The good news and the really bad news about discrete Morse Theory
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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
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Motivation
Smooth Discrete Optimal description
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Collapsing
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Collapsing
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Collapsing
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No free faces!
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Erase (Remove)
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Critical triangle
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Example
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No free faces
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Remove
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Keep collapsing
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No free faces
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Remove
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Collaspe away
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Collapse the graph
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Spanning tree
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One critical vertex left
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Main Theorem of Discrete Morse Theory
Take home message: only critical simplicies matter!
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Torus example
Smooth Discrete(Cell complex)
Optimal description(CW complex)
1 critical vertex2 critical edges1 critical face
Goal: Minimize number of critical cells
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Collapsing surfaces is easy!
Images from J. Erickson 2011Tree-cotree decomposition [von Staudt 1847; Eppstein 2003; Lewiner 2003]
Primal spanning tree Dual spanning tree
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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)?
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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
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How many hard gates? (remove slide ?)
Independent set is W[1]-complete
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W-hierarchy (remove slide?)
Dominating set is W[2]-complete
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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
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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
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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
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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
BA
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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
BA
D
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Axiom set reduces to Erasability
A and B and C => D
D C B A
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Implication gadget
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Implication gadget
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Implication gadget
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Implication gadget
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Implication gadget
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Implication gadget
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Implication gadget
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Implication gadget
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Implication gadget
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Implication gadget
• Lemma: White sphere is collapsible if and only if every other sphere is collapsed.
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Combining the gadgets
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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
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Treewidth
• Tree-width of a graph measures its similarity to a tree
TW(G) = 3
Other examples:TW(tree) = 1TW(cycle) =2
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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
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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)
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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?