We discuss versions of isolation lemma for problems in NP and NL. We then show that improving the success probability of the isolation lemma is equivalent to some complexity theoretic collapses such as NP is in P/poly and NL is in L/poly. Basic familiarity with complexity classes such as NP, P/poly, NL will be assumed. All the other notions used in the talk will be introduced during the talk.

1. Isolation Lemma for Directed Reachability and NL vs. L
Nutan Limaye (IIT Bombay)
Joint work with Vaibhav Krishan
Indian National Mathematics Initiative
Workshop on Complexity Theory
Nov 04 - Nov 06, 2016

2. Isolation lemma for NP
SAT
Given a Boolean formula φ determine whether φ has a satisfying
assignment.

3. Isolation lemma for NP
SAT
Given a Boolean formula φ determine whether φ has a satisfying
assignment.
Isolation Procedure for SAT

4. Isolation lemma for NP
SAT
Given a Boolean formula φ determine whether φ has a satisfying
assignment.
Isolation Procedure for SAT
Given: a Boolean forumla φ on n input variables,

5. Isolation lemma for NP
SAT
Given a Boolean formula φ determine whether φ has a satisfying
assignment.
Isolation Procedure for SAT
Given: a Boolean forumla φ on n input variables,
Output: a new formula ψ on the same n variables such that

6. Isolation lemma for NP
SAT
Given a Boolean formula φ determine whether φ has a satisfying
assignment.
Isolation Procedure for SAT
Given: a Boolean forumla φ on n input variables,
Output: a new formula ψ on the same n variables such that
every satisfying assignment of ψ also satisfies φ

7. Isolation lemma for NP
SAT
Given a Boolean formula φ determine whether φ has a satisfying
assignment.
Isolation Procedure for SAT
Given: a Boolean forumla φ on n input variables,
Output: a new formula ψ on the same n variables such that
every satisfying assignment of ψ also satisfies φ
This implies that if φ is not satisfiable then ψ is also not satisfiable.

8. Isolation lemma for NP
SAT
Given a Boolean formula φ determine whether φ has a satisfying
assignment.
Isolation Procedure for SAT
Given: a Boolean forumla φ on n input variables,
Output: a new formula ψ on the same n variables such that
every satisfying assignment of ψ also satisfies φ
This implies that if φ is not satisfiable then ψ is also not satisfiable.
if φ is satisfiable, then ψ has exactly one satisfying assignment.

9. Isolation lemma for NP
SAT
Given a Boolean formula φ determine whether φ has a satisfying
assignment.
Isolation Procedure for SAT
Given: a Boolean forumla φ on n input variables,
Output: a new formula ψ on the same n variables such that
every satisfying assignment of ψ also satisfies φ
This implies that if φ is not satisfiable then ψ is also not satisfiable.
if φ is satisfiable, then ψ has exactly one satisfying assignment.
Valiant Vazirani Isolation Lemma

10. Isolation lemma for NP
SAT
Given a Boolean formula φ determine whether φ has a satisfying
assignment.
Isolation Procedure for SAT
Given: a Boolean forumla φ on n input variables,
Output: a new formula ψ on the same n variables such that
every satisfying assignment of ψ also satisfies φ
This implies that if φ is not satisfiable then ψ is also not satisfiable.
if φ is satisfiable, then ψ has exactly one satisfying assignment.
Valiant Vazirani Isolation Lemma
There is a randomized polynomial time isolation procedure for SAT
with success probability Ω(1
n )

11. Bumping up the success probability
Determinizing Isolation Procedure for SAT
Can one get rid of the randomness in the Isolation Procedure for
SAT?

12. Bumping up the success probability
Determinizing Isolation Procedure for SAT
Can one get rid of the randomness in the Isolation Procedure for
SAT? Open!
Success probability of the Isolation Procedure for SAT

13. Bumping up the success probability
Determinizing Isolation Procedure for SAT
Can one get rid of the randomness in the Isolation Procedure for
SAT? Open!
Success probability of the Isolation Procedure for SAT
Can one make the success probability of the Isolation Procedure
higher?

