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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

Isolation Lemma for Directed Reachability and NL vs. L

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Page 1: Isolation Lemma for Directed Reachability and NL vs. L

Isolation Lemma for Directed Reachability and NL vs. L

Nutan Limaye (IIT Bombay)Joint work with Vaibhav Krishan

Indian National Mathematics InitiativeWorkshop on Complexity Theory

Nov 04 - Nov 06, 2016

Page 2: Isolation Lemma for Directed Reachability and NL vs. L

Isolation lemma for NP

SAT

Given a Boolean formula φ determine whether φ has a satisfyingassignment.

Isolation Procedure for SAT

Given: a Boolean forumla φ on n input variables,

Output: a new formula ψ on the same n variables such thatI every satisfying assignment of ψ also satisfies φ

This implies that if φ is not satisfiable then ψ is also not satisfiable.I if φ is satisfiable, then ψ has exactly one satisfying assignment.

Valiant Vazirani Isolation Lemma

There is a randomized polynomial time isolation procedure for SATwith success probability Ω( 1

n )

Page 3: Isolation Lemma for Directed Reachability and NL vs. L

Isolation lemma for NP

SAT

Given a Boolean formula φ determine whether φ has a satisfyingassignment.

Isolation Procedure for SAT

Given: a Boolean forumla φ on n input variables,

Output: a new formula ψ on the same n variables such thatI every satisfying assignment of ψ also satisfies φ

This implies that if φ is not satisfiable then ψ is also not satisfiable.I if φ is satisfiable, then ψ has exactly one satisfying assignment.

Valiant Vazirani Isolation Lemma

There is a randomized polynomial time isolation procedure for SATwith success probability Ω( 1

n )

Page 4: Isolation Lemma for Directed Reachability and NL vs. L

Isolation lemma for NP

SAT

Given a Boolean formula φ determine whether φ has a satisfyingassignment.

Isolation Procedure for SAT

Given: a Boolean forumla φ on n input variables,

Output: a new formula ψ on the same n variables such thatI every satisfying assignment of ψ also satisfies φ

This implies that if φ is not satisfiable then ψ is also not satisfiable.I if φ is satisfiable, then ψ has exactly one satisfying assignment.

Valiant Vazirani Isolation Lemma

There is a randomized polynomial time isolation procedure for SATwith success probability Ω( 1

n )

Page 5: Isolation Lemma for Directed Reachability and NL vs. L

Isolation lemma for NP

SAT

Given a Boolean formula φ determine whether φ has a satisfyingassignment.

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

I every satisfying assignment of ψ also satisfies φThis implies that if φ is not satisfiable then ψ is also not satisfiable.

I if φ is satisfiable, then ψ has exactly one satisfying assignment.

Valiant Vazirani Isolation Lemma

There is a randomized polynomial time isolation procedure for SATwith success probability Ω( 1

n )

Page 6: Isolation Lemma for Directed Reachability and NL vs. L

Isolation lemma for NP

SAT

Given a Boolean formula φ determine whether φ has a satisfyingassignment.

Isolation Procedure for SAT

Given: a Boolean forumla φ on n input variables,

Output: a new formula ψ on the same n variables such thatI every satisfying assignment of ψ also satisfies φ

This implies that if φ is not satisfiable then ψ is also not satisfiable.I if φ is satisfiable, then ψ has exactly one satisfying assignment.

Valiant Vazirani Isolation Lemma

There is a randomized polynomial time isolation procedure for SATwith success probability Ω( 1

n )

Page 7: Isolation Lemma for Directed Reachability and NL vs. L

Isolation lemma for NP

SAT

Given a Boolean formula φ determine whether φ has a satisfyingassignment.

Isolation Procedure for SAT

Given: a Boolean forumla φ on n input variables,

Output: a new formula ψ on the same n variables such thatI every satisfying assignment of ψ also satisfies φ

This implies that if φ is not satisfiable then ψ is also not satisfiable.

I if φ is satisfiable, then ψ has exactly one satisfying assignment.

Valiant Vazirani Isolation Lemma

There is a randomized polynomial time isolation procedure for SATwith success probability Ω( 1

n )

Page 8: Isolation Lemma for Directed Reachability and NL vs. L

Isolation lemma for NP

SAT

Given a Boolean formula φ determine whether φ has a satisfyingassignment.

