JASS 2008
Bounded Treewidth
Stephan Holzer Technische Universität München Plan of talk
History
Definitions
Examples
Applications Plan of talk
History
Definitions
Examples
Applications History
• Ohm’s laws:
series R1 R2
parallel R1 R2 History
Series parallel graphs
R1 R2
R3 R4
R5 R6 series parallel
R7 History
Series parallel graphs Tree structure:
R1 R2 Forget graph structure S
Essential information: R8 R3 R4 R8P
S S P
R1 R2 R3 R4 R5 R6 R7 R5 R6 series parallel
R7 Early results (1960s / 70s)
• Tree structures help! restricted to easy! e m • So NP-complete problems on general graphs are hard
• trees • series parallel graphs
Even some PSPACE-complete problems become easy Paul Seymour ZE! Neil Robertson ALI ER GEN
Tree-decomposition, Treewidth Plan of talk
History
Definitions
Examples
Applications Tree-decomposition Treewidth Plan of talk
History
Definitions
Examples
Applications Example
f d k c tw(G)=2 g j e b h i a df ij cde deg hik
abc egh Examples
Graph classes with bounded treewidth include
• Trees tw=1 • Series-parallel graphs tw=2 • Outerplanar graphs tw=3 • Halin graphs tw=3 • Pseudoforests tw=2 • Cactus graphs tw=2 • Control flow graphs arising in the compilation of structured programs
All results for bounded treewidth apply to theses classes! h st rap g fore s o u ud Examples e Cact Ps
O ut erp lan ar Planar graphs?
Grid graph Plan of talk
History
Definitions
Examples
Applications Applications
• Cholesky factorisation • Evolutionary Theory • Expert Systems • VLSI Layouts (via pathwidth) • Natural language processing • Computing Tutte-polynomial of graphs with bounded treewidth in polynomial time
• Linear time algorithms (for Independent Set)* • and other NP-complete problems*
* when the undelying class of graphs has constant bounded tree width Treewidth and Cholesky factorisation
Relation to graph: blackboard Treewidth and Cholesky factorisation Treewidth and evolutionary theory chordal?? Chordal graphs
So every induced cycle is a triangle!
Treewidth: linear time algorithm for IS Treewidth: linear time algorithm for IS
T
X X 2 1
X X X X 3 4 5 6
X X X X 7 8 9 10
X 1 Y2 Treewidth: linear time algorithm for IS Treewidth: linear time algorithm for IS Treewidth: linear time algorithm for IS Treewidth: linear time algorithm for IS Treewidth: linear time algorithm for IS Treewidth: linear time algorithm for IS many problems Treewidth: linear time algorithms for IS
General idea:
• Each table entry gives information about an equivalence class of partial solutions
• Number of equivalence classes is bounded by some constant when treewidth is bounded by a constant
• Tables can be computed using only the tables of the children of the node Treewidth: linear time algorithm for IS
Other results:
• Large classes of problems can be solved in linear time when a tree-decomposition with constant bounded treewidth is available
• Approximation results Finding tree-decompositions
? Treewidth: linear time algorithm for IS Finding tree-decompositions
•For constant treewidth k=1,2,3,4 graphs can be recognized in linear time
• Arbitrary fixed k: O(n logn) Thank You