Algorithms for Non-crossing Spanning Trees

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# Algorithms for Non-crossing Spanning Trees - PowerPoint PPT Presentation

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

Joint with

Christian Knauer Freie U., Berlin

Andreas Spillner Jena

Takeshi Tokuyama Tohoku University

Alexander Wolff University of Karlsruhe

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

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

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Non-Crossing Spanning Tree

Set of edges that:

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

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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
• 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 Dec’95:
• O*(k) time, where 1.9

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

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

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

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

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

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

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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
• 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 <= 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

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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

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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”

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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”

ICE-TCS Theory Day

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
• 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”

ICE-TCS Theory Day

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))
• i^O(i) algorithm, exponential space

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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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