Isolation Procedure for SAT

Given: a Boolean forumla φ on n input variables,

Output: a new formula ψ on the same n variables such thatI every satisfying assignment of ψ also satisfies φ

This implies that if φ is not satisfiable then ψ is also not satisfiable.I if φ is satisfiable, then ψ has exactly one satisfying assignment.

Valiant Vazirani Isolation Lemma

There is a randomized polynomial time isolation procedure for SATwith success probability Ω( 1

n )

Page 9: Isolation Lemma for Directed Reachability and NL vs. L

Isolation lemma for NP

SAT

Given a Boolean formula φ determine whether φ has a satisfyingassignment.

Isolation Procedure for SAT

Given: a Boolean forumla φ on n input variables,

Output: a new formula ψ on the same n variables such thatI every satisfying assignment of ψ also satisfies φ

This implies that if φ is not satisfiable then ψ is also not satisfiable.I if φ is satisfiable, then ψ has exactly one satisfying assignment.

Valiant Vazirani Isolation Lemma

There is a randomized polynomial time isolation procedure for SATwith success probability Ω( 1

n )

Page 10: Isolation Lemma for Directed Reachability and NL vs. L

Isolation lemma for NP

SAT

Given a Boolean formula φ determine whether φ has a satisfyingassignment.

Isolation Procedure for SAT

Given: a Boolean forumla φ on n input variables,

Output: a new formula ψ on the same n variables such thatI every satisfying assignment of ψ also satisfies φ

This implies that if φ is not satisfiable then ψ is also not satisfiable.I if φ is satisfiable, then ψ has exactly one satisfying assignment.

Valiant Vazirani Isolation Lemma

There is a randomized polynomial time isolation procedure for SATwith success probability Ω( 1

n )

Page 11: Isolation Lemma for Directed Reachability and NL vs. L

Bumping up the success probability

Determinizing Isolation Procedure for SAT

Can one get rid of the randomness in the Isolation Procedure forSAT?

Open!

Success probability of the Isolation Procedure for SAT

Can one make the success probability of the Isolation Procedurehigher?

The success probability of the Isolation Procedure for SAT can bemade 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.

Page 12: Isolation Lemma for Directed Reachability and NL vs. L

Bumping up the success probability

Determinizing Isolation Procedure for SAT

Can one get rid of the randomness in the Isolation Procedure forSAT? Open!

Success probability of the Isolation Procedure for SAT

Can one make the success probability of the Isolation Procedurehigher?

The success probability of the Isolation Procedure for SAT can bemade 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.

Page 13: Isolation Lemma for Directed Reachability and NL vs. L

Bumping up the success probability

Determinizing Isolation Procedure for SAT

Can one get rid of the randomness in the Isolation Procedure forSAT? Open!

Success probability of the Isolation Procedure for SAT

Can one make the success probability of the Isolation Procedurehigher?

The success probability of the Isolation Procedure for SAT can bemade 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.

Page 14: Isolation Lemma for Directed Reachability and NL vs. L

Bumping up the success probability

Determinizing Isolation Procedure for SAT

Can one get rid of the randomness in the Isolation Procedure forSAT? Open!

Success probability of the Isolation Procedure for SAT

Can one make the success probability of the Isolation Procedurehigher?

The success probability of the Isolation Procedure for SAT can bemade 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.

Page 15: Isolation Lemma for Directed Reachability and NL vs. L

Bumping up the success probability

Determinizing Isolation Procedure for SAT

Can one get rid of the randomness in the Isolation Procedure forSAT? Open!

Success probability of the Isolation Procedure for SAT

Can one make the success probability of the Isolation Procedurehigher?

The success probability of the Isolation Procedure for SAT can bemade 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.

Page 16: Isolation Lemma for Directed Reachability and NL vs. L

NL, L, Directed Reachability, L/polyComplexity classes NL, L, L/poly

L: the class of decision problems decidable by dereministic Turingmachines using O(log n) space, where n is the length of the input.

NL: the class of decision problems decidable by non-dereministicTuring machine using O(log n) space, where n is the length of theinput.

L/poly: the class of decision problems decidable by dereministicTuring 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 .

Page 17: Isolation Lemma for Directed Reachability and NL vs. L

NL, L, Directed Reachability, L/polyComplexity classes NL, L, L/poly

L: the class of decision problems decidable by dereministic Turingmachines using O(log n) space, where n is the length of the input.