14. Bumping up the success probability
Determinizing Isolation Procedure for SAT
Can one get rid of the randomness in the Isolation Procedure for
SAT? Open!
Success probability of the Isolation Procedure for SAT
Can one make the success probability of the Isolation Procedure
higher?
The success probability of the Isolation Procedure for SAT can be
made greater than 2/3 if and only if NP ⊆ P/poly.

15. Bumping up the success probability
Determinizing Isolation Procedure for SAT
Can one get rid of the randomness in the Isolation Procedure for
SAT? Open!
Success probability of the Isolation Procedure for SAT
Can one make the success probability of the Isolation Procedure
higher?
The success probability of the Isolation Procedure for SAT can be
made greater than 2/3 if and only if NP ⊆ P/poly.
[DKvMW] Is Valiant Vazinai’s isolation probability improvable? Dell, Kabanets, van
Melkebeek, Watanabe, Computational Complexity, 2013.

16. NL, L, Directed Reachability, L/poly
Complexity classes NL, L, L/poly

17. NL, L, Directed Reachability, L/poly
Complexity classes NL, L, L/poly
L: the class of decision problems decidable by dereministic Turing
machines using O(log n) space, where n is the length of the input.

18. NL, L, Directed Reachability, L/poly
Complexity classes NL, L, L/poly
L: the class of decision problems decidable by dereministic Turing
machines using O(log n) space, where n is the length of the input.
NL: the class of decision problems decidable by non-dereministic
Turing machine using O(log n) space, where n is the length of the
input.

19. NL, L, Directed Reachability, L/poly
Complexity classes NL, L, L/poly
L: the class of decision problems decidable by dereministic Turing
machines using O(log n) space, where n is the length of the input.
NL: the class of decision problems decidable by non-dereministic
Turing machine using O(log n) space, where n is the length of the
input.
L/poly: the class of decision problems decidable by dereministic
Turing machine using O(log n) space and poly(n) amount of advice,
where n is the length of the input.

20. NL, L, Directed Reachability, L/poly
Complexity classes NL, L, L/poly
L: the class of decision problems decidable by dereministic Turing
machines using O(log n) space, where n is the length of the input.
NL: the class of decision problems decidable by non-dereministic
Turing machine using O(log n) space, where n is the length of the
input.
L/poly: the class of decision problems decidable by dereministic
Turing machine using O(log n) space and poly(n) amount of advice,
where n is the length of the input.
Directed Reachability, Reach
Given: A directed graph G = (V , E) and two designated vertices s, t

21. NL, L, Directed Reachability, L/poly
Complexity classes NL, L, L/poly
L: the class of decision problems decidable by dereministic Turing
machines using O(log n) space, where n is the length of the input.
NL: the class of decision problems decidable by non-dereministic
Turing machine using O(log n) space, where n is the length of the
input.
L/poly: the class of decision problems decidable by dereministic
Turing machine using O(log n) space and poly(n) amount of advice,
where n is the length of the input.
Directed Reachability, Reach
Given: A directed graph G = (V , E) and two designated vertices s, t
Output: yes if and only if there is a directed path from s to t in G.

22. Our result
Isolation Procedure for Directed Reachability

23. Our result
Isolation Procedure for Directed Reachability
Given: a graph G on n input vertices, and two designated vertices s, t

24. Our result
Isolation Procedure for Directed Reachability
Given: a graph G on n input vertices, and two designated vertices s, t
Output: a new graph H on the same n variables such that

25. Our result
Isolation Procedure for Directed Reachability
Given: a graph G on n input vertices, and two designated vertices s, t
Output: a new graph H on the same n variables such that
every s to t path in H is also a path in G

26. Our result
Isolation Procedure for Directed Reachability
Given: a graph G on n input vertices, and two designated vertices s, t
Output: a new graph H on the same n variables such that
every s to t path in H is also a path in G
This implies that in G if t is not reachable from s then in H as well
there is no s to t path.

27. Our result
Isolation Procedure for Directed Reachability
Given: a graph G on n input vertices, and two designated vertices s, t
Output: a new graph H on the same n variables such that
every s to t path in H is also a path in G
This implies that in G if t is not reachable from s then in H as well
there is no s to t path.
if G has an s to t path, then H has exactly one s to t path.

