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Lecture 13: SOS Lower Bounds for Planted Clique Part II Lecture Outline Part I: Relaxed k-clique Equations and Theorem Statement Part II: Pseudo-Calibration/Moment Matching Part III: Decomposition of Graph Matrices via Minimum Vertex

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SLIDE 1

Lecture 13: SOS Lower Bounds for Planted Clique Part II

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SLIDE 2

Lecture Outline

  • Part I: Relaxed k-clique Equations and Theorem

Statement

  • Part II: Pseudo-Calibration/Moment Matching
  • Part III: Decomposition of Graph Matrices via

Minimum Vertex Separators

  • Part IV: Attempt #1: Bounding with Square Terms
  • Part V: Approximate PSD Decomposition
  • Part VI: Further Work and Open Problems
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SLIDE 3

Part I: Relaxed k-clique Equations and Theorem Statement

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SLIDE 4

Relaxed Planted Clique Equations

  • Flaw in the current analysis: Need to relax the

𝑙-clique equations slightly to make the combinatorics easier to analyze

  • Relaxed 𝑙-clique Equations:

𝑦𝑗

2 = 𝑦𝑗 for all i.

π‘¦π‘—π‘¦π‘˜ = 0 if 𝑗, π‘˜ βˆ‰ 𝐹(𝐻) 1 βˆ’ πœ— 𝑙 ≀ σ𝑗 𝑦𝑗 ≀ (1 + πœ—)𝑙

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SLIDE 5

Planted Clique SOS Lower Bound

  • Theorem 1.1 of [BHK+16]: βˆƒπ‘‘ > 0 such that if

𝑙 ≀ π‘œ

1 2βˆ’π‘‘ 𝑒 π‘šπ‘π‘•π‘œ, with high probability degree 𝑒

SOS cannot prove that the relaxed 𝑙-clique equations are infeasible.

  • Note: For 𝑒 = 4 there is a lower bound of

ΰ·© Ξ© π‘œ for the original 𝑙-clique equations.

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SLIDE 6

High Level Idea

  • High level idea: Show that it is hard to

distinguish between the random distribution 𝐻 π‘œ,

1 2 and the planted distribution where we

put each vertex in the planted clique with probability

𝑙 π‘œ.

  • Remark: We take this planted distribution to

make the combinatorics easier. If we could analyze the planted distribution where the clique has size exactly 𝑙, we would satisfy the constraint σ𝑗 𝑦𝑗 = 𝑙 exactly.

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SLIDE 7

Part II: Pseudo-Calibration/Moment Matching

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SLIDE 8

Choosing Pseudo-Expectation Values

  • Last lecture, Pessimist disproved our first

attempt for pseudo-expectation values, the MW moments.

  • How can we come up with better pseudo-

expectation values?

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SLIDE 9

Pseudo-Calibration/Moment Matching

  • Setup: We are trying to distinguish between a

random distribution (𝐻 π‘œ,

1 2 ) and a planted

distribution (𝐻 π‘œ,

1 2 + planted clique)

  • Pseudo-calibration/moment matching: The

pseudo-expectation values over the random distribution should match the actual expected values over the planted distribution in expectation for all low degree tests.

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SLIDE 10

Review: Discrete Fourier Analysis

  • Requirements for discrete Fourier analysis
  • 1. An inner product
  • 2. An orthonormal basis of Fourier characters
  • This gives us Fourier decompositions and

Parseval’s Theorem

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SLIDE 11

Fourier Analysis over the Hypercube

  • Example: Fourier analysis on {βˆ’1,1}π‘œ
  • Inner product: 𝑔 β‹… 𝑕 = 1

2π‘œ σ𝑦 𝑔 𝑦 𝑕(𝑦)

  • Fourier characters: πœ“π΅(𝑦) = Ο‚π‘—βˆˆπ΅ 𝑦𝑗
  • Fourier decomposition: 𝑔 = Οƒπ‘Š መ

𝑔

𝐡 πœ“π΅ where

መ 𝑔

𝐡 = 𝑔 β‹… πœ“π΅

  • Parseval’s Theorem: σ𝐡 መ

𝑔

𝐡 2 = 𝑔 β‹… 𝑔 =

𝑔 2

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SLIDE 12

Fourier Analysis over 𝐻 π‘œ,

1 2

  • Inner product: 𝑔 β‹… 𝑕 = 𝐹𝐻∼𝐻 π‘œ,1

2

𝑔 𝐻 𝑕(𝐻)

  • Fourier characters: πœ“πΉ(𝐻) = βˆ’1 |𝐹\E 𝐻 |
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SLIDE 13

Pseudo-Calibration Equation

  • Pseudo-Calibration Equation:

𝐹𝐻∼𝐻 π‘œ,1

2

[ ΰ·¨ 𝐹[π‘¦π‘Š] β‹… πœ“πΉ] = πΉπ»βˆΌπ‘žπ‘šπ‘π‘œπ‘’π‘“π‘’ 𝑒𝑗𝑑𝑒 [π‘¦π‘Š β‹… πœ“πΉ]

  • We want this equation to hold for all small π‘Š

and 𝐹

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SLIDE 14

Pseudo-Calibration Calculation

  • To calculate πΉπ»βˆΌπ‘žπ‘šπ‘π‘œπ‘’π‘“π‘’ 𝑒𝑗𝑑𝑒 π‘¦π‘Š β‹… πœ“πΉ , first

choose the planted clique and then choose the rest of the graph

  • π‘¦π‘Š = 0 if any 𝑗 ∈ π‘Š is not in the planted clique
  • 𝐹[πœ“πΉ(𝐻)] = 0 whenever 𝐹 is not fully

contained in the planted clique

  • Def: Define π‘Š 𝐹 = endpoints of edges in 𝐹
  • If π‘Š βˆͺ π‘Š 𝐹 βŠ† π‘žπ‘šπ‘π‘œπ‘’π‘“π‘’ π‘‘π‘šπ‘—π‘Ÿπ‘£π‘“ then π‘¦π‘Šπœ“πΉ = 1
  • πΉπ»βˆΌπ‘žπ‘šπ‘π‘œπ‘’π‘“π‘’ 𝑒𝑗𝑑𝑒 π‘¦π‘Š β‹… πœ“πΉ =

𝑙 π‘œ |π‘Šβˆͺπ‘Š(𝐹)|

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SLIDE 15

Calculation Picture

  • If all the vertices are in the planted clique then

π‘¦π‘Šπœ“πΉ(𝐻) = 1 . Otherwise, either π‘¦π‘Š = 0 (because an 𝑗 ∈ π‘Š) is missing or 𝐹 πœ“πΉ = 0 because each edge outside the clique is present with probability 1

2

π‘Š 𝐹

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SLIDE 16

Fourier Coefficients of ΰ·¨ 𝐹[π‘¦π‘Š]

  • From the pseudo-calibration calculation,

ΰ·£ ΰ·¨ 𝐹[π‘¦π‘Š]𝐹 = 𝐹𝐻∼𝐻 π‘œ,1

2

ΰ·¨ 𝐹[π‘¦π‘Š] β‹… πœ“πΉ =

𝑙 π‘œ |π‘Šβˆͺπ‘Š(𝐹)|

  • We take ΰ·¨

𝐹[π‘¦π‘Š] = σ𝐹: π‘Šβˆͺπ‘Š 𝐹 ≀𝐸

𝑙 π‘œ |π‘Šβˆͺπ‘Š(𝐹)|

where 𝐸 is a truncation parameter and then normalize so that ΰ·¨ 𝐹[π‘¦βˆ…] = ΰ·¨ 𝐹[1] = 1

  • Good exercise: What happens if we don’t

truncate at all?

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SLIDE 17

Graph Matrix Decomposition

  • Ignoring the normalization, 𝑁 = σ𝐼

𝑙 π‘œ |π‘Š(𝐼)|

𝑆𝐼 where we sum over ALL 𝐼 with at most 𝐸 vertices which have no isolated vertices outside

  • f 𝑉 and π‘Š.
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SLIDE 18

Part III: Decomposition of Graph Matrices via Minimum Vertex Separators

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SLIDE 19

Proof Sketch

  • How can we show 𝑁 ≽ 0 with high probability?
  • High level idea:
  • 1. Find an approximate PSD decomposition 𝑁𝑔𝑏𝑑𝑒 of

𝑁

  • 2. Handle the error 𝑁𝑔𝑏𝑑𝑒 βˆ’ 𝑁. Unfortunately, this

error is not small enough to ignore, so we carefully show that 𝑁𝑔𝑏𝑑𝑒 βˆ’ 𝑁 β‰Ό 𝑁𝑔𝑏𝑑𝑒 with high

  • probability. We briefly sketch the ideas for this in

Appendix I. For the full details, see [BHK+16]

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SLIDE 20

Technical Minefield

  • Warning: This analysis is a technical minefield

Mine handled correctly Not quite correct, see Appendix II

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SLIDE 21

Decomposition via Separators

  • How can we handle all of the different 𝑆𝐼?
  • Key idea: Decompose each 𝐼 into three parts

𝜏, 𝜐, πœβ€²π‘ˆ based on the leftmost and rightmost minimum vertex separators 𝑇 and π‘ˆ of 𝐼 U V S T H 𝜏 𝜐 πœβ€²π‘ˆ

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SLIDE 22

Separator Definitions

  • Definition: Given a graph 𝐼 with

distinguished sets of vertices 𝑉 and π‘Š, a vertex separator 𝑇 is a set of vertices such that any path from 𝑉 to π‘Š must intersect 𝑇.