NL: the class of decision problems decidable by non-dereministicTuring machine using O(log n) space, where n is the length of theinput.

L/poly: the class of decision problems decidable by dereministicTuring 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 .

Page 18: Isolation Lemma for Directed Reachability and NL vs. L

NL, L, Directed Reachability, L/polyComplexity classes NL, L, L/poly

L: the class of decision problems decidable by dereministic Turingmachines using O(log n) space, where n is the length of the input.

NL: the class of decision problems decidable by non-dereministicTuring machine using O(log n) space, where n is the length of theinput.

L/poly: the class of decision problems decidable by dereministicTuring 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 .

Page 19: Isolation Lemma for Directed Reachability and NL vs. L

NL, L, Directed Reachability, L/polyComplexity classes NL, L, L/poly

L: the class of decision problems decidable by dereministic Turingmachines using O(log n) space, where n is the length of the input.

NL: the class of decision problems decidable by non-dereministicTuring machine using O(log n) space, where n is the length of theinput.

L/poly: the class of decision problems decidable by dereministicTuring 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 .

Page 20: Isolation Lemma for Directed Reachability and NL vs. L

NL, L, Directed Reachability, L/polyComplexity classes NL, L, L/poly

L: the class of decision problems decidable by dereministic Turingmachines using O(log n) space, where n is the length of the input.

NL: the class of decision problems decidable by non-dereministicTuring machine using O(log n) space, where n is the length of theinput.

L/poly: the class of decision problems decidable by dereministicTuring 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 .

Page 21: Isolation Lemma for Directed Reachability and NL vs. L

NL, L, Directed Reachability, L/polyComplexity classes NL, L, L/poly

L: the class of decision problems decidable by dereministic Turingmachines using O(log n) space, where n is the length of the input.

NL: the class of decision problems decidable by non-dereministicTuring machine using O(log n) space, where n is the length of theinput.

L/poly: the class of decision problems decidable by dereministicTuring 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 .

Page 22: Isolation Lemma for Directed Reachability and NL vs. L

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 thatI 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 wellthere is no s to t path.

I 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 successprobability greater than 2/3 if and only if NL ⊆ L/poly.

Page 23: Isolation Lemma for Directed Reachability and NL vs. L

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 thatI 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 wellthere is no s to t path.

I 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 successprobability greater than 2/3 if and only if NL ⊆ L/poly.

Page 24: Isolation Lemma for Directed Reachability and NL vs. L

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

I every s to t path in H is also a path in GThis implies that in G if t is not reachable from s then in H as wellthere is no s to t path.

I 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 successprobability greater than 2/3 if and only if NL ⊆ L/poly.

Page 25: Isolation Lemma for Directed Reachability and NL vs. L

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 thatI 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 wellthere is no s to t path.

I 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 successprobability greater than 2/3 if and only if NL ⊆ L/poly.

Page 26: Isolation Lemma for Directed Reachability and NL vs. L

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 thatI 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 wellthere is no s to t path.

I 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 successprobability greater than 2/3 if and only if NL ⊆ L/poly.

Page 27: Isolation Lemma for Directed Reachability and NL vs. L

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 thatI 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 wellthere is no s to t path.

I 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 successprobability greater than 2/3 if and only if NL ⊆ L/poly.

Page 28: Isolation Lemma for Directed Reachability and NL vs. L

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 thatI 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 wellthere is no s to t path.

I 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 successprobability greater than 2/3 if and only if NL ⊆ L/poly.

Page 29: Isolation Lemma for Directed Reachability and NL vs. L

Isolation for other classes

Isolation for LogCFL

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

Isolation for NP

There exists a randomized isolation procedure for SAT that runs inL/poly with success probability greater than 2/3 if and only if NP ⊆L/poly.

Page 30: Isolation Lemma for Directed Reachability and NL vs. L

Isolation for other classes

Isolation for LogCFL

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

Isolation for NP

There exists a randomized isolation procedure for SAT that runs inL/poly with success probability greater than 2/3 if and only if NP ⊆L/poly.

Page 31: Isolation Lemma for Directed Reachability and NL vs. L

Isolation for other classes

Isolation for LogCFL

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

Isolation for NP

There exists a randomized isolation procedure for SAT that runs inL/poly with success probability greater than 2/3 if and only if NP ⊆L/poly.