28. Our result
Isolation Procedure for Directed Reachability
Given: a graph G on n input vertices, and two designated vertices s, t
Output: a new graph H on the same n variables such that
every s to t path in H is also a path in G
This implies that in G if t is not reachable from s then in H as well
there is no s to t path.
if G has an s to t path, then H has exactly one s to t path.
Success probability of the Isolation Procedure for Reach
Theorem (Main result)
There exists a randomized isolation procedure for Reach with success
probability greater than 2/3 if and only if NL ⊆ L/poly.

29. Isolation for other classes
Isolation for LogCFL

30. Isolation for other classes
Isolation for LogCFL
There exists a randomized isolation procedure for a hard problem in
LogCFL that runs in L/poly with success probability greater than 2/3
if and only if LogCFL ⊆ L/poly.

31. Isolation for other classes
Isolation for LogCFL
There exists a randomized isolation procedure for a hard problem in
LogCFL that runs in L/poly with success probability greater than 2/3
if and only if LogCFL ⊆ L/poly.
Isolation for NP
There exists a randomized isolation procedure for SAT that runs in
L/poly with success probability greater than 2/3 if and only if NP ⊆
L/poly.

32. Proof outline
The proofs follows a similar structure as in [DKvMW]

33. Proof outline
The proofs follows a similar structure as in [DKvMW]
We will start with some definitions.

34. Proof outline
The proofs follows a similar structure as in [DKvMW]
We will start with some definitions.
Definition (Promise sets for a version of Reach)

35. Proof outline
The proofs follows a similar structure as in [DKvMW]
We will start with some definitions.
Definition (Promise sets for a version of Reach)
YesReach = {(G, s, t) | unique reachable path between s and t},

36. Proof outline
The proofs follows a similar structure as in [DKvMW]
We will start with some definitions.
Definition (Promise sets for a version of Reach)
YesReach = {(G, s, t) | unique reachable path between s and t}, and
NoReach = {(G, s, t) | no path between s and t}

37. Proof outline
The proofs follows a similar structure as in [DKvMW]
We will start with some definitions.
Definition (Promise sets for a version of Reach)
YesReach = {(G, s, t) | unique reachable path between s and t}, and
NoReach = {(G, s, t) | no path between s and t}
Definition (PrUReach)

38. Proof outline
The proofs follows a similar structure as in [DKvMW]
We will start with some definitions.
Definition (Promise sets for a version of Reach)
YesReach = {(G, s, t) | unique reachable path between s and t}, and
NoReach = {(G, s, t) | no path between s and t}
Definition (PrUReach)
Given: G ∈ YesReach ∪ NoReach

39. Proof outline
The proofs follows a similar structure as in [DKvMW]
We will start with some definitions.
Definition (Promise sets for a version of Reach)
YesReach = {(G, s, t) | unique reachable path between s and t}, and
NoReach = {(G, s, t) | no path between s and t}
Definition (PrUReach)
Given: G ∈ YesReach ∪ NoReach
Output: yes if and only if G ∈ YesReach.

40. Proof outline
Recall the statement we wish to prove

41. Proof outline
Recall the statement we wish to prove
Theorem (Main result)
There exists a randomized isolation procedure for Reach with success
probability greater than 2/3 if and only if NL ⊆ L/poly.

42. Proof outline
Recall the statement we wish to prove
Theorem (Main result)
There exists a randomized isolation procedure for Reach with success
probability greater than 2/3 if and only if NL ⊆ L/poly.
Step 1 Prove that NL ⊆ L/poly if and only if PrUReach ∈ L/poly.

43. Proof outline
Recall the statement we wish to prove
Theorem (Main result)
There exists a randomized isolation procedure for Reach with success
probability greater than 2/3 if and only if NL ⊆ L/poly.
Step 1 Prove that NL ⊆ L/poly if and only if PrUReach ∈ L/poly.
Step 2 Prove the following statement:

44. Proof outline
Recall the statement we wish to prove
Theorem (Main result)
There exists a randomized isolation procedure for Reach with success
probability greater than 2/3 if and only if NL ⊆ L/poly.
Step 1 Prove that NL ⊆ L/poly if and only if PrUReach ∈ L/poly.
Step 2 Prove the following statement:
There exists a randomized isolation procedure for Reach with success
probability greater than 2/3 if and only if PrUReach ∈ L/poly.

