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

Parameterized Complexity of Discrete Morse TheoryB. Burton, J. Spreer, J. Paixão, T. Lewiner

University of QueenslandPUC- Rio de Janeiro

Motivation

Smooth Discrete Optimal description

Collapsing

Collapsing

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 vertex2 critical edges1 critical face

Goal: Minimize number of critical cells

Collapsing surfaces is easy!

Images from J. Erickson 2011Tree-cotree decomposition [von Staudt 1847; Eppstein 2003; Lewiner 2003]

Primal spanning tree Dual spanning tree

Collapsing non-surfaces is hard!

• NP-hard• Reduction to Set Cover• Try every set of critical simplicies O(nk)• Can we do better than O(nk)?

How hard is Collapsibility?

If W[1]=FPT then there is something better than brute force for 3-SAT

€

FPT ⊆W [1]⊆W [2]⊆W [3]⊆ ...⊆W [t]⊆W [P]⊆XP

€

O( f (k)n c )

€

O(n k )

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 SetStatements 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 Set2 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 Set2 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 Set2 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 Set2 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

Implication gadget

Implication gadget

Implication gadget

Implication gadget

Implication gadget

Implication gadget

Implication gadget

Implication gadget

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) = 1TW(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

€

O( f (k)n2)

€

O( f (k)n2)

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?