end_inset
+
the Distance Problem
-\layout Definition
+\end_layout
-Let
+\end_inset
+
+
+\begin_inset Separator parbreak
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Definition
+Let
+\color none
+
\color red
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
-\end_inset
+\end_inset
+
+\color none
-\color default
+\color inherit
contain all tuples
\begin_inset Formula $(T,s,t,d)$
-\end_inset
+\end_inset
such that
-\begin_deeper
-\layout Enumerate
+\begin_inset Separator parbreak
+\end_inset
+
+\end_layout
+\begin_deeper
+\begin_layout Enumerate
\begin_inset Formula $T=(V,E)$
-\end_inset
+\end_inset
is a tournament in which
-\layout Enumerate
+\end_layout
+\begin_layout Enumerate
the distance of
\begin_inset Formula $s$
-\end_inset
+\end_inset
+
+ and
+\begin_inset space ~
+\end_inset
- and\SpecialChar ~
\begin_inset Formula $t$
-\end_inset
+\end_inset
+
+ is at most
+\begin_inset space ~
+\end_inset
- is at most\SpecialChar ~
\begin_inset Formula $d$
-\end_inset
+\end_inset
.
-\end_deeper
-\layout BeginFrame
+\end_layout
+
+\end_deeper
+\end_deeper
+\begin_layout Standard
+\begin_inset Separator parbreak
+\end_inset
+
+
+\end_layout
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
The Tournament Distance Problem is Hard
-\layout Theorem
+\end_layout
+
+\end_inset
+
+
+\begin_inset Separator parbreak
+\end_inset
+\end_layout
+
+\begin_deeper
+\begin_layout Theorem
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
-\end_inset
+\end_inset
is
\begin_inset Formula $\Class{NL}$
-\end_inset
+\end_inset
-complete.
-\layout Standard
+\end_layout
+\begin_layout Standard
+\begin_inset space \hfill{}
+\end_inset
-\hfill
\begin_inset ERT
-status Inlined
+status collapsed
+
+\begin_layout Plain Layout
-\layout Standard
-\backslash
+\backslash
hyperlink{hierarchy<6>}{
-\backslash
+\backslash
beamerskipbutton{Skip Proof}}
-\end_inset
+\end_layout
+
+\end_inset
+
+\end_layout
-\layout Pause
+\begin_layout Pause
-\layout Corollary
+\end_layout
+\begin_layout Corollary
Shortest path in tournaments can be constructed
-\newline
+\begin_inset Newline newline
+\end_inset
+
in logarithmic space, iff
\begin_inset Formula $\Class{L}=\Class{NL}$
-\end_inset
+\end_inset
.
-\layout Pause
+\end_layout
-\layout Corollary
+\begin_layout Pause
+\end_layout
+\begin_layout Corollary
\begin_inset Formula $\Lang{distance}\le_{\operatorname{m}}^{\Class{AC}^{0}}\Lang{distance}_{\operatorname{tourn}}$
-\end_inset
+\end_inset
.
-\layout BeginFrame
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+\begin_inset Separator parbreak
+\end_inset
+
+
+\end_layout
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
Proof That
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
-\end_inset
+\end_inset
is NL-complete
-\layout Standard
+\end_layout
+
+\end_inset
+
+
+\begin_inset Separator parbreak
+\end_inset
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
\begin_inset ERT
-status Collapsed
+status collapsed
+
+\begin_layout Plain Layout
-\layout Standard
-\backslash
+\backslash
nointerlineskip
-\end_inset
+\end_layout
+\end_inset
-\layout Columns
+\end_layout
-\begin_inset ERT
-status Inlined
+\begin_layout Columns
+\begin_inset Argument 1
+status open
-\layout Standard
-[t,onlytextwidth]
-\end_inset
+\begin_layout Plain Layout
+t,onlytextwidth
+\end_layout
+\end_inset
-\begin_deeper
-\layout Column
-5.7cm
-\layout Standard
+\begin_inset Separator parbreak
+\end_inset
+\end_layout
+
+\begin_deeper
+\begin_layout Column
+5.7cm
+\end_layout
+
+\begin_layout Standard
\begin_inset ERT
-status Inlined
+status collapsed
+
+\begin_layout Plain Layout
-\layout Standard
-\backslash
+\backslash
setlength
-\backslash
+\backslash
leftmargini{1.5em}
-\end_inset
+\end_layout
+\end_inset
-\layout Block
+\end_layout
-\begin_inset ERT
-status Collapsed
-
-\layout Standard
-{
-\end_inset
+\begin_layout Block
+\begin_inset Argument 2
+status open
+\begin_layout Plain Layout
Reduce
\begin_inset Formula $\Lang{reach}$
-\end_inset
+\end_inset
to
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}$
-\end_inset
+\end_inset
-\begin_inset ERT
-status Collapsed
+\end_layout
-\layout Standard
-}
-\end_inset
+\end_inset
-\begin_deeper
-\layout Enumerate
+\begin_inset Separator parbreak
+\end_inset
-\begin_inset ERT
-status Inlined
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:2
+status open
-\layout Standard
-<alert@1>
-\end_inset
+\begin_layout Plain Layout
+alert@1
+\end_layout
+
+\end_inset
Is input
\begin_inset Formula $(G,s,t)$
-\end_inset
+\end_inset
in
\begin_inset Formula $\Lang{reach}$
-\end_inset
+\end_inset
?