45. Proof outline
Recall the statement we wish to prove
Theorem (Main result)
There exists a randomized isolation procedure for Reach with success
probability greater than 2/3 if and only if NL ⊆ L/poly.
Step 1 Prove that NL ⊆ L/poly if and only if PrUReach ∈ L/poly.
Step 2 Prove the following statement:
There exists a randomized isolation procedure for Reach with success
probability greater than 2/3 if and only if PrUReach ∈ L/poly.

46. Details of Step 1
NL ⊆ L/poly ⇔ PrUReach ∈ L/poly

47. Details of Step 1
NL ⊆ L/poly ⇔ PrUReach ∈ L/poly
⇒ This direction is trivial.

48. Details of Step 1
NL ⊆ L/poly ⇔ PrUReach ∈ L/poly
⇒ This direction is trivial.
⇐ Uses an algorithm developed in the following work.

49. Details of Step 1
NL ⊆ L/poly ⇔ PrUReach ∈ L/poly
⇒ This direction is trivial.
⇐ Uses an algorithm developed in the following work.
[RA] Making Nondeterminism Unambiguous, Reinhardt, Allender, SIAM Journal of
Computing, 2000.

50. Details of Step 1
NL ⊆ L/poly ⇔ PrUReach ∈ L/poly
⇒ This direction is trivial.
⇐ Uses an algorithm developed in the following work.
[RA] Making Nondeterminism Unambiguous, Reinhardt, Allender, SIAM Journal of
Computing, 2000.
We need one more definition.

51. Details of Step 1
NL ⊆ L/poly ⇔ PrUReach ∈ L/poly
⇒ This direction is trivial.
⇐ Uses an algorithm developed in the following work.
[RA] Making Nondeterminism Unambiguous, Reinhardt, Allender, SIAM Journal of
Computing, 2000.
We need one more definition.
Definition (Min-unique graph [GW, RA] )

52. Details of Step 1
NL ⊆ L/poly ⇔ PrUReach ∈ L/poly
⇒ This direction is trivial.
⇐ Uses an algorithm developed in the following work.
[RA] Making Nondeterminism Unambiguous, Reinhardt, Allender, SIAM Journal of
Computing, 2000.
We need one more definition.
Definition (Min-unique graph [GW, RA] )
A weighted directed graph G = (V , E) with weight function w : E → N is said to
be min-unique if between every pair of vertices the minimum weight path is
unique.

53. Breaking down Step 1
Step 1.1: Generating min-unique graph using advice

54. Breaking down Step 1
Step 1.1: Generating min-unique graph using advice
Given a graph G on n vertices a procedure P1 generates graphs
G1, G2, . . . , Gn2
For all 1 ≤ i ≤ n2
, Gi is on the same set of vertices as G.

55. Breaking down Step 1
Step 1.1: Generating min-unique graph using advice
Given a graph G on n vertices a procedure P1 generates graphs
G1, G2, . . . , Gn2
For all 1 ≤ i ≤ n2
, Gi is on the same set of vertices as G.
G has an s to t path iff ∀i ∈ [n2
], Gi has an s to t path.

56. Breaking down Step 1
Step 1.1: Generating min-unique graph using advice
Given a graph G on n vertices a procedure P1 generates graphs
G1, G2, . . . , Gn2
For all 1 ≤ i ≤ n2
, Gi is on the same set of vertices as G.
G has an s to t path iff ∀i ∈ [n2
], Gi has an s to t path.
If G has an s to t path then ∃i ∈ [n2
] : such that Gi is min-unique.

