1. The W-Hierarchy

2. A Parametereized Problem

3. A Parametereized Problem
NP-hard for constant
values of the parameter
“Easy” for small
values of the parameter

4. “Easy” for small
values of the parameter

5. “Easy” for small
values of the parameter
An O(nk) algorithm exists.

6. “Easy” for small
values of the parameter
An O(nk) algorithm exists.
Fixed-Parameter
Tractable: f(k)poly(n)
As hard as solving
Clique

7. “Easy” for small
values of the parameter
An O(nk) algorithm exists.
Polynomial
“Stuck” with
kernel
large instances
As hard as solving
Clique

12. To establish membership, reduce to X.

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

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

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

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

17. Short Turing Machine Acceptance

18. Short Turing Machine Acceptance

19. Short Turing Machine Acceptance

20. Short Turing Machine Acceptance

21. Short Turing Machine Acceptance

22. Short Turing Machine Acceptance

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

24. Short Turing Machine Acceptance
Input
A [ND] Turing machine M, an input x, and an integer k.
Does M accept x in at most k steps?
Question

25. Independent Set
Short Turing Machine Acceptance

26. Independent Set
Short Turing Machine Acceptance
(qa,happy)

27. Independent Set
Short Turing Machine Acceptance
(qa,happy)
Read b and (a,b) is an edge.

28. Independent Set
Short Turing Machine Acceptance
(qa,happy)
Read b and (a,b) is an edge.
(qa,happy) —> (QUIT)

29. Independent Set
Short Turing Machine Acceptance
(qa,happy)
Read b and (a,b) is not an edge.
(qa,happy) —> (QUIT)

30. Independent Set
Short Turing Machine Acceptance
(qa,happy)
Read b and (a,b) is not an edge.
(qa,happy) —> (QUIT)
(qa,happy) —> (qa,happy); move right

31. Independent Set
Short Turing Machine Acceptance
(qa,happy)
(qa,happy) —> (QUIT)
(qa,happy) —> (qa,happy); move right

32. Independent Set
Short Turing Machine Acceptance
(qa,happy)
Read an EOF symbol.
(qa,happy) —> (QUIT)
(qa,happy) —> (qa,happy); move right

33. Independent Set
Short Turing Machine Acceptance
(qa,happy)
Read an EOF symbol.
(qa,happy) —> (QUIT)
(qa,happy) —> (qa,happy); move right
(qa,happy) —> (q[move,a],happy); move left

34. Independent Set
Short Turing Machine Acceptance
(qa,happy)
(qa,happy) —> (QUIT)
(qa,happy) —> (qa,happy); move right
(qa,happy) —> (q[move,a],happy); move left

35. Independent Set
Short Turing Machine Acceptance
(qa,happy)
Read a.
(qa,happy) —> (QUIT)
(qa,happy) —> (qa,happy); move right
(qa,happy) —> (q[move,a],happy); move left

36. Independent Set
Short Turing Machine Acceptance
(qa,happy)
Read a.
(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

37. Independent Set
Short Turing Machine Acceptance
(qa,happy)
(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

38. Independent Set
Short Turing Machine Acceptance
(qa,happy)
(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

39. Dominating Set
Short Turing Machine Acceptance

40. Dominating Set
Short Turing Machine Acceptance

41. Short Turing Machine Acceptance
Independent Set

42. Short Turing Machine Acceptance
Independent Set

43. Short Turing Machine Acceptance
Independent Set

44. Short Turing Machine Acceptance
Independent Set

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

46. Short Turing Machine Acceptance

47. Short Turing Machine Acceptance
Encode every configuration as a vertex.

48. Short Turing Machine Acceptance
Encode every configuration as a vertex.
At step i, the head is at j, and the transition is δ.

49. Short Turing Machine Acceptance
Encode every configuration as a vertex.
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.

50. [Step 3, Position 1, δ]
[Step 3, Position 2, δ]
[Step 3, Position 3, δ]

51. [Step 3, Position 1, a]
[Step 3, Position 1, b]
[Step 3, Position 2, a]
[Step 3, Position 2, b]
[Step 3, Position 3, a]
[Step 3, Position 3, b]
[Step 3, Position 1, δ]
[Step 3, Position 2, δ]
[Step 3, Position 3, δ]

52. [Step 3, Position 1, a]
[Step 3, Position 1, b]
[Step 3, Position 2, a]
[Step 3, Position 2, b]
[Step 3, Position 3, a]
[Step 3, Position 3, b]
[Step 3, Position 1, δ]
[Step 3, Position 2, δ]
[Step 3, Position 3, δ]

53. [Step 3, Position 1, a]
[Step 3, Position 1, b]
[Step 3, Position 2, a]
[Step 3, Position 2, b]
[Step 3, Position 3, a]
[Step 3, Position 3, b]
[Step 3, Position 1, δ]
[Step 3, Position 2, δ]
[Step 3, Position 3, δ]

54. [Step 3, Position 1, a]
[Step 3, Position 1, b]
[Step 3, Position 2, a]
[Step 3, Position 2, b]
[Step 3, Position 3, a]
[Step 3, Position 3, b]
[Step 3, Position 1, δ]
[Step 3, Position 2, δ]
[Step 3, Position 3, δ]

55. [Step 3, Position 1, a]
[Step 3, Position 1, b]
[Step 3, Position 2, a]
[Step 3, Position 2, b]
[Step 3, Position 3, a]
[Step 3, Position 3, b]
[Step 3, Position 1, δ]
[Step 3, Position 2, δ]
[Step 3, Position 3, δ]