Page 32: Isolation Lemma for Directed Reachability and NL vs. L

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, andNoReach = (G , s, t) | no path between s and t

Definition (PrUReach)

Given: G ∈ YesReach ∪ NoReach

Output: yes if and only if G ∈ YesReach.

Page 33: Isolation Lemma for Directed Reachability and NL vs. L

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, andNoReach = (G , s, t) | no path between s and t

Definition (PrUReach)

Given: G ∈ YesReach ∪ NoReach

Output: yes if and only if G ∈ YesReach.

Page 34: Isolation Lemma for Directed Reachability and NL vs. L

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, andNoReach = (G , s, t) | no path between s and t

Definition (PrUReach)

Given: G ∈ YesReach ∪ NoReach

Output: yes if and only if G ∈ YesReach.

Page 35: Isolation Lemma for Directed Reachability and NL vs. L

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,

andNoReach = (G , s, t) | no path between s and t

Definition (PrUReach)

Given: G ∈ YesReach ∪ NoReach

Output: yes if and only if G ∈ YesReach.

Page 36: Isolation Lemma for Directed Reachability and NL vs. L

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, andNoReach = (G , s, t) | no path between s and t

Definition (PrUReach)

Given: G ∈ YesReach ∪ NoReach

Output: yes if and only if G ∈ YesReach.

Page 37: Isolation Lemma for Directed Reachability and NL vs. L

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, andNoReach = (G , s, t) | no path between s and t

Definition (PrUReach)

Given: G ∈ YesReach ∪ NoReach

Output: yes if and only if G ∈ YesReach.

Page 38: Isolation Lemma for Directed Reachability and NL vs. L

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, andNoReach = (G , s, t) | no path between s and t

Definition (PrUReach)

Given: G ∈ YesReach ∪ NoReach

Output: yes if and only if G ∈ YesReach.

Page 39: Isolation Lemma for Directed Reachability and NL vs. L

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, andNoReach = (G , s, t) | no path between s and t

Definition (PrUReach)

Given: G ∈ YesReach ∪ NoReach

Output: yes if and only if G ∈ YesReach.

Page 40: Isolation Lemma for Directed Reachability and NL vs. L

Proof outline

Recall the statement we wish to prove

Theorem (Main result)

There exists a randomized isolation procedure for Reach with successprobability 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 successprobability greater than 2/3 if and only if PrUReach ∈ L/poly.

Page 41: Isolation Lemma for Directed Reachability and NL vs. L

Proof outline

Recall the statement we wish to prove

Theorem (Main result)

There exists a randomized isolation procedure for Reach with successprobability 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 successprobability greater than 2/3 if and only if PrUReach ∈ L/poly.

Page 42: Isolation Lemma for Directed Reachability and NL vs. L

Proof outline

Recall the statement we wish to prove

Theorem (Main result)

There exists a randomized isolation procedure for Reach with successprobability 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 successprobability greater than 2/3 if and only if PrUReach ∈ L/poly.

Page 43: Isolation Lemma for Directed Reachability and NL vs. L

Proof outline

Recall the statement we wish to prove

Theorem (Main result)

There exists a randomized isolation procedure for Reach with successprobability 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 successprobability greater than 2/3 if and only if PrUReach ∈ L/poly.

Page 44: Isolation Lemma for Directed Reachability and NL vs. L

Proof outline

Recall the statement we wish to prove

Theorem (Main result)

There exists a randomized isolation procedure for Reach with successprobability 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 successprobability greater than 2/3 if and only if PrUReach ∈ L/poly.

Page 45: Isolation Lemma for Directed Reachability and NL vs. L

Proof outline

Recall the statement we wish to prove

Theorem (Main result)

There exists a randomized isolation procedure for Reach with successprobability 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 successprobability greater than 2/3 if and only if PrUReach ∈ L/poly.

Page 46: Isolation Lemma for Directed Reachability and NL vs. L

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 tobe min-unique if between every pair of vertices the minimum weight path isunique.

Page 47: Isolation Lemma for Directed Reachability and NL vs. L

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 tobe min-unique if between every pair of vertices the minimum weight path isunique.