-\layout Enumerate
+\end_layout
+\begin_layout Enumerate
+\begin_inset Argument item:2
+status open
-\begin_inset ERT
-status Inlined
+\begin_layout Plain Layout
+2-| alert@2-8
+\end_layout
-\layout Standard
-<2-| alert@2-8>
-\end_inset
+\end_inset
Map
\begin_inset Formula $G$
-\end_inset
+\end_inset
to
\begin_inset Formula $G'$
-\end_inset
+\end_inset
.
-\layout Enumerate
+\end_layout
+\begin_layout Enumerate
+\begin_inset Argument item:2
+status open
-\begin_inset ERT
-status Inlined
+\begin_layout Plain Layout
+9-| alert@9
+\end_layout
-\layout Standard
-<9-| alert@9>
-\end_inset
+\end_inset
Query:
-\newline
+\begin_inset Newline newline
+\end_inset
+
\begin_inset Formula $(G',s',t',3)\in\Lang{distance}_{\operatorname{tourn}}$
-\end_inset
+\end_inset
?
-\end_deeper
-\layout Separator
+\end_layout
-\layout Block
+\end_deeper
+\begin_layout Standard
+\begin_inset Separator parbreak
+\end_inset
-\begin_inset ERT
-status Collapsed
+\end_layout
-\layout Standard
-{
-\end_inset
+\begin_layout Block
+\begin_inset Argument 2
+status open
+\begin_layout Plain Layout
Correctness
-\begin_inset ERT
-status Collapsed
+\end_layout
-\layout Standard
-}
-\end_inset
+\end_inset
-\begin_inset ERT
-status Collapsed
+\begin_inset Argument 1
+status open
-\layout Standard
-<10->
-\end_inset
+\begin_layout Plain Layout
+10-
+\end_layout
+\end_inset
-\begin_deeper
-\layout Enumerate
+\begin_inset Separator parbreak
+\end_inset
-\begin_inset ERT
-status Inlined
-\layout Standard
-<10-| alert@10-11>
-\end_inset
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:2
+status open
+
+\begin_layout Plain Layout
+10-| alert@10-11
+\end_layout
+
+\end_inset
+
+A path in
+\begin_inset space ~
+\end_inset
-A path in\SpecialChar ~
\begin_inset Formula $G$
-\end_inset
+\end_inset
induces
-\newline
-a length-3 path in\SpecialChar ~
+\begin_inset Newline newline
+\end_inset
+
+a length-3 path in
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $G'$
-\end_inset
+\end_inset
.
-\layout Enumerate
+\end_layout
+\begin_layout Enumerate
+\begin_inset Argument item:2
+status open
-\begin_inset ERT
-status Inlined
+\begin_layout Plain Layout
+12-| alert@12-13
+\end_layout
-\layout Standard
-<12-| alert@12-13>
-\end_inset
+\end_inset
+
+A length-3 path in
+\begin_inset space ~
+\end_inset
-A length-3 path in\SpecialChar ~
\begin_inset Formula $G'$
-\end_inset
+\end_inset
induces
-\newline
-a path in\SpecialChar ~
+\begin_inset Newline newline
+\end_inset
+
+a path in
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $G'$
-\end_inset
+\end_inset
.
