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

Motivation Smooth Discrete Optimal description

Collapsing

No free faces!

Erase (Remove)

Critical triangle

Example

Collapse

No free faces

Remove

Collapse

Keep collapsing

No free faces

Remove

Collaspe away

Collapse the graph

Spanning tree

One critical vertex left

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

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

Collapsing surfaces is easy! Images from J. Erickson 2011 Tree-cotree decomposition [von Staudt 1847; Eppstein 2003; Lewiner 2003] Primal spanning treeDual spanning tree

Collapsing non-surfaces is hard! NP-hard Reduction to Set Cover Try every set of critical simplicies O(n k ) Can we do better than O(n k )?

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

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

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

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

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

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

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

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

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

Implication gadget

Lemma: White sphere is collapsible if and only if every other sphere is collapsed.

Combining the gadgets

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

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

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

Good news before the coffee break If adjacency graph of the 2-complex is a k-tree, then HALF-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

Future Directions Improve on f(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?