Joint with

1
Algorithms for Non-crossing Spanning Trees
Magnús M. Halldórsson

• Joint with
• Christian Knauer Freie U., Berlin
• Andreas Spillner Jena
• Takeshi Tokuyama Tohoku University
• Alexander Wolff University of Karlsruhe

2
Geometric graphs

• Points (vertices), and
• lines (edges)embedded in the plane

3
Topological graphs

• Points (vertices), and
• curves (edges)embedded in the plane

4
Non-Crossing Spanning Tree

• Set of edges that
• No two overlap
• Involve all vertices
• Form a tree

5
NP-hardness

• Does topological graph G contain a NCST is
  an NP-complete problem Kratochvil,
  Lubiw, Nesetril, 91
• Same for geometric graphs
  Jansen, Woeginger, 9x

• ERGO We (almost surely) cant find efficient
  algorithms

THEN WHAT?
Parameterize
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Input parameters

• Crossing pair of edges that cross
• k crossings
• Crossedge edge that crosses other edges
• ? crossedges

k 2 ? 2
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Recent results for NCST

• Knauer,Schramm,Spillner,Wolff, 2005
• FPT
• O(2k) time algorithm
• Approximation
• k1-? ratio is NP-hard!
• k ratio is trivial

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O(2k) algorithm

• Pick an edge e that crosses other edges
• Either e is in the solution or not in.
• Try both possibilities, recursively!

Original problem instance and its measure
Recurrence tree
k
k-1
k-1
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Improved results

• Knauer,Schramm,Spillner,Wolff Dec95
• O(?k) time, where ??1.9

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Improved results

• Knauer,Schramm,Spillner,Wolff Dec95
• O(?k) time, where ??1.99

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Improved results

• Knauer,Schramm,Spillner,Wolff Dec95
• O(?k) time, where ??1.999

12
Improved results

• Knauer,Schramm,Spillner,Wolff Dec95
• O(?k) time, where ??1.9999

13
Improved results

• Knauer,Schramm,Spillner,Wolff Dec95
• O(?k) time, where ??1.99999

14
Improved results

• Knauer,Schramm,Spillner,Wolff Dec95
• O(?k) time, where ??1.999999

15
Improved results

• Knauer,Schramm,Spillner,Wolff Dec95
• O(?k) time, where ??1.9999992

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Improved results

• Knauer,Schramm,Spillner,Wolff Dec95
• O(?k) time, where ??1.9999992
• Here
• c?k time
• Matching lower bound

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Outline of our approach

• Simplify the instance
• Find a small graph separator
• Guess which edges to use
• Guess their configuration how they connect the
  rest of the graph
• Recursively solve left half
• Recursively solve right half

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Outline of our approach

• Simplify the instance
• Kernelize Obtain an equivalent graph on O(k)
  vertices (only those involved in crossing edges)
• Degree reduction Obtain equivalent graph where
  each vertex has degree lt 3
• Multiplicity reduction Only two edges cross in
  the same point in ?2

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Outline of our approach

• Simplify the instance
• Find a small graph separator

S? c??n, G1? 2n/3, G2? 2n/3 Lipton, Tarjan
79
S
G1
G2
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Outline of our approach

• Simplify the instance
• Find a small graph separator

Edge-cut C
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Outline of our approach

• Simplify the instance
• Find a small graph separator
• Guess which edges to use

22
Outline of our approach

• Simplify the instance
• Find a small graph separator
• Guess which edges to use
• Guess their configuration how they connect the
  rest of the graph

23
Outline of our approach

• Simplify the instance
• Find a small graph separator
• Guess which edges to use
• Guess their configuration how they connect the
  rest of the graph
• Recursively solve left half

24
Outline of our approach

• Simplify the instance
• Find a small graph separator
• Guess which edges to use
• Guess their configuration how they connect the
  rest of the graph
• Recursively solve left half

25
Outline of our approach

• Simplify the instance
• Find a small graph separator
• Guess which edges to use
• Guess their configuration how they connect the
  rest of the graph
• Recursively solve left half
• Recursively solve right half

26
Outline of our approach

• Simplify the instance
• Find a small graph separator
• Guess which edges to use
• Guess their configuration how they connect the
  rest of the graph
• Recursively solve left half
• Recursively solve right half

27
Outline of our approach

• Simplify the instance
• Find a small graph separator
• Guess which edges to use
• Guess their configuration how they connect the
  rest of the graph
• Recursively solve left half
• Recursively solve right half

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Sketch of analysis

• Kernelization implies n O(k)
• Let s O(?n) be vertex separator size
• s O(s) O(?n) is edge separator size
• Time complexity
• T(n) ? separator edge subsets
  spanning forests of left half cost of
  recursive problems ? 2s ss T(n)
  T(n-n) ? nO(?n) T(n/3) T(2n/3) ?
  nO(?n)

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Sketch of analysis, improved

• spanning plane forests of s points is only
  exp(s)
• Time complexity
• T(n) ? separator edge subsets
  spanning forests of left half cost of
  recursive problems ? 2s exp(s) T(n)
  T(n-n) ? c?n T(n/3) T(2n/3) ? cO(?n)

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Lower bound

• If we can solve NCST in time exp(f(n)), then we
  can solve SAT in time exp(f(n)2)
• Reduction, through Planar SAT
• Cor c?k time is the best we can hope for

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Further results

• Several generalizations possible
• Various non-crossing problems (paths, cycles)
• Optimization crossings left, components
• Similar measures crossing edges, crossing
  points
• Different measure i, nodes inside convex hull
• tw(G) O(sqrt(i))
• iO(i) algorithm, exponential space

32
Further results

• Several generalizations possible
• Various non-crossing problems (paths, cycles)
• Optimization crossings left, components
• Measure crossing edges, crossing points
• Can apply technique to other problem
• Min Connected Dominating Set in planar graphs
  (but already done by Fomin et al. 06)