Page 48: Isolation Lemma for Directed Reachability and NL vs. L

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 tobe min-unique if between every pair of vertices the minimum weight path isunique.

Page 49: Isolation Lemma for Directed Reachability and NL vs. L

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 tobe min-unique if between every pair of vertices the minimum weight path isunique.

Page 50: Isolation Lemma for Directed Reachability and NL vs. L

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 tobe min-unique if between every pair of vertices the minimum weight path isunique.

Page 51: Isolation Lemma for Directed Reachability and NL vs. L

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 tobe min-unique if between every pair of vertices the minimum weight path isunique.

Page 52: Isolation Lemma for Directed Reachability and NL vs. L

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 tobe min-unique if between every pair of vertices the minimum weight path isunique.

Page 53: Isolation Lemma for Directed Reachability and NL vs. L

Breaking down Step 1

Step 1.1: Generating min-unique graph using advice

Given a graph G on n vertices a procedure P1 generates graphsG1,G2, . . . ,Gn2

I For all 1 ≤ i ≤ n2, Gi is on the same set of vertices as G .

I G has an s to t path iff ∀i ∈ [n2], Gi has an s to t path.

I If G has an s to t path then ∃i ∈ [n2] : such that Gi is min-unique.

I P1 has an L/poly algorithm.

P1

advice string

〈G1,G2, . . . ,Gn2〉G

Page 54: Isolation Lemma for Directed Reachability and NL vs. L

Breaking down Step 1

Step 1.1: Generating min-unique graph using advice

Given a graph G on n vertices a procedure P1 generates graphsG1,G2, . . . ,Gn2

I For all 1 ≤ i ≤ n2, Gi is on the same set of vertices as G .

I G has an s to t path iff ∀i ∈ [n2], Gi has an s to t path.

I If G has an s to t path then ∃i ∈ [n2] : such that Gi is min-unique.

I P1 has an L/poly algorithm.

P1

advice string

〈G1,G2, . . . ,Gn2〉G

Page 55: Isolation Lemma for Directed Reachability and NL vs. L

Breaking down Step 1

Step 1.1: Generating min-unique graph using advice

Given a graph G on n vertices a procedure P1 generates graphsG1,G2, . . . ,Gn2

I For all 1 ≤ i ≤ n2, Gi is on the same set of vertices as G .

I G has an s to t path iff ∀i ∈ [n2], Gi has an s to t path.

I If G has an s to t path then ∃i ∈ [n2] : such that Gi is min-unique.

I P1 has an L/poly algorithm.

P1

advice string

〈G1,G2, . . . ,Gn2〉G

Page 56: Isolation Lemma for Directed Reachability and NL vs. L

Breaking down Step 1

Step 1.1: Generating min-unique graph using advice

Given a graph G on n vertices a procedure P1 generates graphsG1,G2, . . . ,Gn2

I For all 1 ≤ i ≤ n2, Gi is on the same set of vertices as G .

I G has an s to t path iff ∀i ∈ [n2], Gi has an s to t path.

I If G has an s to t path then ∃i ∈ [n2] : such that Gi is min-unique.

I P1 has an L/poly algorithm.

P1

advice string

〈G1,G2, . . . ,Gn2〉G

Page 57: Isolation Lemma for Directed Reachability and NL vs. L

Breaking down Step 1

Step 1.1: Generating min-unique graph using advice

Given a graph G on n vertices a procedure P1 generates graphsG1,G2, . . . ,Gn2

I For all 1 ≤ i ≤ n2, Gi is on the same set of vertices as G .

I G has an s to t path iff ∀i ∈ [n2], Gi has an s to t path.

I If G has an s to t path then ∃i ∈ [n2] : such that Gi is min-unique.

I P1 has an L/poly algorithm.

P1

advice string

〈G1,G2, . . . ,Gn2〉G

Page 58: Isolation Lemma for Directed Reachability and NL vs. L

Breaking down Step 1

Step 1.1: Generating min-unique graph using advice

Given a graph G on n vertices a procedure P1 generates graphsG1,G2, . . . ,Gn2

I For all 1 ≤ i ≤ n2, Gi is on the same set of vertices as G .

I G has an s to t path iff ∀i ∈ [n2], Gi has an s to t path.

I If G has an s to t path then ∃i ∈ [n2] : such that Gi is min-unique.

I P1 has an L/poly algorithm.