-\end_deeper
-\layout Column
+\end_layout
+\end_deeper
+\begin_layout Column
4.5cm
-\layout Example
-
+\end_layout
+\begin_layout Example
\begin_inset ERT
-status Inlined
+status collapsed
-\layout Standard
+\begin_layout Plain Layout
-\backslash
+
+\backslash
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-
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+
+\begin_layout Plain Layout
+
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-
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+\end_layout
+
+\begin_layout Plain Layout
+
+
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+
+\begin_layout Plain Layout
+
+
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+
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+\end_layout
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+
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-
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+\end_layout
+
+\begin_layout Plain Layout
+
+
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pgfputat{
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+
+\begin_layout Plain Layout
+
+
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+\backslash
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-
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-
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+
+\begin_layout Plain Layout
+
+
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-\layout Standard
-
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+\end_layout
+
+\begin_layout Plain Layout
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+
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-\layout Standard
-
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+
+\begin_layout Plain Layout
+
+
+\backslash
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-\layout Standard
-
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+
+
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-
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+
+\begin_layout Plain Layout
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+
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alert<4>{
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-\layout Standard
-
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+
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-
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+
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-
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-
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+
+
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pgfputat{
-\backslash
+\backslash
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-\backslash
+\backslash
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-\backslash
+\backslash
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-\layout Standard
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+
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+
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+
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-
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+
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+
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-
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-
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-
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-
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+\begin_layout AgainFrame
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+status open
+
+\begin_layout Plain Layout
+6
+\end_layout
-\layout Standard
-<6>
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+\end_inset
hierarchy
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+\end_layout
+\begin_layout Subsection
Complexity of: Approximating the Shortest Path
-\layout BeginFrame
+\end_layout
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
Approximators Compute Paths that Are Nearly As Short As a Shortest Path
-\layout Definition
+\end_layout
+
+\end_inset
+
+
+\begin_inset Separator parbreak
+\end_inset
-An
+
+\end_layout
+
+\begin_deeper
+\begin_layout Definition
+An
+\color none
+
\color red
approximation scheme for
\begin_inset Formula $\Lang{tournament-shortest-path}$
-\end_inset
+\end_inset
+
+
+\color none
+
+\color inherit
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+\begin_inset Separator parbreak
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-\color default
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-\begin_deeper
-\layout Enumerate
+\end_layout
+\begin_deeper
+\begin_layout Enumerate
a tuple
\begin_inset Formula $(T,s,t)\in\Lang{reach}_{\operatorname{tourn}}$
-\end_inset
+\end_inset
and
-\layout Enumerate
+\end_layout
+\begin_layout Enumerate
a number
\begin_inset Formula $r>1$
-\end_inset
+\end_inset
.
-\layout Standard
+\end_layout
+\begin_layout Standard
It outputs
-\layout Itemize
+\end_layout
+\begin_layout Itemize
a path from
\begin_inset Formula $s$
-\end_inset
+\end_inset
+
+ to
+\begin_inset space ~
+\end_inset
- to\SpecialChar ~
\begin_inset Formula $t$
-\end_inset
+\end_inset
of length at most
\begin_inset Formula $r\operatorname{d_{T}}(s,t)$
-\end_inset
+\end_inset
.
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+\end_deeper
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+\begin_inset Separator parbreak
+\end_inset
+
+\end_layout
+
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
There Exists a Logspace Approximation Scheme for
-\newline
+\begin_inset Newline newline
+\end_inset
+
the Tournament Shortest Path Problem
-\layout Theorem
+\end_layout
+
+\end_inset
+
+
+\begin_inset Separator parbreak
+\end_inset
+
+\end_layout
+
+\begin_deeper
+\begin_layout Theorem
There exists an approximation scheme for
\begin_inset Formula $\Lang{tournament-shortest-path}$
-\end_inset
+\end_inset
that for
\begin_inset Formula $1<r<2$
-\end_inset
+\end_inset
needs space
-\begin_inset Formula \[
-O\left(\log|V|\log\frac{1}{r-1}\right).\]
+\begin_inset Formula
+\[
+O\left(\log|V|\log\frac{1}{r-1}\right).
+\]
+
+\end_inset
-\end_inset
+\end_layout
-\layout Pause
+\begin_layout Pause
-\layout Corollary
+\end_layout
+\begin_layout Corollary
In tournaments, paths can be constructed in logarithmic space.
