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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. Points ( vertices ), and

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Algorithms for Non-crossing Spanning Trees

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  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 ICE-TCS Theory Day

  3. Topological graphs • Points (vertices), and • curves (edges)embedded in the plane ICE-TCS Theory Day

  4. Non-Crossing Spanning Tree Set of edges that: • No two overlap • Involve all vertices • Form a tree ICE-TCS Theory Day

  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) can’t find efficient algorithms THEN WHAT? Parameterize ICE-TCS Theory Day

  6. Input parameters • Crossing: pair of edges that cross • k = # crossings • Crossedge: edge that crosses other edges •  = # crossedges k = 2  = 2 ICE-TCS Theory Day

  7. Recent results for NCST [Knauer,Schramm,Spillner,Wolff, 2005] • FPT: • O*(2k) time algorithm • Approximation: • k1- ratio is NP-hard! • k ratio is trivial ICE-TCS Theory Day

  8. 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 ICE-TCS Theory Day

  9. Improved results • Knauer,Schramm,Spillner,Wolff Dec’95: • O*(k) time, where 1.9 ICE-TCS Theory Day

  10. Improved results • Knauer,Schramm,Spillner,Wolff Dec’95: • O*(k) time, where 1.99 ICE-TCS Theory Day

  11. Improved results • Knauer,Schramm,Spillner,Wolff Dec’95: • O*(k) time, where 1.999 ICE-TCS Theory Day

  12. Improved results • Knauer,Schramm,Spillner,Wolff Dec’95: • O*(k) time, where 1.9999 ICE-TCS Theory Day

  13. Improved results • Knauer,Schramm,Spillner,Wolff Dec’95: • O*(k) time, where 1.99999 ICE-TCS Theory Day

  14. Improved results • Knauer,Schramm,Spillner,Wolff Dec’95: • O*(k) time, where 1.999999 ICE-TCS Theory Day

  15. Improved results • Knauer,Schramm,Spillner,Wolff Dec’95: • O*(k) time, where 1.9999992 ICE-TCS Theory Day

  16. Improved results • Knauer,Schramm,Spillner,Wolff Dec’95: • O*(k) time, where 1.9999992 • [Here:] • ck time • Matching lower bound ICE-TCS Theory Day

  17. 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

  18. 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 ICE-TCS Theory Day

  19. 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 ICE-TCS Theory Day

  20. Outline of our approach • Simplify the instance • Find a small graph separator Edge-cut C ICE-TCS Theory Day

  21. Outline of our approach • Simplify the instance • Find a small graph separator • Guess which edges to use ICE-TCS Theory Day

  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 ICE-TCS Theory Day

  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” ICE-TCS Theory Day

  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” ICE-TCS Theory Day

  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” ICE-TCS Theory Day

  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” ICE-TCS Theory Day

  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” ICE-TCS Theory Day

  28. 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) ICE-TCS Theory Day

  29. 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) ICE-TCS Theory Day

  30. 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 ICE-TCS Theory Day

  31. 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 ICE-TCS Theory Day

  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) ICE-TCS Theory Day

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