57. Breaking down Step 1
Step 1.1: Generating min-unique graph using advice
Given a graph G on n vertices a procedure P1 generates graphs
G1, G2, . . . , Gn2
For all 1 ≤ i ≤ n2
, Gi is on the same set of vertices as G.
G has an s to t path iff ∀i ∈ [n2
], Gi has an s to t path.
If G has an s to t path then ∃i ∈ [n2
] : such that Gi is min-unique.
P1 has an L/poly algorithm.

58. Breaking down Step 1
Step 1.1: Generating min-unique graph using advice
Given a graph G on n vertices a procedure P1 generates graphs
G1, G2, . . . , Gn2
For all 1 ≤ i ≤ n2
, Gi is on the same set of vertices as G.
G has an s to t path iff ∀i ∈ [n2
], Gi has an s to t path.
If G has an s to t path then ∃i ∈ [n2
] : such that Gi is min-unique.
P1 has an L/poly algorithm.
P1
advice string
G1, G2, . . . , Gn2
G

59. Breaking down Step 1
Step 1.2: Generating G ∈ YesReach ∪ NoReach given a min-unique G

60. Breaking down Step 1
Step 1.2: Generating G ∈ YesReach ∪ NoReach given a min-unique G
Given a min-unique graph G on n vertices a procedure P2 generates a
graph CG

61. Breaking down Step 1
Step 1.2: Generating G ∈ YesReach ∪ NoReach given a min-unique G
Given a min-unique graph G on n vertices a procedure P2 generates a
graph CG
On input (G, s, t), (CG , s , t ) is produced.

62. Breaking down Step 1
Step 1.2: Generating G ∈ YesReach ∪ NoReach given a min-unique G
Given a min-unique graph G on n vertices a procedure P2 generates a
graph CG
On input (G, s, t), (CG , s , t ) is produced.
G has an s to t path if and only if CG has an s to t path.

63. Breaking down Step 1
Step 1.2: Generating G ∈ YesReach ∪ NoReach given a min-unique G
Given a min-unique graph G on n vertices a procedure P2 generates a
graph CG
On input (G, s, t), (CG , s , t ) is produced.
G has an s to t path if and only if CG has an s to t path.
If G is min-unique and has an s to t path then there is a unique path
from s to t .

64. Breaking down Step 1
Step 1.2: Generating G ∈ YesReach ∪ NoReach given a min-unique G
Given a min-unique graph G on n vertices a procedure P2 generates a
graph CG
On input (G, s, t), (CG , s , t ) is produced.
G has an s to t path if and only if CG has an s to t path.
If G is min-unique and has an s to t path then there is a unique path
from s to t .
P2 has an L/poly algorithm.

65. Breaking down Step 1
Step 1.2: Generating G ∈ YesReach ∪ NoReach given a min-unique G
Given a min-unique graph G on n vertices a procedure P2 generates a
graph CG
On input (G, s, t), (CG , s , t ) is produced.
G has an s to t path if and only if CG has an s to t path.
If G is min-unique and has an s to t path then there is a unique path
from s to t .
P2 has an L/poly algorithm.
P2
advice string
(CG , s , t )
(G, s, t)

66. Breaking down Step 1
Step 1.3: Using P1, P2 to solve Reach.

67. Breaking down Step 1
Step 1.3: Using P1, P2 to solve Reach.
P be the algorithm that solves PrUReach in L/poly.

68. Breaking down Step 1
Step 1.3: Using P1, P2 to solve Reach.
P be the algorithm that solves PrUReach in L/poly.
G P1
G1
G2
P2 CG1
P2 CG2
...
...
...
Gn2 P2 CGn2
P
Yes iff P ac-
cepts one of
the graphs

69. Breaking down Step 1
Step 1.3: Using P1, P2 to solve Reach.
P be the algorithm that solves PrUReach in L/poly.
G P1
G1
G2
P2 CG1
P2 CG2
...
...
...
Gn2 P2 CGn2
P
Yes iff P ac-
cepts one of
the graphs
If G does not have an s to t path, we reject.