-\layout Standard
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+\begin_layout Standard
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beamergotobutton{More Details}}
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+\end_layout
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+
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+This is some program code:
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+
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+\begin_inset listings
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+status open
+
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+
+def func(param):
+\end_layout
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+
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+\begin_layout Plain Layout
+
+ pass
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+
+\begin_layout Plain Layout
+
+def func(param):
+\end_layout
+\begin_layout Plain Layout
+
+'This is a German word: Tschüs'
+\end_layout
+
+\begin_layout Plain Layout
+
+pass
+\end_layout
+
+\begin_layout Plain Layout
+
+def func(param):
+\end_layout
+
+\begin_layout Plain Layout
+
+'this is a python function'
+\end_layout
+
+\begin_layout Plain Layout
+
+pass
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Section*
Summary
-\layout Subsection*
+\end_layout
+\begin_layout Subsection*
Summary
-\layout BeginFrame
+\end_layout
+
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+\begin_layout Plain Layout
Summary
-\layout Block
+\end_layout
+\end_inset
-\begin_inset ERT
-status Inlined
-\layout Standard
-{Summary}
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+\begin_inset Separator parbreak
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+
+\end_layout
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+\begin_deeper
+\begin_layout Block
+\begin_inset Argument 2
+status collapsed
-Tournament
+\begin_layout Plain Layout
+Summary
+\end_layout
+
+\end_inset
+
+
+\begin_inset Separator parbreak
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+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+Tournament
+\color none
+
\color red
reachability
-\color default
- is in
-\color red
+\color none
+
+\color inherit
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+\color none
+\color red
+
\begin_inset Formula $\Class{AC}^{0}$
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-\color default
+\color inherit
.
-\layout Itemize
+\end_layout
-There exists a
+\begin_layout Itemize
+There exists a
+\color none
+
\color red
logspace approximation scheme
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+\color none
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+\color inherit
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+\color none
+
\color red
approximating
-\color default
- shortest paths in tournaments.
-\layout Itemize
+\color none
+
+\color inherit
+shortest paths in tournaments.
+\end_layout
-Finding
+\begin_layout Itemize
+Finding
+\color none
+
\color red
shortest paths
-\color default
- in tournaments is
-\color red
+\color none
+\color inherit
+in tournaments is
+\color none
+
+\color red
+
\begin_inset Formula $\Class{NL}$
-\end_inset
+\end_inset
-complete
-\color default
+\color inherit
.
-\end_deeper
-\layout Separator
+\end_layout
-\layout Block
+\end_deeper
+\begin_layout Standard
+\begin_inset Separator parbreak
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+
+\begin_layout Block
+\begin_inset Argument 2
+status collapsed
+
+\begin_layout Plain Layout
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-{Outlook}
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+\end_inset
-\begin_deeper
-\layout Itemize
+\begin_inset Separator parbreak
+\end_inset
+
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
The same results apply to graphs with
-\newline
+\begin_inset Newline newline
+\end_inset
+
bounded independence number.
-\hfill
+\begin_inset space \hfill{}
+\end_inset
+
\begin_inset ERT
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+status collapsed
-\layout Standard
+\begin_layout Plain Layout
-\backslash
+
+\backslash
hyperlink{independence}{
-\backslash
+\backslash
beamergotobutton{More Details}}
-\end_inset
+\end_layout
+
+\end_inset
-\layout Itemize
+\end_layout
+\begin_layout Itemize
The complexity of finding paths in undirected graphs
-\newline
+\begin_inset Newline newline
+\end_inset
+
is partly open.
-\hfill
+\begin_inset space \hfill{}
+\end_inset
+
\begin_inset ERT
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+status collapsed
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hyperlink{undirected}{
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beamergotobutton{More Details}}
-\end_inset
+\end_layout
+
+\end_inset
-\end_deeper
-\layout Subsection*
+\end_layout
+\end_deeper
+\end_deeper
+\begin_layout Subsection*
For Further Reading
-\layout BeginFrame
+\end_layout
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
For Further Reading
-\layout Standard
+\end_layout
+
+\end_inset
+\begin_inset Separator parbreak
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
\begin_inset ERT
-status Inlined
+status open
+
+\begin_layout Plain Layout
-\layout Standard
-\backslash
+\backslash
beamertemplatebookbibitems
-\end_inset
+\end_layout
+\end_inset
-\layout Bibliography
-\bibitem {Moon1968}
-\SpecialChar ~
+\end_layout
+
+\begin_layout Bibliography
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+key "Moon1968"
+
+\end_inset
+
+
+\begin_inset space ~
+\end_inset
+
John Moon.
