Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G)...
Transcript of Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G)...
![Page 1: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/1.jpg)
On page 6 . . .
Lower Bounds for
Clique vs. Independent Set
Mika GoosUniversity of Toronto
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 1 / 14
![Page 2: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/2.jpg)
On page 6 . . .
Lower Bounds for
Clique vs. Independent Set
Mika GoosUniversity of Toronto
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 1 / 14
![Page 3: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/3.jpg)
CIS problem [Yannakakis, STOC’88]
G = ([n], E)
Alice
Clique x ⊆ [n] of G
Bob
Independent set y ⊆ [n] of G
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 2 / 14
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CIS problem [Yannakakis, STOC’88]
G = ([n], E)
AliceClique x ⊆ [n] of G
BobIndependent set y ⊆ [n] of G
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 2 / 14
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CIS problem [Yannakakis, STOC’88]
G = ([n], E)
AliceClique x ⊆ [n] of G
BobIndependent set y ⊆ [n] of G
Compute: CISG(x, y) = |x ∩ y|
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 2 / 14
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Background
Yannakakis’s motivation:Size of LPs for the vertex packing polytope of GBreakthrough: [Fiorini et al., STOC’12]
Known bounds:NPcc(CISG) = dlog ne (guess x ∩ y)∀G :
coNPcc(CISG) ≤ O(log2 n)∀G :
Yannakakis’s question:
coNPcc(CISG) ≤ O(log n) ?∀G :
Alon–Saks–Seymour conjecture:
χ(G) ≤ bp(G) + 1 ?∀G :
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 3 / 14
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Background
Yannakakis’s motivation:Size of LPs for the vertex packing polytope of GBreakthrough: [Fiorini et al., STOC’12]
Known bounds:NPcc(CISG) = dlog ne (guess x ∩ y)∀G :
coNPcc(CISG) ≤ O(log2 n)∀G :
Yannakakis’s question:
coNPcc(CISG) ≤ O(log n) ?∀G :
Alon–Saks–Seymour conjecture:
χ(G) ≤ bp(G) + 1 ?∀G :
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 3 / 14
![Page 8: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/8.jpg)
Background
Yannakakis’s motivation:Size of LPs for the vertex packing polytope of GBreakthrough: [Fiorini et al., STOC’12]
Known bounds:NPcc(CISG) = dlog ne (guess x ∩ y)∀G :
coNPcc(CISG) ≤ O(log2 n)∀G :
Yannakakis’s question:
coNPcc(CISG) ≤ O(log n) ?∀G :
Alon–Saks–Seymour conjecture:
χ(G) ≤ bp(G) + 1 ?∀G :
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 3 / 14
![Page 9: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/9.jpg)
Background
Yannakakis’s motivation:Size of LPs for the vertex packing polytope of GBreakthrough: [Fiorini et al., STOC’12]
Known bounds:NPcc(CISG) = dlog ne (guess x ∩ y)∀G :
coNPcc(CISG) ≤ O(log2 n)∀G :
Yannakakis’s question:
coNPcc(CISG) ≤ O(log n) ?∀G :
Alon–Saks–Seymour conjecture:
χ(G) ≤ bp(G) + 1 ?∀G :
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 3 / 14
![Page 10: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/10.jpg)
Background
Yannakakis’s motivation:Size of LPs for the vertex packing polytope of GBreakthrough: [Fiorini et al., STOC’12]
Known bounds:NPcc(CISG) = dlog ne (guess x ∩ y)∀G :
coNPcc(CISG) ≤ O(log2 n)∀G :
Yannakakis’s question:
coNPcc(CISG) ≤ O(log n) ?∀G :
Alon–Saks–Seymour conjecture:
χ(G) ≤ bp(G) + 1 ?∀G :
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 3 / 14
![Page 11: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/11.jpg)
Background
Yannakakis’s motivation:Size of LPs for the vertex packing polytope of GBreakthrough: [Fiorini et al., STOC’12]
Known bounds:NPcc(CISG) = dlog ne (guess x ∩ y)∀G :
coNPcc(CISG) ≤ O(log2 n)∀G :
Yannakakis’s question:
coNPcc(CISG) ≤ O(log n) ?∀G :
Alon–Saks–Seymour conjecture:
χ(G) ≤ bp(G) + 1 ?∀G :