56. Introducing…
CIRCUITS

57. CIRCUITS

58. CIRCUITS

59. Weighted Circuit Satisfiablity

60. Weighted Circuit Satisfiablity
Input
A circuit C, and an integer k.

61. Weighted Circuit Satisfiablity
Input
A circuit C, and an integer k.
Is there an assignment setting EXACTLY k variables
to one, that satisfies C?
Question

62. Independent Set
Weighted Circuit Satisfiability
e
b
c
a
t
d
h
f
g

63. Independent Set
Weighted Circuit Satisfiability
e
b
c
a
t
d
t
h
f
g
h

64. Independent Set
Weighted Circuit Satisfiability
e
b
c
a
t
d
t
h
f
g
h
AND

65. Independent Set
Weighted Circuit Satisfiability
e
b
c
a
t
d
t
h
f
g
h
AND

66. Independent Set
Weighted Circuit Satisfiability
e
a
t
t
h
NOT
b
c
d
h
f
g
AND

67. Independent Set
Weighted Circuit Satisfiability

68. Independent Set
Weighted Circuit Satisfiability

69. Independent Set
Weighted Circuit Satisfiability

70. Independent Set
Weighted Circuit Satisfiability

71. Independent Set
Weighted Circuit Satisfiability
Vertices as
Input Gates
NOT Gates
Edges as AND Gates
Output Gate [AND]

72. Dominating Set
Weighted Circuit Satisfiability

73. Dominating Set
Weighted Circuit Satisfiability
OR Gates

74. Dominating Set
Weighted Circuit Satisfiability
OR Gates
AND Gate

75. Dominating Set
Weighted Circuit Satisfiability
OR Gates
AND Gate

76. Dominating Set
Weighted Circuit Satisfiability
OR Gates
AND Gate

77. Dominating Set
Weighted Circuit Satisfiability
OR Gates
AND Gate
Set a Dominating Set to 1 and the rest to 0…

78. Dominating Set
Weighted Circuit Satisfiability
OR Gates
AND Gate
Set a Dominating Set to 1 and the rest to 0…

79. Dominating Set
Weighted Circuit Satisfiability
OR Gates
AND Gate
Otherwise:

80. Weighted Circuit Satisfiability
Independent Set

81. Weighted Circuit Satisfiability
Independent Set

82. Weighted Circuit Satisfiability
Independent Set

83. Weighted Circuit Satisfiability
Independent Set

84. 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.

85. Independent Set
Weighted Circuit Satisfiability
Vertices as
Input Gates
NOT Gates
Edges as AND Gates
Output Gate [AND]

86. Dominating Set
Weighted Circuit Satisfiability
OR Gates
AND Gate

87. Weighted Circuit Satisfiability
restricted to circuits of weft 1.
Independent Set, Clique, etc.
W[1]

88. Weighted Circuit Satisfiability
restricted to circuits of weft 2.
W[2]

89. Weighted Circuit Satisfiability
restricted to circuits of weft 2.
Dominating Set, Set Cover, Hitting Set
W[2]

90. Weighted Circuit Satisfiability
restricted to circuits of weft 2.
Dominating Set, Set Cover, Hitting Set
W[2]

91. Weighted Circuit Satisfiability
restricted to circuits of weft 2.
Dominating Set, Set Cover, Hitting Set
W[2]

92. CIRCUITS & SATISFIABILITY
(as we know it)

93. Independent Set
Weighted Circuit Satisfiability
Vertices as
Input Gates
NOT Gates
Edges as AND Gates
Output Gate [AND]

94. Input Gates
NOT Gates
AND Gates
Output Gate [AND]

95. (x
y)
(z
x)
···
(z
y)
Input Gates
NOT Gates
AND Gates
Output Gate [AND]

96. Dominating Set
Weighted Circuit Satisfiability
OR Gates
AND Gate

97. OR Gates
AND Gate

98. (x
y
a
b
p
q)
(z
OR Gates
AND Gate
x
r)
···
(z
y
p
q
z)

101. Weighted Circuit Satisfiability
restricted to circuits of weft 2.

102. Weighted Circuit Satisfiability
restricted to circuits of weft 2.
Weighted SAT for Appropriately Normalized & Monotone Formulas

103. Weighted Circuit Satisfiability
restricted to circuits of weft 2.
Weighted SAT for Appropriately Normalized & Monotone Formulas

104. Weighted Circuit Satisfiability
restricted to circuits of weft 2.
Weighted SAT for Appropriately Normalized & Monotone Formulas

105. Weighted Circuit Satisfiability
restricted to circuits of weft 2.
Weighted SAT for Appropriately Normalized & Monotone Formulas
Dominating Set, Set Cover, Hitting Set

106. Weighted Circuit Satisfiability
restricted to circuits of weft 2.
Weighted SAT for Appropriately Normalized & Monotone Formulas
Dominating Set, Set Cover, Hitting Set

107. Weighted Circuit Satisfiability
restricted to circuits of weft 2.
Weighted SAT for Appropriately Normalized & Monotone Formulas
Dominating Set, Set Cover, Hitting Set

108. Weighted Circuit Satisfiability
restricted to circuits of weft 2.
Weighted SAT for Appropriately Normalized & Monotone Formulas
Dominating Set, Set Cover, Hitting Set

109. (x
y
a
b
p
q)
(z
x
r)
···
(z
y
p
q
z)

110. (x
y
a
b
p
q)
(z
x
r)
···
(z
y
p
q
z)

111. (x
y
a
b
p
q)
(z
p
x
r)
···
(z
y
p
q
z)

112. (x
y
a
b
p
q)
(z
p
x
r)
···
(z
y
p
q
z)