JASS 2008

Bounded

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 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 (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 is a triangle!

Treewidth: linear time 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