[Huang–Sudakov, 2010]: ∃G : χ(G) ≥ bp(G)6/5
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 3 / 14
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Background
Yannakakis’s motivation:Size of LPs for the vertex packing polytope of GBreakthrough: [Fiorini et al., STOC’12]
Known bounds:NPcc(CISG) = dlog ne (guess x ∩ y)∀G :
coNPcc(CISG) ≤ O(log2 n)∀G :
Yannakakis’s question:
coNPcc(CISG) ≤ O(log n) ?∀G :
Polynomial Alon–Saks–Seymour conjecture:
χ(G) ≤ poly( bp(G)) ?∀G :
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 3 / 14
![Page 13: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/13.jpg)
Background
Yannakakis’s motivation:Size of LPs for the vertex packing polytope of GBreakthrough: [Fiorini et al., STOC’12]
Known bounds:NPcc(CISG) = dlog ne (guess x ∩ y)∀G :
coNPcc(CISG) ≤ O(log2 n)∀G :
Yannakakis’s question:
coNPcc(CISG) ≤ O(log n) ?∀G :
Polynomial Alon–Saks–Seymour conjecture:
χ(G) ≤ poly( bp(G)) ?∀G :
[Alon–Haviv]
=⇒
=⇒ [Bousquet et al.]
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 3 / 14
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Background
Yannakakis’s motivation:Size of LPs for the vertex packing polytope of GBreakthrough: [Fiorini et al., STOC’12]
Known bounds:NPcc(CISG) = dlog ne (guess x ∩ y)∀G :
coNPcc(CISG) ≤ O(log2 n)∀G :
Yannakakis’s question:
coNPcc(CISG) ≤ O(log n) ?∀G :
Polynomial Alon–Saks–Seymour conjecture:
χ(G) ≤ poly( bp(G)) ?∀G :
[Alon–Haviv]
=⇒
=⇒ [Bousquet et al.]
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 3 / 14
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Our result
Main theorem
∃G : coNPcc(CISG) ≥ Ω(log1.128 n)
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 4 / 14
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Our result
Main theorem
∃G : coNPcc(CISG) ≥ Ω(log1.128 n)
Prior boundsMeasure Lower bound Reference
Pcc 2 · log n Kushilevitz, Linial, and Ostrovsky (1999)coNPcc 6/5 · log n Huang and Sudakov (2010)coNPcc 3/2 · log n Amano (2014)coNPcc 2 · log n Shigeta and Amano (2014)
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 4 / 14
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Our result
Main theorem
∃G : coNPcc(CISG) ≥ Ω(log1.128 n)
Proof strategy:
Query complexity −→ Communication complexity
Cf. lower bounds for log-rank[Nisan–Wigderson, 1995]
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 4 / 14
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Models of communication
1
1
1
1
1
1
1
1
1
1
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1 1
1
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1 1
1
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1
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1
1 1
1
1
0
0
0
0
0
0
0
0
0
0 0
0
0
0 0 0 0
0
0
0
0 F : X ×Y → 0, 1
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 5 / 14
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Models of communication
1
1
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1 1
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1 1
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1 1
1
1
0
0
0
0
0
0
0
0
0
0 0
0
0
0 0 0 0
0
0
0
0 NPcc
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 5 / 14
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Models of communication
1
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1 1
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0
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0
0
0
0
0
0
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0
0
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0
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0
0 0
0
0
0
0
0
0 UPcc
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 5 / 14
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Models of communication
1
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1 1
1
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0