70. Breaking down Step 1
Step 1.3: Using P1, P2 to solve Reach.
P be the algorithm that solves PrUReach in L/poly.
G P1
G1
G2
P2 CG1
P2 CG2
...
...
...
Gn2 P2 CGn2
P
Yes iff P ac-
cepts one of
the graphs
If G does not have an s to t path, we reject.
If G has an s to t path,

71. Breaking down Step 1
Step 1.3: Using P1, P2 to solve Reach.
P be the algorithm that solves PrUReach in L/poly.
G P1
G1
G2
P2 CG1
P2 CG2
...
...
...
Gn2 P2 CGn2
P
Yes iff P ac-
cepts one of
the graphs
If G does not have an s to t path, we reject.
If G has an s to t path,
at least one of the Gi is min-unique.

72. Breaking down Step 1
Step 1.3: Using P1, P2 to solve Reach.
P be the algorithm that solves PrUReach in L/poly.
G P1
G1
G2
P2 CG1
P2 CG2
...
...
...
Gn2 P2 CGn2
P
Yes iff P ac-
cepts one of
the graphs
If G does not have an s to t path, we reject.
If G has an s to t path,
at least one of the Gi is min-unique.
The corresponding CGi
∈ YesReach.

73. Details about Step 1.2
Step 1.2: Generating G ∈ YesReach ∪ NoReach given a min-unique G

74. Details about Step 1.2
Step 1.2: Generating G ∈ YesReach ∪ NoReach given a min-unique G
Given a min-unique graph G on n vertices
On input (G, s, t), (CG , s , t ) is produced.
G has an s to t path if and only if CG has an s to t path.

75. Details about Step 1.2
Step 1.2: Generating G ∈ YesReach ∪ NoReach given a min-unique G
Given a min-unique graph G on n vertices
On input (G, s, t), (CG , s , t ) is produced.
G has an s to t path if and only if CG has an s to t path.
If G is min-unique then there is a unique path from s to t .

76. Details about Step 1.2
Step 1.2: Generating G ∈ YesReach ∪ NoReach given a min-unique G
Given a min-unique graph G on n vertices
On input (G, s, t), (CG , s , t ) is produced.
G has an s to t path if and only if CG has an s to t path.
If G is min-unique then there is a unique path from s to t .
[RA] If G is min-unqiue then there is a UL algorithm that decides the
reachability in G.

77. Details about Step 1.2
Step 1.2: Generating G ∈ YesReach ∪ NoReach given a min-unique G
Given a min-unique graph G on n vertices
On input (G, s, t), (CG , s , t ) is produced.
G has an s to t path if and only if CG has an s to t path.
If G is min-unique then there is a unique path from s to t .
[RA] If G is min-unqiue then there is a UL algorithm that decides the
reachability in G.
CG is the configuration graph of the UL algorithm on input G.

78. Details about Step 1.2
Step 1.2: Generating G ∈ YesReach ∪ NoReach given a min-unique G
Given a min-unique graph G on n vertices
On input (G, s, t), (CG , s , t ) is produced.
G has an s to t path if and only if CG has an s to t path.
If G is min-unique then there is a unique path from s to t .
[RA] If G is min-unqiue then there is a UL algorithm that decides the
reachability in G.
CG is the configuration graph of the UL algorithm on input G.
CG can be computed in L.

79. Proof outline
Recall the statement we wish to prove

80. Proof outline
Recall the statement we wish to prove
Theorem (Main result)
There exists a randomized isolation procedure for Reach with success
probability greater than 2/3 if and only if NL ⊆ L/poly.

81. Proof outline
Recall the statement we wish to prove
Theorem (Main result)
There exists a randomized isolation procedure for Reach with success
probability greater than 2/3 if and only if NL ⊆ L/poly.
Step 1 Prove that NL ⊆ L/poly if and only if PrUReach ∈ L/poly.

82. Proof outline
Recall the statement we wish to prove
Theorem (Main result)
There exists a randomized isolation procedure for Reach with success
probability greater than 2/3 if and only if NL ⊆ L/poly.
Step 1 Prove that NL ⊆ L/poly if and only if PrUReach ∈ L/poly.
Step 2 Prove the following statement:

83. Proof outline
Recall the statement we wish to prove
Theorem (Main result)
There exists a randomized isolation procedure for Reach with success
probability greater than 2/3 if and only if NL ⊆ L/poly.
Step 1 Prove that NL ⊆ L/poly if and only if PrUReach ∈ L/poly.
Step 2 Prove the following statement:
There exists a randomized isolation procedure for Reach with success
probability greater than 2/3 if and only if PrUReach ∈ L/poly.