P1

advice string

〈G1,G2, . . . ,Gn2〉G

Page 59: Isolation Lemma for Directed Reachability and NL vs. L

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 agraph CG

I On input (G , s, t), (CG , s′, t ′) is produced.

I G has an s to t path if and only if CG has an s ′ to t ′ path.

I If G is min-unique and has an s to t path then there is a unique pathfrom s ′ to t ′.

I P2 has an L/poly algorithm.

P2

advice string

〈(CG , s′, t ′)〉

(G , s, t)

Page 60: Isolation Lemma for Directed Reachability and NL vs. L

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 agraph CG

I On input (G , s, t), (CG , s′, t ′) is produced.

I G has an s to t path if and only if CG has an s ′ to t ′ path.

I If G is min-unique and has an s to t path then there is a unique pathfrom s ′ to t ′.

I P2 has an L/poly algorithm.

P2

advice string

〈(CG , s′, t ′)〉

(G , s, t)

Page 61: Isolation Lemma for Directed Reachability and NL vs. L

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 agraph CG

I On input (G , s, t), (CG , s′, t ′) is produced.

I G has an s to t path if and only if CG has an s ′ to t ′ path.

I If G is min-unique and has an s to t path then there is a unique pathfrom s ′ to t ′.

I P2 has an L/poly algorithm.

P2

advice string

〈(CG , s′, t ′)〉

(G , s, t)

Page 62: Isolation Lemma for Directed Reachability and NL vs. L

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 agraph CG

I On input (G , s, t), (CG , s′, t ′) is produced.

I G has an s to t path if and only if CG has an s ′ to t ′ path.

I If G is min-unique and has an s to t path then there is a unique pathfrom s ′ to t ′.

I P2 has an L/poly algorithm.

P2

advice string

〈(CG , s′, t ′)〉

(G , s, t)

Page 63: Isolation Lemma for Directed Reachability and NL vs. L

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 agraph CG

I On input (G , s, t), (CG , s′, t ′) is produced.

I G has an s to t path if and only if CG has an s ′ to t ′ path.

I If G is min-unique and has an s to t path then there is a unique pathfrom s ′ to t ′.

I P2 has an L/poly algorithm.

P2

advice string

〈(CG , s′, t ′)〉

(G , s, t)

Page 64: Isolation Lemma for Directed Reachability and NL vs. L

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 agraph CG

I On input (G , s, t), (CG , s′, t ′) is produced.

I G has an s to t path if and only if CG has an s ′ to t ′ path.

I If G is min-unique and has an s to t path then there is a unique pathfrom s ′ to t ′.

I P2 has an L/poly algorithm.

P2

advice string

〈(CG , s′, t ′)〉

(G , s, t)

Page 65: Isolation Lemma for Directed Reachability and NL vs. L

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 agraph CG

I On input (G , s, t), (CG , s′, t ′) is produced.

I G has an s to t path if and only if CG has an s ′ to t ′ path.

I If G is min-unique and has an s to t path then there is a unique pathfrom s ′ to t ′.

I P2 has an L/poly algorithm.

P2

advice string

〈(CG , s′, t ′)〉

(G , s, t)

Page 66: Isolation Lemma for Directed Reachability and NL vs. L

Breaking down Step 1Step 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.

Page 67: Isolation Lemma for Directed Reachability and NL vs. L

Breaking down Step 1Step 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.

Page 68: Isolation Lemma for Directed Reachability and NL vs. L

Breaking down Step 1Step 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.

Page 69: Isolation Lemma for Directed Reachability and NL vs. L

Breaking down Step 1Step 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.

Page 70: Isolation Lemma for Directed Reachability and NL vs. L

Breaking down Step 1Step 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.

Page 71: Isolation Lemma for Directed Reachability and NL vs. L

Breaking down Step 1Step 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.

Page 72: Isolation Lemma for Directed Reachability and NL vs. L

Breaking down Step 1Step 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.

Page 73: Isolation Lemma for Directed Reachability and NL vs. L

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 thereachability in G .

CG is the configuration graph of the UL algorithm on input G .

CG can be computed in L.

Page 74: Isolation Lemma for Directed Reachability and NL vs. L

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 thereachability in G .

CG is the configuration graph of the UL algorithm on input G .

CG can be computed in L.