0
0
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0
0
0
0
0
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0
0
0
0
0
0
0
0
0 0
0
0
0
0
0
0;
CISG is complete for UPcc: F ≤ CISG
UPcc(F) = UPcc(CISG) = log n
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 5 / 14
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Proof strategy
Restatement of Main theorem:
coNPcc(F) ≥ UPcc(F)1.128∃F : X ×Y → 0, 1
Query separation:
coNPdt( f ) ≥ UPdt( f )1.128∃ f : 0, 1n → 0, 1
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 6 / 14
![Page 23: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/23.jpg)
Proof strategy
Restatement of Main theorem:
coNPcc(F) ≥ UPcc(F)1.128∃F : X ×Y → 0, 1
Query separation:
coNPdt( f ) ≥ UPdt( f )1.128∃ f : 0, 1n → 0, 1
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 6 / 14
![Page 24: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/24.jpg)
Proof strategy
Restatement of Main theorem:
coNPcc(F) ≥ UPcc(F)1.128∃F : X ×Y → 0, 1
Query separation:
coNPdt( f ) ≥ UPdt( f )1.128∃ f : 0, 1n → 0, 1
Decision tree complexity measures:
NPdt = DNF width = 1-certificate complexitycoNPdt = CNF width = 0-certificate complexity
UPdt = Unambiguous DNF width
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 6 / 14
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Proof strategy
Restatement of Main theorem:
coNPcc(F) ≥ UPcc(F)1.128∃F : X ×Y → 0, 1
Query separation:
coNPdt( f ) ≥ UPdt( f )1.128∃ f : 0, 1n → 0, 1
Agenda:
Step 1: Query separation
Step 2: Simulation theorem [GLMWZ, 2015]
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 6 / 14
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Step 1: Query separation
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 7 / 14
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Warm-up
Example: Let f (x1, x2, x3) = 1 iff x1 + x2 + x3 ∈ 1, 2
UPdt( f ) = 2 because f ≡ x1 x2 ∨ x2 x3 ∨ x3 x1
coNPdt( f ) = 3 because 0-input~0 is fully sensitive
Recursive composition:
f 1( · ) := f ( · )f i+1( · ) := f ( f i( · ), f i( · ), f i( · ))
Hope: coNPdt( f i)
UPdt( f i)≥(
32
)i
Problem!
In order to certify “ f i( · ) = 1”, (should be easy)might need to certify “ f i−1( · ) = 0” (should be hard)
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 8 / 14
![Page 28: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/28.jpg)
Warm-up
Example: Let f (x1, x2, x3) = 1 iff x1 + x2 + x3 ∈ 1, 2
UPdt( f ) = 2 because f ≡ x1 x2 ∨ x2 x3 ∨ x3 x1
coNPdt( f ) = 3 because 0-input~0 is fully sensitive
Recursive composition:
f
x1 x2 x3
f 1( · ) := f ( · )f i+1( · ) := f ( f i( · ), f i( · ), f i( · ))
Hope: coNPdt( f i)
UPdt( f i)≥(
32
)i
Problem!
In order to certify “ f i( · ) = 1”, (should be easy)might need to certify “ f i−1( · ) = 0” (should be hard)
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 8 / 14
![Page 29: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/29.jpg)
Warm-up
Example: Let f (x1, x2, x3) = 1 iff x1 + x2 + x3 ∈ 1, 2
UPdt( f ) = 2 because f ≡ x1 x2 ∨ x2 x3 ∨ x3 x1
coNPdt( f ) = 3 because 0-input~0 is fully sensitive
Recursive composition:
f
f
x1 x2 x3
f
x4 x5 x6
f
x7 x8 x9
f 1( · ) := f ( · )f i+1( · ) := f ( f i( · ), f i( · ), f i( · ))
Hope: coNPdt( f i)
UPdt( f i)≥(
32
)i
Problem!
In order to certify “ f i( · ) = 1”, (should be easy)might need to certify “ f i−1( · ) = 0” (should be hard)
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 8 / 14
![Page 30: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/30.jpg)
Warm-up
Example: Let f (x1, x2, x3) = 1 iff x1 + x2 + x3 ∈ 1, 2
UPdt( f ) = 2 because f ≡ x1 x2 ∨ x2 x3 ∨ x3 x1
coNPdt( f ) = 3 because 0-input~0 is fully sensitive
Recursive composition:
f
f
x1 x2 x3
f
x4 x5 x6
f
x7 x8 x9
f 1( · ) := f ( · )f i+1( · ) := f ( f i( · ), f i( · ), f i( · ))
Hope: coNPdt( f i)
UPdt( f i)≥(
32
)i
Problem!