84. Proof outline
Recall the statement we wish to prove
Theorem (Main result)
There exists a randomized isolation procedure for Reach with success
probability greater than 2/3 if and only if NL ⊆ L/poly.
Step 1 Prove that NL ⊆ L/poly if and only if PrUReach ∈ L/poly.
Step 2 Prove the following statement:
There exists a randomized isolation procedure for Reach with success
probability greater than 2/3 if and only if PrUReach ∈ L/poly.

85. Details about Step 2
From the hypothesis we can show that there is an L/poly procedure,
say B s.t.

86. Details about Step 2
From the hypothesis we can show that there is an L/poly procedure,
say B s.t.
Given a graph G as input, it outputs G1, G2, . . . , Gt such that
> 2/3 fraction of the Gi s have unique s to t paths.

87. Details about Step 2
From the hypothesis we can show that there is an L/poly procedure,
say B s.t.
Given a graph G as input, it outputs G1, G2, . . . , Gt such that
> 2/3 fraction of the Gi s have unique s to t paths.
B
advice string
G1, G2, . . . , Gt
G

88. Details about Step 2
From the hypothesis we can show that there is an L/poly procedure,
say B s.t.
Given a graph G as input, it outputs G1, G2, . . . , Gt such that
> 2/3 fraction of the Gi s have unique s to t paths.
B
advice string
G1, G2, . . . , Gt
G
H be a graph with π as its s to t path.

89. Details about Step 2
From the hypothesis we can show that there is an L/poly procedure,
say B s.t.
Given a graph G as input, it outputs G1, G2, . . . , Gt such that
> 2/3 fraction of the Gi s have unique s to t paths.
B
advice string
G1, G2, . . . , Gt
G
H be a graph with π as its s to t path.
For two graph G, H we say that (G, H) is good if π is a reachable
path in < 2/3 fraction of graphs in B(G + H).

90. Details about Step 2
From the hypothesis we can show that there is an L/poly procedure,
say B s.t.
Given a graph G as input, it outputs G1, G2, . . . , Gt such that
> 2/3 fraction of the Gi s have unique s to t paths.
B
advice string
G1, G2, . . . , Gt
G
H be a graph with π as its s to t path.
For two graph G, H we say that (G, H) is good if π is a reachable
path in < 2/3 fraction of graphs in B(G + H).
Let B(G + H) = G1, G2, . . . , Gt .

91. Details about Step 2
Properties of a good pair (G, H)

92. Details about Step 2
Properties of a good pair (G, H)
If (G, H) is good then G ∈ YesReach.

93. Details about Step 2
Properties of a good pair (G, H)
If (G, H) is good then G ∈ YesReach.
If (G, H) good then there is a ρ such that ρ = π

94. Details about Step 2
Properties of a good pair (G, H)
If (G, H) is good then G ∈ YesReach.
If (G, H) good then there is a ρ such that ρ = π and ρ is a unique
reachable path in some Gi

95. Details about Step 2
Properties of a good pair (G, H)
If (G, H) is good then G ∈ YesReach.
If (G, H) good then there is a ρ such that ρ = π and ρ is a unique
reachable path in some Gi , that ρ is a reachable path in G.

96. Details about Step 2
Properties of a good pair (G, H)
If (G, H) is good then G ∈ YesReach.
If (G, H) good then there is a ρ such that ρ = π and ρ is a unique
reachable path in some Gi , that ρ is a reachable path in G. Therefore
G ∈ YesReach.

97. Details about Step 2
Properties of a good pair (G, H)
If (G, H) is good then G ∈ YesReach.
If (G, H) good then there is a ρ such that ρ = π and ρ is a unique
reachable path in some Gi , that ρ is a reachable path in G. Therefore
G ∈ YesReach.
If G, H are both in YesReach then either (G, H) or (H, G) is good.
Let π, ρ be unique s to t paths in H, G, respectively.

