Hardness Amplification within NP against Deterministic Algorithms

1
Hardness Amplification within NP against
Deterministic Algorithms
2
Why Hardness Amplification

• Goal Show there are hard problems in NP.
• Lower bounds out of reach.
• Cryptography, Derandomization require average
  case hardness.
• Revised Goal Relate various kinds of hardness
  assumptions.
• Hardness Amplification Start with mild hardness,
  amplify.

3
Hardness Amplification

• Generic Amplification Theorem
• If there are problems in class A that are mildly
  hard for algorithms in Z, then there are problems
  in A that are very hard for Z.

NP, EXP, PSPACE
P/poly, BPP, P
4
PSPACE versus P/poly, BPP

• Long line of work
• Theorem If there are problems in PSPACE that are
  worst case hard for P/poly (BPP), then there are
  problems that are ½ ? hard for P/poly(BPP).

Yao, Nisan-Wigderson, Babai-Fortnow-Nisan-Wigderso
n, Impagliazzo, Impagliazzo-Wigderson1,
Impagliazzo-Wigderson2, Sudan-Trevisan-Vadhan,
Trevisan-Vadhan, Impagliazzo-Jaiswal-Kabanets,
Impagliazzo-Jaiswal-Kabanets-Wigderson.
5
NP versus P/poly

• ODonnell.
• Theorem If there are problems in NP that are 1 -
  ? hard for P/poly, then there are problems that
  are ½ ? hard.
• Starts from average-case assumption.
• Healy-Vadhan-Viola.

6
NP versus BPP

• Trevisan03.
• Theorem If there are problems in NP that are 1 -
  ? hard for BPP, then there are problems that are
  ¾ ? hard.

7
NP versus BPP

• Trevisan05.
• Theorem If there are problems in NP that are 1 -
  ? hard for BPP, then there are problems that are
  ½ ? hard.
• BureshOppenheim-Kabanets-Santhanam alternate
  proof via monotone codes.
• Optimal up to ?.

8
Our resultsAmplification against P.

• Theorem 1 If there is a problem in NP that is 1
  - ? hard for P, then there is a problem which is
  ¾ ? hard.
• Theorem 2 If there is a problem in PSPACE that
  is1 - ? hard for P, then there is a problem which
  is ¾ ? hard.
• Trevisan 1 - ? hardness to 7/8 ? for PSPACE.
• Goldreich-Wigderson Unconditional hardness for
  EXP against P.

? 1/(log n)100
? 1/n100
9
Outline of This Talk

• Amplification via Decoding.
• Deterministic Local Decoding.
• Amplification within NP.

10
Outline of This Talk

• Amplification via Decoding.
• Deterministic Local Decoding.
• Amplification within NP.

11
Amplification via DecodingTrevisan,
Sudan-Trevisan-Vadhan
1 0 1 1 0 0 1 0 1
1 0 0 1 1 0 0 1 1
1 0 1 1 0 0
1 0 1 1 0 0
Encode
Decode
f
g Wildly hard
Approx. to g
f Mildly hard
12
Amplification via Decoding.
Case Study PSPACE versus BPP.
1 0 1 1 0 0 1 0 1

• fs table has size 2n.
• gs table has size 2n2.
• Encoding in space n100.

1 0 1 1 0 0
Encode
PSPACE
f Mildly hard
g Wildly hard
13
Amplification via Decoding.
Case Study PSPACE versus BPP.
1 0 0 1 1 0 0 1 1

• Randomized local decoder.
• List-decoding beyond ¼ error.

1 0 1 1 0 0
Decode
BPP
f
Approx. to g
14
Amplification via Decoding.
Case Study NP versus BPP.
1 0 1 1 0 0 1 0 1

• g is a monotone function M of f.
• M is computable in NTIME(n100)
• M needs to be noise-sensitive.

1 0 1 1 0 0
Encode
NP
f Mildly hard
g Wildly hard
15
Amplification via Decoding.
Case Study NP versus BPP.

• Randomized local decoder.
• Monotone codes are bad codes.
• Can only approximate f.

1 0 0 1 1 0 0 1 1
1 0 1 0 0 0
Decode
BPP
Approx. to f
Approx. to g
16
Outline of This Talk

• Amplification via Decoding.
• Deterministic Local Decoding.
• Amplification within NP.

17
Deterministic Amplification.
Deterministic local decoding?
1 0 0 1 1 0 0 1 1
1 0 1 1 0 0
Decode
P
18
Deterministic Amplification.
Deterministic local decoding?

• Can force an error on any bit.
• Need near-linear length encoding.
• Monotone codes for NP.

1 0 0 1 1 0 0 1 1
2nn100
1 0 1 1 0 0
Decode
2n
P
19
Deterministic Local Decoding

• up to unique decoding radius.
• Deterministic local decoding up to 1 - ? from ¾
  ? agreement.
• Monotone code construction with similar
  parameters.
• Main tool ABNNR codes GMD decoding.
  Guruswami-Indyk, Akavia-Venkatesan
• Open Problem Go beyond Unique Decoding.