In order to certify “ f i( · ) = 1”, (should be easy)might need to certify “ f i−1( · ) = 0” (should be hard)
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 8 / 14
![Page 31: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/31.jpg)
Warm-up
Example: Let f (x1, x2, x3) = 1 iff x1 + x2 + x3 ∈ 1, 2
UPdt( f ) = 2 because f ≡ x1 x2 ∨ x2 x3 ∨ x3 x1
coNPdt( f ) = 3 because 0-input~0 is fully sensitive
Recursive composition:
f
f
x1 x2 x3
f
x4 x5 x6
f
x7 x8 x9
f 1( · ) := f ( · )f i+1( · ) := f ( f i( · ), f i( · ), f i( · ))
Hope: coNPdt( f i)
UPdt( f i)≥(
32
)i
Problem!
In order to certify “ f i( · ) = 1”, (should be easy)might need to certify “ f i−1( · ) = 0” (should be hard)
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 8 / 14
![Page 32: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/32.jpg)
Warm-up
Example: Let f (x1, x2, x3) = 1 iff x1 + x2 + x3 ∈ 1, 2
UPdt( f ) = 2 because f ≡ x1 x2 ∨ x2 x3 ∨ x3 x1
coNPdt( f ) = 3 because 0-input~0 is fully sensitive
Recursive composition:
f
f
x1 x2 x3
f
x4 x5 x6
f
x7 x8 x9
f 1( · ) := f ( · )f i+1( · ) := f ( f i( · ), f i( · ), f i( · ))
Hope: coNPdt( f i)
UPdt( f i)≥(
32
)i
Problem!
In order to certify “ f i( · ) = 1”, (should be easy)might need to certify “ f i−1( · ) = 0” (should be hard)
Solution: Enlarge input/output alphabets
f : (0 ∪ Σ)n → 0 ∪ Σ
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 8 / 14
![Page 33: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/33.jpg)
Warm-up
Example: Let f (x1, x2, x3) = 1 iff x1 + x2 + x3 ∈ 1, 2
UPdt( f ) = 2 because f ≡ x1 x2 ∨ x2 x3 ∨ x3 x1
coNPdt( f ) = 3 because 0-input~0 is fully sensitive
Recursive composition:
f
f
x1 x2 x3
f
x4 x5 x6
f
x7 x8 x9
f 1( · ) := f ( · )f i+1( · ) := f ( f i( · ), f i( · ), f i( · ))
Hope: coNPdt( f i)
UPdt( f i)≥(
32
)i
Problem!
In order to certify “ f i( · ) = 1”, (should be easy)might need to certify “ f i−1( · ) = 0” (should be hard)
Solution: Enlarge input/output alphabets
f : (0 ∪ Σ)n → 0 ∪ Σ
Now: In order to certify “ f i( · ) = σ” for σ ∈ Σ,only need to certify “ f i−1( · ) = σ′” for σ′ ∈ Σ
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 8 / 14
![Page 34: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/34.jpg)
Defining f
Any two certificates in an UPdt
decision tree intersect in variables
=⇒ Finite projective planes!
Recursive composition
Key trick:
(0 ∪ Σ)n → 0, 1From(0 ∪ Σ)n → 0 ∪ pointersConstruct
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 9 / 14
![Page 35: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/35.jpg)
Defining f
123
2
1 2
3
12
1
31
3
2
3
2
1
3
1
2 3
Incidence ordering:
Each node orders its incidentedges using numbers from [3]
Recursive composition
Key trick:
(0 ∪ Σ)n → 0, 1From(0 ∪ Σ)n → 0 ∪ pointersConstruct
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 9 / 14
![Page 36: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/36.jpg)
Defining f
0
0
Inputs to nodes:
Pointer values from 0 ∪ [3]︸︷︷︸=Σ(0 is a null pointer)
Recursive composition
Key trick:
(0 ∪ Σ)n → 0, 1From(0 ∪ Σ)n → 0 ∪ pointersConstruct
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 9 / 14
![Page 37: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/37.jpg)
Defining f
0
0
Defining f : (0 ∪ [3])7 → 0, 1
Say edge e is satisfied on input x iffall nodes v ∈ e point to e under x
f (x) = 1 iff x satisfies an edge
Clearly UPdt( f ) = 3
Recursive composition
Key trick:
(0 ∪ Σ)n → 0, 1From(0 ∪ Σ)n → 0 ∪ pointersConstruct
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 9 / 14
![Page 38: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/38.jpg)
Defining f
0 0
0
0 0 0
0
Problem!
Certifying “ f (~0) = 0” too easy!
