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 Neil Robertson GENERALIZE! 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! Cactus graph Examples Pseudoforest O u t e r p l a n a r 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.