Page 75: Isolation Lemma for Directed Reachability and NL vs. L

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 thereachability in G .

CG is the configuration graph of the UL algorithm on input G .

CG can be computed in L.

Page 76: Isolation Lemma for Directed Reachability and NL vs. L

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 thereachability in G .

CG is the configuration graph of the UL algorithm on input G .

CG can be computed in L.

Page 77: Isolation Lemma for Directed Reachability and NL vs. L

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 thereachability in G .

CG is the configuration graph of the UL algorithm on input G .

CG can be computed in L.

Page 78: Isolation Lemma for Directed Reachability and NL vs. L

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 thereachability in G .

CG is the configuration graph of the UL algorithm on input G .

CG can be computed in L.

Page 79: Isolation Lemma for Directed Reachability and NL vs. L

Proof outline

Recall the statement we wish to prove

Theorem (Main result)

There exists a randomized isolation procedure for Reach with successprobability 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 successprobability greater than 2/3 if and only if PrUReach ∈ L/poly.

Page 80: Isolation Lemma for Directed Reachability and NL vs. L

Proof outline

Recall the statement we wish to prove

Theorem (Main result)

There exists a randomized isolation procedure for Reach with successprobability 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 successprobability greater than 2/3 if and only if PrUReach ∈ L/poly.

Page 81: Isolation Lemma for Directed Reachability and NL vs. L

Proof outline

Recall the statement we wish to prove

Theorem (Main result)

There exists a randomized isolation procedure for Reach with successprobability 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 successprobability greater than 2/3 if and only if PrUReach ∈ L/poly.

Page 82: Isolation Lemma for Directed Reachability and NL vs. L

Proof outline

Recall the statement we wish to prove

Theorem (Main result)

There exists a randomized isolation procedure for Reach with successprobability 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 successprobability greater than 2/3 if and only if PrUReach ∈ L/poly.

Page 83: Isolation Lemma for Directed Reachability and NL vs. L

Proof outline

Recall the statement we wish to prove

Theorem (Main result)

There exists a randomized isolation procedure for Reach with successprobability 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 successprobability greater than 2/3 if and only if PrUReach ∈ L/poly.

Page 84: Isolation Lemma for Directed Reachability and NL vs. L

Proof outline

Recall the statement we wish to prove

Theorem (Main result)

There exists a randomized isolation procedure for Reach with successprobability 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 successprobability greater than 2/3 if and only if PrUReach ∈ L/poly.

Page 85: Isolation Lemma for Directed Reachability and NL vs. L

Details about Step 2From 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 reachablepath in < 2/3 fraction of graphs in B(G + H).

Let B(G + H) = 〈G1,G2, . . . ,Gt〉.

Page 86: Isolation Lemma for Directed Reachability and NL vs. L

Details about Step 2From 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 reachablepath in < 2/3 fraction of graphs in B(G + H).

Let B(G + H) = 〈G1,G2, . . . ,Gt〉.

Page 87: Isolation Lemma for Directed Reachability and NL vs. L

Details about Step 2From 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 reachablepath in < 2/3 fraction of graphs in B(G + H).

Let B(G + H) = 〈G1,G2, . . . ,Gt〉.

Page 88: Isolation Lemma for Directed Reachability and NL vs. L

Details about Step 2From 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 reachablepath in < 2/3 fraction of graphs in B(G + H).

Let B(G + H) = 〈G1,G2, . . . ,Gt〉.

Page 89: Isolation Lemma for Directed Reachability and NL vs. L

Details about Step 2From 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 reachablepath in < 2/3 fraction of graphs in B(G + H).

Let B(G + H) = 〈G1,G2, . . . ,Gt〉.

Page 90: Isolation Lemma for Directed Reachability and NL vs. L

Details about Step 2From 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 reachablepath in < 2/3 fraction of graphs in B(G + H).

Let B(G + H) = 〈G1,G2, . . . ,Gt〉.

Page 91: Isolation Lemma for Directed Reachability and NL vs. L

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 6= π and ρ is a uniquereachable path in some Gi , that ρ is a reachable path in G . ThereforeG ∈ 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 6= ρ).

If neither good, then each π and ρ are reachable paths in 2/3 of theGi 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 notcan be decided in L/poly.

Page 92: Isolation Lemma for Directed Reachability and NL vs. L

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 6= π and ρ is a uniquereachable path in some Gi , that ρ is a reachable path in G . ThereforeG ∈ 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 6= ρ).