Recursive composition
Key trick:
(0 ∪ Σ)n → 0, 1From(0 ∪ Σ)n → 0 ∪ pointersConstruct
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 9 / 14
![Page 39: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/39.jpg)
Defining f
12
3
2
1 2
3
12
1
31
3
2
3
2
1
3
1
2 3
Add input weights: f g7
Gadget g is such that deciding ifg( · ) = i for i ∈ [3] costs i queries
Recursive composition
Key trick:
(0 ∪ Σ)n → 0, 1From(0 ∪ Σ)n → 0 ∪ pointersConstruct
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 9 / 14
![Page 40: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/40.jpg)
Defining f
12
3
2
1 2
3
12
1
31
3
2
3
2
1
3
1
2 3
Add input weights: f g7
Gadget g is such that deciding ifg( · ) = i for i ∈ [3] costs i queries
1∗∗ 7→ 1
2∗∗ 7→ 20 2 ∗ 7→ 2
3∗∗ 7→ 30 3 ∗ 7→ 30 0 3 7→ 3
Else 7→ 0
Recursive composition
Key trick:
(0 ∪ Σ)n → 0, 1From(0 ∪ Σ)n → 0 ∪ pointersConstruct
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 9 / 14
![Page 41: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/41.jpg)
Defining f
12
3
2
1 2
3
12
1
31
3
2
3
2
1
3
1
2 3
Key properties:
UPdt( f g7) = 1 + 2 + 3 = 6
Certifying “( f g7)(~0) = 0”requires (# edges) = 7 queries
Recursive composition
Key trick:
(0 ∪ Σ)n → 0, 1From(0 ∪ Σ)n → 0 ∪ pointersConstruct
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 9 / 14
![Page 42: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/42.jpg)
Defining f
12
3
2
1 2
3
12
1
31
3
2
3
2
1
3
1
2 3
Key properties:
UPdt( f g7) = 1 + 2 + 3 = 6
Certifying “( f g7)(~0) = 0”requires (# edges) = 7 queries(
Magic numerology: 57 ≈ 361.128)
Recursive composition
Key trick:
(0 ∪ Σ)n → 0, 1From(0 ∪ Σ)n → 0 ∪ pointersConstruct
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 9 / 14
![Page 43: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/43.jpg)
Defining f
12
3
2
1 2
3
12
1
31
3
2
3
2
1
3
1
2 3
Key properties:
UPdt( f g7) = 1 + 2 + 3 = 6
Certifying “( f g7)(~0) = 0”requires (# edges) = 7 queries(
Magic numerology: 57 ≈ 361.128)
Recursive composition
Key trick:
(0 ∪ Σ)n → 0, 1From(0 ∪ Σ)n → 0 ∪ pointersConstruct
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 9 / 14
![Page 44: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/44.jpg)
Query separation:
coNPdt( f ) ≥ UPdt( f )1.128∃ f : 0, 1n → 0, 1
Step 2: Simulation theorem from
“Rectangles Are Nonnegative Juntas”
Mika Goos, Shachar Lovett, Raghu Meka, Thomas Watson,and David Zuckerman (STOC’15)
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 10 / 14
![Page 45: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/45.jpg)
Composed functions f gn
f f
z1 z2 z3 z4 z5 g g g g g
x1 y1 x2 y2 x3 y3 x4 y4 x5 y5
Compose with gn
Examples: Set-disjointness: OR ANDn
Inner-product: XOR ANDn
In general: g : 0, 1b × 0, 1b → 0, 1 is a small gadget
Alice holds x ∈ 0, 1bn
Bob holds y ∈ 0, 1bn
We choose: g = inner-product with b = Θ(log n) bits per party
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 11 / 14
![Page 46: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/46.jpg)
Composed functions f gn
f f
z1 z2 z3 z4 z5 g g g g g
x1 y1 x2 y2 x3 y3 x4 y4 x5 y5
Compose with gn
Examples: Set-disjointness: OR ANDn
Inner-product: XOR ANDn
In general: g : 0, 1b × 0, 1b → 0, 1 is a small gadget
Alice holds x ∈ 0, 1bn
Bob holds y ∈ 0, 1bn
We choose: g = inner-product with b = Θ(log n) bits per party
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 11 / 14
![Page 47: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/47.jpg)
Approximation by juntas