\begin_inset ERT
-status Collapsed
+status collapsed
+
+\begin_layout Plain Layout
-\layout Standard
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+\backslash
newblock
-\end_inset
+\end_layout
+
+\end_inset
-\emph on
+\emph on
Topics on Tournaments.
-\emph default
+\emph default
\begin_inset ERT
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+status collapsed
+
+\begin_layout Plain Layout
-\layout Standard
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+\backslash
newblock
-\end_inset
+\end_layout
+
+\end_inset
Holt, Rinehart, and Winston, 1968.
\begin_inset ERT
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-\layout Standard
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+\backslash
beamertemplatearticlebibitems
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+\end_layout
+
+\end_inset
+
+\end_layout
-\layout Bibliography
-\bibitem {NickelsenT2002}
+\begin_layout Bibliography
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+key "NickelsenT2002"
+
+\end_inset
+
+
+\begin_inset space ~
+\end_inset
-\SpecialChar ~
Arfst Nickelsen and Till Tantau.
\begin_inset ERT
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+status collapsed
+
+\begin_layout Plain Layout
-\layout Standard
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+\backslash
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-\end_inset
+\end_layout
+
+\end_inset
On reachability in graphs with bounded independence number.
\begin_inset ERT
-status Collapsed
+status collapsed
+
+\begin_layout Plain Layout
-\layout Standard
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+\backslash
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-\end_inset
+\end_layout
+
+\end_inset
In
-\emph on
+\emph on
Proc.
of COCOON 2002
-\emph default
+\emph default
, Springer-Verlag, 2002.
-\layout Bibliography
-\bibitem {Tantau2004b}
+\end_layout
+
+\begin_layout Bibliography
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
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+
+\end_inset
+
+
+\begin_inset space ~
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-\SpecialChar ~
Till Tantau
\begin_inset ERT
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+status collapsed
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+\begin_layout Plain Layout
-\layout Standard
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newblock
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+\end_layout
+
+\end_inset
A logspace approximation scheme for the shortest path problem for graphs
with bounded independence number.
\begin_inset ERT
-status Collapsed
+status collapsed
+
+\begin_layout Plain Layout
-\layout Standard
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+\backslash
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-\end_inset
+\end_layout
+
+\end_inset
In
-\emph on
+\emph on
Proc.
of STACS 2004
-\emph default
+\emph default
, Springer-Verlag, 2004.
\begin_inset ERT
-status Collapsed
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+
+\begin_layout Plain Layout
-\layout Standard
-\backslash
+\backslash
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-\end_inset
+\end_layout
- In press.
-\layout EndFrame
+\end_inset
-\layout Standard
-\start_of_appendix
+ In press.
+\end_layout
+\end_deeper
+\begin_layout Standard
+\start_of_appendix
\begin_inset ERT
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+status open
+
+\begin_layout Plain Layout
+
-\layout Standard
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+AtBeginSubsection[]{}
+\end_layout
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-AtBeginSubsection[]{}
-\end_inset
+\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Appendix
-\layout Subsection
+\end_layout
+\begin_layout Subsection
Graphs With Bounded Independence Number
-\layout BeginFrame
+\end_layout
+\begin_layout Frame
+\begin_inset Argument 3
+status collapsed
-\begin_inset ERT
-status Inlined
+\begin_layout Plain Layout
+label=independence
+\end_layout
+
+\end_inset
-\layout Standard
-[label=independence]
-\end_inset
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
Definition of Independence Number of a Graph
-\layout Definition
+\end_layout
+
+\end_inset
+
+
+\begin_inset Separator parbreak
+\end_inset
+
+
+\end_layout
-The
+\begin_deeper
+\begin_layout Definition
+The
+\color none
+
\color red
independence number
-\color default
+\color none
+\color inherit
+
\begin_inset Formula $\alpha(G)$
-\end_inset
+\end_inset
of a directed graph
-\newline
+\begin_inset Newline newline
+\end_inset
+
is the maximum number of vertices we can pick,
-\newline
+\begin_inset Newline newline
+\end_inset
+
such that there is no edge between them.
-\layout Example
+\end_layout
+\begin_layout Example
Tournaments have independence number 1.
