The good news and the really bad news about discrete Morse Theory

1. The good news and the really bad news about discrete Morse Theory Parameterized Complexity of Discrete Morse Theory B. Burton, J. Spreer, J. Paixão, T. Lewiner University of Queensland PUC- Rio de Janeiro

2. Motivation Optimal description Discrete Smooth

3. Collapsing

5. Collapsing

6. Collapsing

7. No free faces!

8. Erase (Remove)

9. Criticaltriangle

10. Example

11. Collapse

12. No free faces

13. Remove

14. Collapse

15. Keepcollapsing

16. No free faces

17. Remove

18. Collaspeaway

19. Collapsethegraph

20. Spanning tree

21. Onecriticalvertexleft

22. Main Theorem of Discrete Morse Theory Take home message: only critical simplicies matter!

23. Torus example Optimal description (CW complex) 1 critical vertex 2 critical edges 1 critical face Discrete (Cell complex) Smooth Goal: Minimize number of critical cells

24. Collapsing surfaces is easy! Primalspanningtree Dual spanningtree Tree-cotree decomposition [von Staudt 1847; Eppstein 2003; Lewiner 2003] Images from J. Erickson 2011

25. Collapsing non-surfaces is hard! • NP-hard • Reduction to Set Cover • Try every set of critical simpliciesO(nk) • Can we do better than O(nk)?

26. How hard is Collapsibility? k-Collapsibility is at least as hard as k-Set Cover If W[1]=FPT then there is something better than brute force for 3-SAT

27. How many hard gates? (remove slide ?) Independent set is W[1]-complete

28. W-hierarchy (remove slide?) Dominating set is W[2]-complete

29. Axiom Set Implications Statements • Choose k statements to be the axioms • Make every other statement true A B and E => A C B A and B and C => D D E C and E => B

30. Axiom Set Implications 2 Axioms • Choose k statements to be the axioms • Make every other statement true B and E => A C A and B and C => D E C and E => B

31. Axiom Set Implications 2 Axioms • Choose k statements to be the axioms • Make every other statement true B and E => A C B A and B and C => D E C and E => B

32. Axiom Set Implications 2 Axioms • Choose k statements to be the axioms • Make every other statement true A B and E => A C B A and B and C => D E C and E => B

33. Axiom Set Implications 2 Axioms • Choose k statements to be the axioms • Make every other statement true A B and E => A C B A and B and C => D D E C and E => B

34. Axiom set reduces to Erasability A and B and C => D D C B A

35. Implication gadget

36. Implication gadget

37. Implication gadget

38. Implication gadget

39. Implication gadget

40. Implication gadget

41. Implication gadget

42. Implication gadget

43. Implication gadget

44. Implication gadget • Lemma: White sphere is collapsible if and only if every other sphere is collapsed.

45. Combining the gadgets

46. Really Bad News • When parameter K = # of critical triangles • Erasability is W[P]-complete “All bad news must be accepted calmly, as if one already knew and didn't care.” Michael Korda

47. Treewidth • Tree-width of a graph measures its similarity to a tree TW(G) = 3 Other examples: TW(tree) = 1 TW(cycle) =2

48. Graphs • Adjacency graph of 2-complex • Triangles and edges of 2-complex are vertices of adjacency graph • Dual graph of 3-manifold • Tetrahedra of 3-manifold are vertices of dual graph • Triangles of 3-manifold are edges are edges if dual graph

49. Good news before the coffee break • If adjacency graph of the 2-complex is a k-tree, thenHALF-COLLAPSIBILITY is polynomial • If dual graph of 3-manifold is a k-tree, then COLLAPSIBILITY is polynomial “The good news is it’s curable, the bad news is you can’t afford it.” Doctor to patient

50. Future Directions • Improve onf(k) • If the graph is planar is still NP-complete or W[P]-complete? • Topological restriction  Forbidden Minors • What topological restriction makes the problems NP-complete • Can you always triangulate a 3-manifold such that the dual graph has bounded treewidth?