Conical d-junta:Nonnegative combination of d-conjunctions(Example: 0.4 · z1z2 + 0.66 · z2z3 + 0.35 · z3z1)
Main Structure Theorem:
Suppose Π is cost-d randomised protocol for f gn
Then there exists a conical d-junta h s.t. ∀z ∈ dom f :
Pr(x,y)∼(gn)−1(z)
[Π(x,y) accepts ] ≈ h(z)
Cf. Polynomial approximation [Razborov, Sherstov, Shi–Zhu,. . . ]:
Approximate poly-degree of AND = Θ(√
n)Approximate junta-degree of AND = Θ(n)
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 12 / 14
![Page 48: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/48.jpg)
Approximation by juntas
Conical d-junta:Nonnegative combination of d-conjunctions(Example: 0.4 · z1z2 + 0.66 · z2z3 + 0.35 · z3z1)
Main Structure Theorem:
Suppose Π is cost-d randomised protocol for f gn
Then there exists a conical d-junta h s.t. ∀z ∈ dom f :
Pr(x,y)∼(gn)−1(z)
[Π(x,y) accepts ] ≈ h(z)
Cf. Polynomial approximation [Razborov, Sherstov, Shi–Zhu,. . . ]:
Approximate poly-degree of AND = Θ(√
n)Approximate junta-degree of AND = Θ(n)
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 12 / 14
![Page 49: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/49.jpg)
Approximation by juntas
Conical d-junta:Nonnegative combination of d-conjunctions(Example: 0.4 · z1z2 + 0.66 · z2z3 + 0.35 · z3z1)
Main Structure Theorem:
Suppose Π is cost-d randomised protocol for f gn
Then there exists a conical d-junta h s.t. ∀z ∈ dom f :
Pr(x,y)∼(gn)−1(z)
[Π(x,y) accepts ] ≈ h(z)
Cf. Polynomial approximation [Razborov, Sherstov, Shi–Zhu,. . . ]:
Approximate poly-degree of AND = Θ(√
n)Approximate junta-degree of AND = Θ(n)
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 12 / 14
![Page 50: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/50.jpg)
Corollaries
Simulation for NP:
NPcc( f gn) = Θ(NPdt( f ) · b). . . recall b = Θ(log n)
Conical d-junta: 0.4 · z1z2 + 0.66 · z2z3 + 0.35 · z3z1;
d-DNF: z1z2 ∨ z2z3 ∨ z3z1
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 13 / 14
![Page 51: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/51.jpg)
Corollaries
Simulation for NP:
NPcc( f gn) = Θ(NPdt( f ) · b). . . recall b = Θ(log n)
Trivially: UPcc( f gn) ≤ O(UPdt( f ) · b)
Main theorem follows!
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 13 / 14
![Page 52: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/52.jpg)
Corollaries
Also covered!
Simulation for NP:
NPcc( f gn) = Θ(NPdt( f ) · b). . . recall b = Θ(log n)
P
BPP
NP MA
SBPWAPP PostBPP PPcorruptionsmooth rectangle ext. discrepancy discrepancy
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 13 / 14
![Page 53: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/53.jpg)
Summary
Main result
∃G : coNPcc(CISG) ≥ Ω(log1.128 n)
Open problems
Better separation for coNPdt vs. UPdt?Simulation theorems for new models (e.g., BPP)Improve gadget size down to b = O(1)(Would give new proof of Ω(n) bound for set-disjointness)
Cheers!
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 14 / 14
![Page 54: Clique vs. Independent Setmika/cis_slides.pdf · Our result Main theorem 9G : coNPcc(CIS G) W(log1.128 n) Prior bounds Measure Lower bound Reference Pcc 2 logn Kushilevitz, Linial,](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb87c0d7c71fb17eb738152/html5/thumbnails/54.jpg)
Summary
Main result
∃G : coNPcc(CISG) ≥ Ω(log1.128 n)
Open problems
Better separation for coNPdt vs. UPdt?Simulation theorems for new models (e.g., BPP)Improve gadget size down to b = O(1)(Would give new proof of Ω(n) bound for set-disjointness)
Cheers!
Mika Goos (Univ. of Toronto) Clique vs. Independent Set 23rd February 2015 14 / 14