The W Hierarchy

The W-Hierarchy

A Parametereized Problem

A Parametereized Problem

NP-hard for constant values of the parameter

“Easy” for small values of the parameter


An O(nk) algorithm exists.


Fixed-Parameter Tractable: f(k)poly(n)

As hard as solving Clique



As hard as solving Clique

Polynomial kernel“Stuck” with

large instances


To establish membership, reduce to X.

A problem is “hard” for a class if every problem in the class reduces to it.

Short Turing Machine Acceptance


Input A [ND] Turing machine M, an input x, and an integer k.


InputQuestionDoes M accept x in at most k steps?

A [ND] Turing machine M, an input x, and an integer k.


Independent Set


Independent Set

(qa,happy)


Independent Set

Read b and (a,b) is an edge.

(qa,happy)


Independent Set

(qa,happy) —> (QUIT)

Read b and (a,b) is an edge.

(qa,happy)


Independent Set


Read b and (a,b) is not an edge.

(qa,happy)


Independent Set


(qa,happy) —> (qa,happy); move right

Read b and (a,b) is not an edge.

(qa,happy)


Independent Set



(qa,happy)


Independent Set



Read an EOF symbol.

(qa,happy)


Independent Set



(qa,happy) —> (q[move,a],happy); move left

Read an EOF symbol.

(qa,happy)


Independent Set




(qa,happy)


Independent Set




(qa,happy)

Read a.


Independent Set




(q[move,a],happy) —> (q[read],happy); move right

(qa,happy)

Read a.


Independent Set





(qa,happy)


Independent Set





(q[read],happy) —> (qb,happy); move right

(qa,happy)


Dominating Set


Independent Set

A problem is “hard” for a class if every problem in the class reduces to it.


Encode every configuration as a vertex.


At step i, the head is at j, and the transition is δ.



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.


[Step 3, Position 1, δ]



[Step 3, Position 1, a]



[Step 3, Position 1, b]






CIRCUITS

Introducing…

CIRCUITS

Weighted Circuit Satisfiablity


Input A circuit C, and an integer k.


Input

QuestionIs there an assignment setting EXACTLY k variables to one, that satisfies C?

A circuit C, and an integer k.

Weighted Circuit Satisfiability

Independent Set

b

c

a

d

e

f g

h

t


Independent Set

b

c

a

d

e

f g

h

t t h


Independent Set

b

c

a

d

e

f g

h

t t h

AND


Independent Set

b

c

a

d

e

f g

h

t t h

NOT

AND


Independent Set


Independent Set

Vertices as Input Gates

NOT Gates

Edges as AND Gates

Output Gate [AND]


Dominating Set


Dominating Set

OR Gates


Dominating Set

OR Gates

AND Gate


Dominating Set

OR Gates

AND Gate

Set a Dominating Set to 1 and the rest to 0…


Dominating Set

OR Gates

AND Gate

Otherwise:


Independent Set

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.


Independent Set


NOT Gates

Edges as AND Gates

Output Gate [AND]


Dominating Set

OR Gates

AND Gate

Weighted Circuit Satisfiability restricted to circuits of weft 1.

Independent Set, Clique, etc.

W[1]


W[2]


W[2]

Dominating Set, Set Cover, Hitting Set

CIRCUITS & SATISFIABILITY (as we know it)


Independent Set


NOT Gates

Edges as AND Gates

Output Gate [AND]

Input Gates

NOT Gates

AND Gates

Output Gate [AND]

Input Gates

NOT Gates

AND Gates

Output Gate [AND]

(x � y) � (z � x) � · · · � (z � y)


Dominating Set

OR Gates

AND Gate

OR Gates

AND Gate

OR Gates

AND Gate

(x � y � a � b � p � q) � (z � x � r) � · · · � (z � y � p � q � z)


Weighted SAT for Appropriately Normalized & Monotone Formulas


Weighted SAT for Appropriately Normalized & Monotone Formulas

Dominating Set, Set Cover, Hitting Set

(x � y � a � b � p � q) � (z � x � r) � · · · � (z � y � p � q � z)

(x � y � a � b � p � q) � (z � x � r) � · · · � (z � y � p � q � z)

p

The W Hierarchy

Education

Transcript of The W Hierarchy