the w hierarchy
TRANSCRIPT
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The W-Hierarchy
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A Parametereized Problem
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A Parametereized Problem
NP-hard for constant values of the parameter
“Easy” for small values of the parameter
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“Easy” for small values of the parameter
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“Easy” for small values of the parameter
An O(nk) algorithm exists.
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“Easy” for small values of the parameter
Fixed-Parameter Tractable: f(k)poly(n)
As hard as solving Clique
An O(nk) algorithm exists.
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“Easy” for small values of the parameter
As hard as solving Clique
Polynomial kernel“Stuck” with
large instances
An O(nk) algorithm exists.
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To establish membership, reduce to X.
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A problem is “hard” for a class if every problem in the class reduces to it.
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A problem is “hard” for a class if every problem in the class reduces to it.
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A problem is “hard” for a class if every problem in the class reduces to it.
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A problem is “hard” for a class if every problem in the class reduces to it.
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Short Turing Machine Acceptance
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Short Turing Machine Acceptance
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Short Turing Machine Acceptance
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Short Turing Machine Acceptance
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Short Turing Machine Acceptance
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Short Turing Machine Acceptance
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Short Turing Machine Acceptance
Input A [ND] Turing machine M, an input x, and an integer k.
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Short Turing Machine Acceptance
InputQuestionDoes M accept x in at most k steps?
A [ND] Turing machine M, an input x, and an integer k.
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Short Turing Machine Acceptance
Independent Set
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Short Turing Machine Acceptance
Independent Set
(qa,happy)
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Short Turing Machine Acceptance
Independent Set
Read b and (a,b) is an edge.
(qa,happy)
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Short Turing Machine Acceptance
Independent Set
(qa,happy) —> (QUIT)
Read b and (a,b) is an edge.
(qa,happy)
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Short Turing Machine Acceptance
Independent Set
(qa,happy) —> (QUIT)
Read b and (a,b) is not an edge.
(qa,happy)
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Short Turing Machine Acceptance
Independent Set
(qa,happy) —> (QUIT)
(qa,happy) —> (qa,happy); move right
Read b and (a,b) is not an edge.
(qa,happy)
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Short Turing Machine Acceptance
Independent Set
(qa,happy) —> (QUIT)
(qa,happy) —> (qa,happy); move right
(qa,happy)
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Short Turing Machine Acceptance
Independent Set
(qa,happy) —> (QUIT)
(qa,happy) —> (qa,happy); move right
Read an EOF symbol.
(qa,happy)
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Short Turing Machine Acceptance
Independent Set
(qa,happy) —> (QUIT)
(qa,happy) —> (qa,happy); move right
(qa,happy) —> (q[move,a],happy); move left
Read an EOF symbol.
(qa,happy)
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Short Turing Machine Acceptance
Independent Set
(qa,happy) —> (QUIT)
(qa,happy) —> (qa,happy); move right
(qa,happy) —> (q[move,a],happy); move left
(qa,happy)
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Short Turing Machine Acceptance
Independent Set
(qa,happy) —> (QUIT)
(qa,happy) —> (qa,happy); move right
(qa,happy) —> (q[move,a],happy); move left
(qa,happy)
Read a.
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Short Turing Machine Acceptance
Independent Set
(qa,happy) —> (QUIT)
(qa,happy) —> (qa,happy); move right
(qa,happy) —> (q[move,a],happy); move left
(q[move,a],happy) —> (q[read],happy); move right
(qa,happy)
Read a.
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Short Turing Machine Acceptance
Independent Set
(qa,happy) —> (QUIT)
(qa,happy) —> (qa,happy); move right
(qa,happy) —> (q[move,a],happy); move left
(q[move,a],happy) —> (q[read],happy); move right
(qa,happy)
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Short Turing Machine Acceptance
Independent Set
(qa,happy) —> (QUIT)
(qa,happy) —> (qa,happy); move right
(qa,happy) —> (q[move,a],happy); move left
(q[move,a],happy) —> (q[read],happy); move right
(q[read],happy) —> (qb,happy); move right
(qa,happy)
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Short Turing Machine Acceptance
Dominating Set
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Short Turing Machine Acceptance
Dominating Set
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Short Turing Machine Acceptance
Independent Set
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Short Turing Machine Acceptance
Independent Set
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Short Turing Machine Acceptance
Independent Set
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Short Turing Machine Acceptance
Independent Set
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A problem is “hard” for a class if every problem in the class reduces to it.