-\layout BeginFrame
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+\begin_inset Separator parbreak
+\end_inset
+
+
+\end_layout
+
+\begin_layout Frame
+\begin_inset Argument 4
+status open
+\begin_layout Plain Layout
The Results for Tournaments also Apply to
-\newline
+\begin_inset Newline newline
+\end_inset
+
Graphs With Bounded Independence Number
-\layout Theorem
+\end_layout
+
+\end_inset
+
+
+\begin_inset Separator parbreak
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Theorem
+For each
+\begin_inset space ~
+\end_inset
-For each\SpecialChar ~
\begin_inset Formula $k$
-\end_inset
+\end_inset
-,
+,
+\color none
+
\color red
reachability
-\color default
- in graphs with independence number
-\newline
-at most\SpecialChar ~
+\color none
+
+\color inherit
+in graphs with independence number
+\begin_inset Newline newline
+\end_inset
+
+at most
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $k$
-\end_inset
+\end_inset
is in
\begin_inset Formula $\Class{AC}^{0}$
-\end_inset
+\end_inset
.
-\layout Separator
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator parbreak
+\end_inset
+
+
+\end_layout
-\layout Theorem
+\begin_layout Theorem
+For each
+\begin_inset space ~
+\end_inset
-For each\SpecialChar ~
\begin_inset Formula $k$
-\end_inset
+\end_inset
-, there exists a
+, there exists a
+\color none
+
\color red
logspace approximation scheme
-\color default
- for approximating the shortest path in graphs with independence number
- at most\SpecialChar ~
+\color none
+
+\color inherit
+for approximating the shortest path in graphs with independence number at
+ most
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $k$
-\end_inset
+\end_inset
+
+\end_layout
-\layout Separator
+\begin_layout Standard
+\begin_inset Separator parbreak
+\end_inset
-\layout Theorem
-For each\SpecialChar ~
+\end_layout
+
+\begin_layout Theorem
+For each
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $k$
-\end_inset
+\end_inset
-, finding the
+, finding the
+\color none
+
\color red
shortest path
-\color default
- in graphs with independence number at most\SpecialChar ~
+\color none
+
+\color inherit
+in graphs with independence number at most
+\begin_inset space ~
+\end_inset
+
\begin_inset Formula $k$
-\end_inset
+\end_inset
- is
+ is
+\color none
+
\color red
\begin_inset Formula $\Class{NL}$
-\end_inset
+\end_inset
-complete
-\color default
+\color inherit
.
-\layout Subsection
+\end_layout
+\end_deeper
+\begin_layout Subsection
Finding Paths in Undirected Graphs
-\layout BeginFrame
+\end_layout
+\begin_layout Frame
+\begin_inset Argument 1
+status collapsed
-\begin_inset ERT
-status Inlined
+\begin_layout Plain Layout
+1-2
+\end_layout
+
+\end_inset
+
+
+\begin_inset Argument 3
+status collapsed
+
+\begin_layout Plain Layout
+label=undirected
+\end_layout
-\layout Standard
-<1-2>[label=undirected]
-\end_inset
+\end_inset
+
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
The Complexity of Finding Paths in Undirected Graphs
-\newline
+\begin_inset Newline newline
+\end_inset
+
Is Party Unknown.
-\layout Fact
+\end_layout
+
+\end_inset
+\begin_inset Separator parbreak
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Fact
\begin_inset Formula $\Lang{reach}_{\operatorname{undirected}}$
-\end_inset
+\end_inset
is
\begin_inset Formula $\Class{SL}$
-\end_inset
+\end_inset
-complete.
-\layout Corollary
+\end_layout
+\begin_layout Corollary
For undirected graphs, we can solve
-\begin_deeper
-\layout Itemize
+\begin_inset Separator parbreak
+\end_inset
+
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
the reachability problem in logspace iff
\begin_inset Formula $\Class L=\Class{SL}$
-\end_inset
+\end_inset
,
-\layout Itemize
+\end_layout
+\begin_layout Itemize
the construction problem in logspace iff
-\begin_inset ERT
-status Inlined
+\begin_inset Flex Alternative
+status open
+
+\begin_layout Plain Layout
+\begin_inset Argument 1
+status open
+
+\begin_layout Plain Layout
+1
+\end_layout
+
+\end_inset
+
+
+\begin_inset Argument 2
+status open
+
+\begin_layout Plain Layout
+?