20
The ABNNR Construction.

• Expander graph.
• 2n vertices.
• Degree n100.

21
The ABNNR Construction.

• Expander graph.
• 2n vertices.
• Degree n100.

1
0
0
1
0
22
The ABNNR Construction.

• Expander graph.
• 2n vertices.
• Degree n100.

1 0 0
1
1 0 1
0

• Start with a binary code with small distance.
• Gives a code of large distance over large
  alphabet.

0 0 0
0
1 0 1
1
0 1 0
0
23
Concatenated ABNNR Codes.
Inner code of distance ½.
1 0 0
1 0 1 0 1 1
1
1 0 1
0 1 1 0 0 1
0

• Binary code of distance ½.
• GI ¼ error, not local.
• T 1/8 error, local.

0 0 0
0 0 0 0 0 0
0
1 0 1
1
0 1 1 0 0 1
0 1 0
0 1 0 1 1 0
0
24
Decoding ABNNR Codes.
1 1 1 0 0 1
0 1 0 0 0 1
0 0 1 0 0 0
0 1 0 0 1 1
0 1 1 1 0 0
25
Decoding ABNNR Codes.
1 0 0
1 1 1 0 0 1

• Decode inner codes.
• Works if error lt ¼.
• Fails if error gt ¼.

0 0 1
0 1 0 0 0 1
0 0 0
0 0 1 0 0 0
0 0 1
0 1 0 0 1 1
0 1 0
0 1 1 1 0 0
26
Decoding ABNNR Codes.
1 0 0
1 1 1 0 0 1
Majority vote on the LHS. Trevisan Corrects
1/8 fraction of errors.
0
0 0 1
0 1 0 0 0 1
0
0 0 0
0 0 1 0 0 0
0
0 0 1
1
0 1 0 0 1 1
0 1 0
0 1 1 1 0 0
0
27
GMD decoding Forney67
c 2 0,1
1 0 0
1 1 1 0 0 1

• If decoding succeeds, error ? 2 0, ¼.
• If 0 error, confidence is 1.
• If ¼ error, confidence is 0.
• c (1 4?).

Could return wrong answer with high confidence
but this requires ? close to ½.
28
GMD Decoding for ABNNR Codes.
GMD decoding Pick threshold, erase, decode.
Non-local. Our approach Weighted Majority. Thm
Corrects ¼ fraction of errors locally.
1 0 0 c1
1 1 1 0 0 1
0 0 1 c2
0 1 0 0 0 1
0 0 0 c3
0 0 1 0 0 0
0 0 1 c4
0 1 0 0 1 1
0 1 0 c5
0 1 1 1 0 0
29
GMD Decoding for ABNNR Codes.

• Thm GMD decoding corrects ¼ fraction of error.
• Proof Sketch
• Globally, good nodes have more confidence than
  bad nodes.
• Locally, this holds for most neighborhoods of
  vertices on LHS.

1 0 0 c1
1
0 0 1 c2
0
0 0 0 c3
0
0 0 1 c4
1
Proof similar to Expander Mixing Lemma.
0 1 0 c5
0
30
Outline of This Talk

• Amplification via Decoding.
• Deterministic Local Decoding.
• Amplification within NP.

• Finding an inner monotone code BOKS.
• Implementing GMD decoding.

31
The BOKS construction.
1 0 1 1 0 0 1 0 1

• T(x) Sample an r-tuple from x, apply the
  Tribes function.
• If x, y are balanced, and ?(x,y) gt ?,
  ?(T(x),T(y)) ¼ ½.
• If x, y are very close, so are T(x), T(y).
• Decoding brute force.

1 0 1 1 0 0
k
kr
x
T(x)
32
GMD Decoding for Monotone codes.

• Start with a balanced f, apply concatenated
  ABNNR.
• Inner decoder returns closest balanced message.
• Apply GMD decoding.
• Thm Decoder corrects ¼ fraction of error
  approximately.
• Analysis becomes harder.

1 0 1 0 c1
1
0 1 1 0 c2
0
1 1 0 0 c3
0
0 1 1 0 c4
1
1 0 1 0 c5
0
33
GMD Decoding for Monotone codes.

• Inner decoder finds the closest balanced
  message.
• Assume 0 error Decoder need not return message.
• Good nodes have few errors, Bad nodes have many.
• Thm Decoder corrects ¼ fraction of error
  approximately.

1 0 1 0 c1
1
0 1 1 0 c2
0
1 1 0 0 c3
0
0 1 1 0 c4
1
1 0 1 0 c5
0
34
Beyond Unique Decoding

• Deterministic local list-decoder
• Set L of machines such that
• - For any received word
• Every nearby codeword is computed by some M 2 L.
• Is this possible?

1 0 0 1 1 0 0 1 1
Thank You!