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Short Turing Machine Acceptance
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Short Turing Machine Acceptance
Encode every configuration as a vertex.
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Short Turing Machine Acceptance
At step i, the head is at j, and the transition is δ.
Encode every configuration as a vertex.
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Short Turing Machine Acceptance
At step i, the head is at j, and the transition is δ.
At step i, the symbol at j is t, and the head is not at j.
Encode every configuration as a vertex.
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[Step 3, Position 1, δ]
[Step 3, Position 2, δ]
[Step 3, Position 3, δ]
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[Step 3, Position 1, a]
[Step 3, Position 2, a]
[Step 3, Position 3, a]
[Step 3, Position 1, b]
[Step 3, Position 2, b]
[Step 3, Position 3, b]
[Step 3, Position 1, δ]
[Step 3, Position 2, δ]
[Step 3, Position 3, δ]
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[Step 3, Position 1, a]
[Step 3, Position 2, a]
[Step 3, Position 3, a]
[Step 3, Position 1, b]
[Step 3, Position 2, b]
[Step 3, Position 3, b]
[Step 3, Position 1, δ]
[Step 3, Position 2, δ]
[Step 3, Position 3, δ]
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[Step 3, Position 1, a]
[Step 3, Position 2, a]
[Step 3, Position 3, a]
[Step 3, Position 1, b]
[Step 3, Position 2, b]
[Step 3, Position 3, b]
[Step 3, Position 1, δ]
[Step 3, Position 2, δ]
[Step 3, Position 3, δ]
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[Step 3, Position 1, a]
[Step 3, Position 2, a]
[Step 3, Position 3, a]
[Step 3, Position 1, b]
[Step 3, Position 2, b]
[Step 3, Position 3, b]
[Step 3, Position 1, δ]
[Step 3, Position 2, δ]
[Step 3, Position 3, δ]
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[Step 3, Position 1, a]
[Step 3, Position 2, a]
[Step 3, Position 3, a]
[Step 3, Position 1, b]
[Step 3, Position 2, b]
[Step 3, Position 3, b]
[Step 3, Position 1, δ]
[Step 3, Position 2, δ]
[Step 3, Position 3, δ]
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CIRCUITS
Introducing…
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CIRCUITS
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CIRCUITS
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Weighted Circuit Satisfiablity
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Weighted Circuit Satisfiablity
Input A circuit C, and an integer k.
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Weighted Circuit Satisfiablity
Input
QuestionIs there an assignment setting EXACTLY k variables to one, that satisfies C?
A circuit C, and an integer k.
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Weighted Circuit Satisfiability
Independent Set
b
c
a
d
e
f g
h
t
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Weighted Circuit Satisfiability
Independent Set
b
c
a
d
e
f g
h
t t h
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Weighted Circuit Satisfiability
Independent Set
b
c
a
d
e
f g
h
t t h
AND
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Weighted Circuit Satisfiability
Independent Set
b
c
a
d
e
f g
h
t t h
AND
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Weighted Circuit Satisfiability
Independent Set
b
c
a
d
e
f g
h
t t h
NOT
AND
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Weighted Circuit Satisfiability
Independent Set
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Weighted Circuit Satisfiability
Independent Set
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Weighted Circuit Satisfiability
Independent Set
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Weighted Circuit Satisfiability
Independent Set
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Weighted Circuit Satisfiability
Independent Set
Vertices as Input Gates
NOT Gates
Edges as AND Gates
Output Gate [AND]
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Weighted Circuit Satisfiability
Dominating Set
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Weighted Circuit Satisfiability
Dominating Set
OR Gates
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Weighted Circuit Satisfiability
Dominating Set
OR Gates
AND Gate
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Weighted Circuit Satisfiability
Dominating Set
OR Gates
AND Gate
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Weighted Circuit Satisfiability
Dominating Set
OR Gates
AND Gate
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Weighted Circuit Satisfiability
Dominating Set
OR Gates
AND Gate
Set a Dominating Set to 1 and the rest to 0…
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Weighted Circuit Satisfiability
Dominating Set
OR Gates
AND Gate
Set a Dominating Set to 1 and the rest to 0…
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Weighted Circuit Satisfiability
Dominating Set
OR Gates
AND Gate
Otherwise:
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Weighted Circuit Satisfiability
Independent Set
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Weighted Circuit Satisfiability
Independent Set
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Weighted Circuit Satisfiability
Independent Set
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Weighted Circuit Satisfiability
Independent Set
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The weft of a circuit is the maximum number of large nodes on
a path from an input node to the output node.