+\end_layout
+
+\end_inset
+
+
+\begin_inset Flex Alert
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\Class L=\Class{SL}$
+\end_inset
+
+
+\end_layout
-\layout Standard
+\end_inset
-\backslash
-alt<1>{?}{
-\backslash
-alert{$
-\backslash
-Class L =
-\backslash
-Class{SL}$}}
-\end_inset
+
+\end_layout
+
+\end_inset
,
-\layout Itemize
+\end_layout
+\begin_layout Itemize
the optimization problem in logspace iff
-\begin_inset ERT
-status Inlined
+\begin_inset Flex Alternative
+status open
+
+\begin_layout Plain Layout
+\begin_inset Argument 1
+status open
+
+\begin_layout Plain Layout
+1
+\end_layout
+
+\end_inset
+
+
+\begin_inset Argument 2
+status open
+
+\begin_layout Plain Layout
+?
+\end_layout
+
+\end_inset
+
+
+\begin_inset Flex Alert
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\Class L=\Class{NL}$
+\end_inset
+
+
+\end_layout
-\layout Standard
+\end_inset
-\backslash
-alt<1>{?}{
-\backslash
-alert{$
-\backslash
-Class L =
-\backslash
-Class{NL}$}}
-\end_inset
+
+\end_layout
+
+\end_inset
,
-\layout Itemize
+\end_layout
+\begin_layout Itemize
the approximation problem in logspace iff ?.
-\end_deeper
-\layout Subsection
+\end_layout
+\end_deeper
+\end_deeper
+\begin_layout Subsection
The Approximation Scheme is Optimal
-\layout BeginFrame
+\end_layout
+\begin_layout Frame
+\begin_inset Argument 3
+status collapsed
-\begin_inset ERT
-status Inlined
+\begin_layout Plain Layout
+label=optimality
+\end_layout
+
+\end_inset
-\layout Standard
-[label=optimality]
-\end_inset
+\begin_inset Argument 4
+status open
+
+\begin_layout Plain Layout
The Approximation Scheme is Optimal
-\layout Theorem
+\end_layout
+
+\end_inset
+
+
+\begin_inset Separator parbreak
+\end_inset
+
+\end_layout
+
+\begin_deeper
+\begin_layout Theorem
Suppose there exists an approximation scheme for
\begin_inset Formula $\Lang{tournament-shortest-path}$
-\end_inset
+\end_inset
that needs space
\begin_inset Formula $O\bigl(\log|V|\log^{1-\epsilon}\frac{1}{r-1}\bigr)$
-\end_inset
+\end_inset
.
Then
\begin_inset Formula $\Class{NL}\subseteq\Class{DSPACE}\bigl[\log^{2-\epsilon}n\bigr]$
-\end_inset
+\end_inset
.
-\layout Proof
+\end_layout
+
+\begin_layout Proof
+\begin_inset Separator parbreak
+\end_inset
+
-\begin_deeper
-\layout Enumerate
+\end_layout
+\begin_deeper
+\begin_layout Enumerate
Suppose the approximation scheme exists.
-\newline
+\begin_inset Newline newline
+\end_inset
+
We show
\begin_inset Formula $\Lang{distance}_{\operatorname{tourn}}\in\Class{DSPACE}\bigl[\log^{2-\epsilon}n\bigr]$
-\end_inset
+\end_inset
.
-\layout Enumerate
+\end_layout
+\begin_layout Enumerate
Let
\begin_inset Formula $(T,s,t)$
-\end_inset
+\end_inset
be an input.
Let
\begin_inset Formula $n$
-\end_inset
+\end_inset
be the number of vertices.
-\layout Enumerate
+\end_layout
+\begin_layout Enumerate
Run the approximation scheme for
\begin_inset Formula $r:=1+\smash{\frac{1}{n+1}}$
-\end_inset
+\end_inset
.
-\newline
+\begin_inset Newline newline
+\end_inset
+
This needs space
\begin_inset Formula $\smash{O(\log^{2-\epsilon}n)}$
-\end_inset
+\end_inset
.
-\layout Enumerate
+\end_layout
+\begin_layout Enumerate
The resulting path has optimal length.
\begin_inset ERT
-status Collapsed
+status collapsed
-\layout Standard
+\begin_layout Plain Layout
-\backslash
+
+\backslash
qedhere
-\end_inset
+\end_layout
+
+\end_inset
-\end_deeper
-\layout EndFrame
+\end_layout
-\the_end
+\end_deeper
+\end_deeper
+\end_body
+\end_document