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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. Geometric graphs. Points ( vertices ), and

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algorithms for non crossing spanning trees

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

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

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

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n on c rossing s panning t ree
Non-Crossing Spanning Tree

Set of edges that:

  • No two overlap
  • Involve all vertices
  • Form a tree

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np hardness
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) can’t find efficient algorithms

THEN WHAT?

Parameterize

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input parameters
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
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 2 k algorithm
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
Improved results
  • Knauer,Schramm,Spillner,Wolff Dec’95:
    • O*(k) time, where 1.9

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improved results1
Improved results
  • Knauer,Schramm,Spillner,Wolff Dec’95:
    • O*(k) time, where 1.99

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improved results2
Improved results
  • Knauer,Schramm,Spillner,Wolff Dec’95:
    • O*(k) time, where 1.999

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improved results3
Improved results
  • Knauer,Schramm,Spillner,Wolff Dec’95:
    • O*(k) time, where 1.9999

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improved results4
Improved results
  • Knauer,Schramm,Spillner,Wolff Dec’95:
    • O*(k) time, where 1.99999

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improved results5
Improved results
  • Knauer,Schramm,Spillner,Wolff Dec’95:
    • O*(k) time, where 1.999999

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improved results6
Improved results
  • Knauer,Schramm,Spillner,Wolff Dec’95:
    • O*(k) time, where 1.9999992

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improved results7
Improved results
  • Knauer,Schramm,Spillner,Wolff Dec’95:
    • O*(k) time, where 1.9999992
  • [Here:]
    • ck time
    • Matching lower bound

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outline of our approach
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 approach1
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 <= 3
  • [Multiplicity reduction] Only two edges cross in the same point in 2

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outline of our approach2
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 approach3
Outline of our approach
  • Simplify the instance
  • Find a small graph separator

Edge-cut C

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outline of our approach4
Outline of our approach
  • Simplify the instance
  • Find a small graph separator
  • Guess which edges to use

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outline of our approach5
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

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outline of our approach6
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”

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outline of our approach7
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”

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outline of our approach8
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 approach9
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 approach10
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
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
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
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
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))
    • i^O(i) algorithm, exponential space

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further results1
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)

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