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
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FPT ⊆W [1]⊆W [2]⊆W [3]⊆ ...⊆W [t]⊆W [P]⊆XP
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O( f (k)n c )
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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
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O( f (k)n2)
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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?