98. Details about Step 2
Properties of a good pair (G, H)
If (G, H) is good then G ∈ YesReach.
If (G, H) good then there is a ρ such that ρ = π and ρ is a unique
reachable path in some Gi , that ρ is a reachable path in G. Therefore
G ∈ YesReach.
If G, H are both in YesReach then either (G, H) or (H, G) is good.
Let π, ρ be unique s to t paths in H, G, respectively. (π = ρ).

99. Details about Step 2
Properties of a good pair (G, H)
If (G, H) is good then G ∈ YesReach.
If (G, H) good then there is a ρ such that ρ = π and ρ is a unique
reachable path in some Gi , that ρ is a reachable path in G. Therefore
G ∈ YesReach.
If G, H are both in YesReach then either (G, H) or (H, G) is good.
Let π, ρ be unique s to t paths in H, G, respectively. (π = ρ).
If neither good, then each π and ρ are reachable paths in 2/3 of the
Gi s.

100. Details about Step 2
Properties of a good pair (G, H)
If (G, H) is good then G ∈ YesReach.
If (G, H) good then there is a ρ such that ρ = π and ρ is a unique
reachable path in some Gi , that ρ is a reachable path in G. Therefore
G ∈ YesReach.
If G, H are both in YesReach then either (G, H) or (H, G) is good.
Let π, ρ be unique s to t paths in H, G, respectively. (π = ρ).
If neither good, then each π and ρ are reachable paths in 2/3 of the
Gi s.
This means > 1/3 of Gi s have two distinct s to t paths.

101. Details about Step 2
Properties of a good pair (G, H)
If (G, H) is good then G ∈ YesReach.
If (G, H) good then there is a ρ such that ρ = π and ρ is a unique
reachable path in some Gi , that ρ is a reachable path in G. Therefore
G ∈ YesReach.
If G, H are both in YesReach then either (G, H) or (H, G) is good.
Let π, ρ be unique s to t paths in H, G, respectively. (π = ρ).
If neither good, then each π and ρ are reachable paths in 2/3 of the
Gi s.
This means > 1/3 of Gi s have two distinct s to t paths.
This contradicts the hypothesis of B.

102. Details about Step 2
Properties of a good pair (G, H)
If (G, H) is good then G ∈ YesReach.
If (G, H) good then there is a ρ such that ρ = π and ρ is a unique
reachable path in some Gi , that ρ is a reachable path in G. Therefore
G ∈ YesReach.
If G, H are both in YesReach then either (G, H) or (H, G) is good.
Let π, ρ be unique s to t paths in H, G, respectively. (π = ρ).
If neither good, then each π and ρ are reachable paths in 2/3 of the
Gi s.
This means > 1/3 of Gi s have two distinct s to t paths.
This contradicts the hypothesis of B.
Given H, π as advice and G as input, whether (G, H) is good or not
can be decided in L/poly.

103. Wrap-up
Design the advice strings
As advice we need (H1, π1), (H2, π2), . . . , (H , π ).

104. Wrap-up
Design the advice strings
As advice we need (H1, π1), (H2, π2), . . . , (H , π ).
The advice ensures that

105. Wrap-up
Design the advice strings
As advice we need (H1, π1), (H2, π2), . . . , (H , π ).
The advice ensures thatif G ∈ YesReach then there is an Hi such that
Hi ∈ YesReach and πi is corresponding path.

106. Wrap-up
Design the advice strings
As advice we need (H1, π1), (H2, π2), . . . , (H , π ).
The advice ensures thatif G ∈ YesReach then there is an Hi such that
Hi ∈ YesReach and πi is corresponding path.
If G ∈ NoReach, then each Hi ∈ NoReach.

107. Wrap-up
Design the advice strings
As advice we need (H1, π1), (H2, π2), . . . , (H , π ).
The advice ensures thatif G ∈ YesReach then there is an Hi such that
Hi ∈ YesReach and πi is corresponding path.
If G ∈ NoReach, then each Hi ∈ NoReach.
Putting it together
Overall, this gives a L/poly algorithm for PrUReach.

108. Thank You!