Joint with Christian KnauerFreie U., Berlin Andreas SpillnerJena Takeshi TokuyamaTohoku University Alexander WolffUniversity of Karlsruhe Algorithms for Non-crossing Spanning Trees Magnús M. Halldórsson

31 May 2006ICE-TCS Theory Day2 Geometric graphs Points (vertices), and lines (edges) embedded in the plane

31 May 2006ICE-TCS Theory Day3 Topological graphs Points (vertices), and curves (edges) embedded in the plane

31 May 2006ICE-TCS Theory Day4 Non-Crossing Spanning Tree Set of edges that: A)No two overlap B)Involve all vertices C)Form a tree

31 May 2006ICE-TCS Theory Day5 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

31 May 2006ICE-TCS Theory Day6 Input parameters Crossing: pair of edges that cross k = # crossings Crossedge: edge that crosses other edges  = # crossedges k = 2  = 2

31 May 2006ICE-TCS Theory Day7 Recent results for NCST [Knauer,Schramm,Spillner,Wolff, 2005] FPT: –O*(2 k ) time algorithm Approximation: –k 1-  ratio is NP-hard! k ratio is trivial

31 May 2006ICE-TCS Theory Day8 O*(2 k ) algorithm Pick an edge e that crosses other edges Either e is in the solution or not in. Try both possibilities, recursively! k Original problem instance and its measure k-1 Recurrence tree

31 May 2006ICE-TCS Theory Day9 Improved results Knauer,Schramm,Spillner,Wolff Dec’95: –O*(  k ) time, where  1.9

31 May 2006ICE-TCS Theory Day10 Improved results Knauer,Schramm,Spillner,Wolff Dec’95: –O*(  k ) time, where  1.99

31 May 2006ICE-TCS Theory Day11 Improved results Knauer,Schramm,Spillner,Wolff Dec’95: –O*(  k ) time, where  1.999

31 May 2006ICE-TCS Theory Day12 Improved results Knauer,Schramm,Spillner,Wolff Dec’95: –O*(  k ) time, where 

31 May 2006ICE-TCS Theory Day13 Improved results Knauer,Schramm,Spillner,Wolff Dec’95: –O*(  k ) time, where 

31 May 2006ICE-TCS Theory Day14 Improved results Knauer,Schramm,Spillner,Wolff Dec’95: –O*(  k ) time, where 

31 May 2006ICE-TCS Theory Day15 Improved results Knauer,Schramm,Spillner,Wolff Dec’95: –O*(  k ) time, where 

31 May 2006ICE-TCS Theory Day16 Improved results Knauer,Schramm,Spillner,Wolff Dec’95: –O*(  k ) time, where  [Here:] –c  k time –Matching lower bound

31 May 2006ICE-TCS Theory Day17 Outline of our approach 1.Simplify the instance 2.Find a small graph separator 3.Guess which edges to use 4.Guess their configuration: how they connect the rest of the graph 5.Recursively solve “left half” 6.Recursively solve “right half”

31 May 2006ICE-TCS Theory Day18 Outline of our approach 1.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

31 May 2006ICE-TCS Theory Day19 Outline of our approach 1.Simplify the instance 2.Find a small graph separator G1G1 S G2G2 |S|  c  n, | G 1 |  2n/3, | G 2 |  2n/3 [Lipton, Tarjan ’79]

31 May 2006ICE-TCS Theory Day20 Outline of our approach 1.Simplify the instance 2.Find a small graph separator Edge-cut C

31 May 2006ICE-TCS Theory Day21 Outline of our approach 1.Simplify the instance 2.Find a small graph separator 3.Guess which edges to use

31 May 2006ICE-TCS Theory Day22 Outline of our approach 1.Simplify the instance 2.Find a small graph separator 3.Guess which edges to use 4.Guess their configuration: how they connect the rest of the graph

31 May 2006ICE-TCS Theory Day23 Outline of our approach 1.Simplify the instance 2.Find a small graph separator 3.Guess which edges to use 4.Guess their configuration: how they connect the rest of the graph 5.Recursively solve “left half”

31 May 2006ICE-TCS Theory Day24 Outline of our approach 1.Simplify the instance 2.Find a small graph separator 3.Guess which edges to use 4.Guess their configuration: how they connect the rest of the graph 5.Recursively solve “left half”

31 May 2006ICE-TCS Theory Day25 Outline of our approach 1.Simplify the instance 2.Find a small graph separator 3.Guess which edges to use 4.Guess their configuration: how they connect the rest of the graph 5.Recursively solve “left half” 6.Recursively solve “right half”

31 May 2006ICE-TCS Theory Day26 Outline of our approach 1.Simplify the instance 2.Find a small graph separator 3.Guess which edges to use 4.Guess their configuration: how they connect the rest of the graph 5.Recursively solve “left half” 6.Recursively solve “right half”

31 May 2006ICE-TCS Theory Day27 Outline of our approach 1.Simplify the instance 2.Find a small graph separator 3.Guess which edges to use 4.Guess their configuration: how they connect the rest of the graph 5.Recursively solve “left half” 6.Recursively solve “right half”

31 May 2006ICE-TCS Theory Day28 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  2 s * s s * [T(n’) + T(n-n’)]  n O(  n) * [T(n/3) + T(2n/3)]  n O(  n)

31 May 2006ICE-TCS Theory Day29 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  2 s * exp(s) * [T(n’) + T(n-n’)]  c  n * [T(n/3) + T(2n/3)]  c O(  n)

31 May 2006ICE-TCS Theory Day30 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

31 May 2006ICE-TCS Theory Day31 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

31 May 2006ICE-TCS Theory Day32 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)