If neither good, then each π and ρ are reachable paths in 2/3 of theGi 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 notcan be decided in L/poly.

Page 93: Isolation Lemma for Directed Reachability and NL vs. L

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 6= π

and ρ is a uniquereachable path in some Gi , that ρ is a reachable path in G . ThereforeG ∈ 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 6= ρ).

If neither good, then each π and ρ are reachable paths in 2/3 of theGi 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 notcan be decided in L/poly.

Page 94: Isolation Lemma for Directed Reachability and NL vs. L

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 6= π and ρ is a uniquereachable path in some Gi

, that ρ is a reachable path in G . ThereforeG ∈ 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 6= ρ).

If neither good, then each π and ρ are reachable paths in 2/3 of theGi 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 notcan be decided in L/poly.

Page 95: Isolation Lemma for Directed Reachability and NL vs. L

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 6= π and ρ is a uniquereachable path in some Gi , that ρ is a reachable path in G .

ThereforeG ∈ 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 6= ρ).

If neither good, then each π and ρ are reachable paths in 2/3 of theGi 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 notcan be decided in L/poly.

Page 96: Isolation Lemma for Directed Reachability and NL vs. L

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 6= π and ρ is a uniquereachable path in some Gi , that ρ is a reachable path in G . ThereforeG ∈ 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 6= ρ).

If neither good, then each π and ρ are reachable paths in 2/3 of theGi 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 notcan be decided in L/poly.

Page 97: Isolation Lemma for Directed Reachability and NL vs. L

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 6= π and ρ is a uniquereachable path in some Gi , that ρ is a reachable path in G . ThereforeG ∈ 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 6= ρ).

If neither good, then each π and ρ are reachable paths in 2/3 of theGi 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 notcan be decided in L/poly.

Page 98: Isolation Lemma for Directed Reachability and NL vs. L

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 6= π and ρ is a uniquereachable path in some Gi , that ρ is a reachable path in G . ThereforeG ∈ 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 6= ρ).

If neither good, then each π and ρ are reachable paths in 2/3 of theGi 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 notcan be decided in L/poly.

Page 99: Isolation Lemma for Directed Reachability and NL vs. L

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 6= π and ρ is a uniquereachable path in some Gi , that ρ is a reachable path in G . ThereforeG ∈ 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 6= ρ).

If neither good, then each π and ρ are reachable paths in 2/3 of theGi 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 notcan be decided in L/poly.

Page 100: Isolation Lemma for Directed Reachability and NL vs. L

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 6= π and ρ is a uniquereachable path in some Gi , that ρ is a reachable path in G . ThereforeG ∈ 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 6= ρ).

If neither good, then each π and ρ are reachable paths in 2/3 of theGi 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 notcan be decided in L/poly.

Page 101: Isolation Lemma for Directed Reachability and NL vs. L

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 6= π and ρ is a uniquereachable path in some Gi , that ρ is a reachable path in G . ThereforeG ∈ 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 6= ρ).

If neither good, then each π and ρ are reachable paths in 2/3 of theGi 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 notcan be decided in L/poly.

Page 102: Isolation Lemma for Directed Reachability and NL vs. L

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 6= π and ρ is a uniquereachable path in some Gi , that ρ is a reachable path in G . ThereforeG ∈ 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 6= ρ).

If neither good, then each π and ρ are reachable paths in 2/3 of theGi 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 notcan be decided in L/poly.

Page 103: Isolation Lemma for Directed Reachability and NL vs. L

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 thatHi ∈ 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.

Page 104: Isolation Lemma for Directed Reachability and NL vs. L

Wrap-up

Design the advice strings

As advice we need (H1, π1), (H2, π2), . . . , (H`, π`).

The advice ensures that

if G ∈ YesReach then there is an Hi such thatHi ∈ 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.

Page 105: Isolation Lemma for Directed Reachability and NL vs. L

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 thatHi ∈ 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.

Page 106: Isolation Lemma for Directed Reachability and NL vs. L

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 thatHi ∈ 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.

Page 107: Isolation Lemma for Directed Reachability and NL vs. L

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 thatHi ∈ 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.

Page 108: Isolation Lemma for Directed Reachability and NL vs. L

Thank You!