*Nodes with in-degree more than two.
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Weighted Circuit Satisfiability
Independent Set
Vertices as Input Gates
NOT Gates
Edges as AND Gates
Output Gate [AND]
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Weighted Circuit Satisfiability
Dominating Set
OR Gates
AND Gate
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Weighted Circuit Satisfiability restricted to circuits of weft 1.
Independent Set, Clique, etc.
W[1]
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Weighted Circuit Satisfiability restricted to circuits of weft 2.
W[2]
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Weighted Circuit Satisfiability restricted to circuits of weft 2.
W[2]
Dominating Set, Set Cover, Hitting Set
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Weighted Circuit Satisfiability restricted to circuits of weft 2.
W[2]
Dominating Set, Set Cover, Hitting Set
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Weighted Circuit Satisfiability restricted to circuits of weft 2.
W[2]
Dominating Set, Set Cover, Hitting Set
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CIRCUITS & SATISFIABILITY (as we know it)
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Weighted Circuit Satisfiability
Independent Set
Vertices as Input Gates
NOT Gates
Edges as AND Gates
Output Gate [AND]
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Input Gates
NOT Gates
AND Gates
Output Gate [AND]
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Input Gates
NOT Gates
AND Gates
Output Gate [AND]
(x � y) � (z � x) � · · · � (z � y)
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Weighted Circuit Satisfiability
Dominating Set
OR Gates
AND Gate
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OR Gates
AND Gate
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OR Gates
AND Gate
(x � y � a � b � p � q) � (z � x � r) � · · · � (z � y � p � q � z)
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Weighted Circuit Satisfiability restricted to circuits of weft 2.
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Weighted Circuit Satisfiability restricted to circuits of weft 2.
Weighted SAT for Appropriately Normalized & Monotone Formulas
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Weighted Circuit Satisfiability restricted to circuits of weft 2.
Weighted SAT for Appropriately Normalized & Monotone Formulas
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Weighted Circuit Satisfiability restricted to circuits of weft 2.
Weighted SAT for Appropriately Normalized & Monotone Formulas
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Weighted Circuit Satisfiability restricted to circuits of weft 2.
Weighted SAT for Appropriately Normalized & Monotone Formulas
Dominating Set, Set Cover, Hitting Set
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Weighted Circuit Satisfiability restricted to circuits of weft 2.
Weighted SAT for Appropriately Normalized & Monotone Formulas
Dominating Set, Set Cover, Hitting Set
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Weighted Circuit Satisfiability restricted to circuits of weft 2.
Weighted SAT for Appropriately Normalized & Monotone Formulas
Dominating Set, Set Cover, Hitting Set
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Weighted Circuit Satisfiability restricted to circuits of weft 2.
Weighted SAT for Appropriately Normalized & Monotone Formulas
Dominating Set, Set Cover, Hitting Set
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(x � y � a � b � p � q) � (z � x � r) � · · · � (z � y � p � q � z)
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(x � y � a � b � p � q) � (z � x � r) � · · · � (z � y � p � q � z)
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(x � y � a � b � p � q) � (z � x � r) � · · · � (z � y � p � q � z)
p
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(x � y � a � b � p � q) � (z � x � r) � · · · � (z � y � p � q � z)
p