  • Definition: A leftmost minimum vertex

separator 𝑇 is a set of vertices such that for any vertex sepator 𝑇′ of minimum size, any path from 𝑉 to 𝑇′ intersects 𝑇.

  • A rightmost minimum vertex separator is

defined analogously.

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SLIDE 23

Existence of Minimum Separators

  • Lemma 6.3 of [BHK+16]: Leftmost and

rightmost minimum vertex separators always exist and are unique.

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SLIDE 24

Left, Middle, and Right Parts

  • Let 𝑇, π‘ˆ be the leftmost and rightmost

minimum vertex separators of 𝐼

  • Definition: We take the left part 𝜏 of 𝐼 to be

the part of 𝐼 between 𝑉 and 𝑇, we take the middle part 𝜐 of 𝐼 to be the part of 𝐼 between 𝑇 and π‘ˆ, and we take the right part πœβ€²π‘ˆ of 𝐼 to be the part of 𝐼 between π‘ˆ and π‘Š

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SLIDE 25

Conditions on Parts

  • 𝜏, 𝜐, πœβ€²π‘ˆ satisfy the following:
  • The unique minimum vertex separator of 𝜏 is

π‘Š

𝜏 = 𝑇 (where π‘Š 𝜏 is the right side of 𝜏)

  • The leftmost and rightmost minimum vertex

separators of 𝜐 are π‘‰πœ = 𝑇 and π‘Š

𝜐 = π‘ˆ (where

π‘‰πœ and π‘Š

𝜐 are the left and right sides of 𝜐)

  • The unique minimum vertex separator of πœβ€²π‘ˆ is

π‘‰πœβ€²π‘ˆ = π‘ˆ (where π‘‰πœβ€²π‘ˆ is the left side of πœβ€²π‘ˆ)

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SLIDE 26

Approximate Decomposition

  • Claim: If 𝑠 is the size of the minimum vertex

separator of 𝐼, 𝑆𝐼 β‰ˆ π‘†πœπ‘†πœπ‘†πœβ€²π‘ˆ

  • Idea: There is a bijection between injective

mappings 𝜚: π‘Š 𝐼 β†’ π‘Š(𝐻) and injective mappings 𝜚1: π‘Š 𝜏 β†’ π‘Š(𝐻), 𝜚2: π‘Š 𝜐 β†’ π‘Š(𝐻), and 𝜚3: π‘Š(πœβ€²π‘ˆ) β†’ π‘Š(𝐻) such that

1. 𝜚1, 𝜚2 agree on 𝑇 and 𝜚2, 𝜚3 agree on π‘ˆ 2. Collectively, 𝜚1, 𝜚2, 𝜚3 don’t map two different vertices of 𝐼 to the same vertex of 𝐻

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SLIDE 27

Approximate Decomposition

  • Claim: If 𝑠 is the size of the minimum vertex

separator of 𝐼, 𝑆𝐼 β‰ˆ π‘†πœπ‘†πœπ‘†πœβ€²π‘ˆ

  • Corollary:

𝑙 π‘œ |π‘Š(𝐼)|

𝑆𝐼 β‰ˆ

𝑙 π‘œ π‘Š 𝐼 βˆ’π‘ 

2 π‘†πœ

𝑙 π‘œ π‘Š 𝐼 βˆ’π‘ 

π‘†πœ

𝑙 π‘œ π‘Š 𝐼 βˆ’π‘ 

2 π‘†πœβ€²π‘ˆ

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SLIDE 28

U V S T 𝑆𝐼 U S S T V T β‰ˆ Γ— Γ— π‘†πœ π‘†πœ π‘†πœβ€²π‘ˆ

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SLIDE 29

Intersection Terms

  • Warning! There will be terms where 𝜚1, 𝜚2, 𝜚3

map multiple vertices to the same vertex. We call these intersection terms.

  • We sketch how to handle intersection terms in

Appendix I. For now, we sweep this under the rug.

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SLIDE 30

Part IV: Attempt #1: Bounding With Square Terms

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SLIDE 31

Bounding With Square Terms

  • How can we handle all of the π‘†πœπ‘†πœπ‘†πœβ€²π‘ˆ terms?
  • One idea: Can bound π‘†πœπ‘†πœπ‘†πœβ€²π‘ˆ + π‘†πœπ‘†πœπ‘†πœβ€²π‘ˆ

π‘ˆ

as follows.

  • π‘π‘†πœ βˆ’ 𝑐𝑆

πœβ€²π‘ˆ π‘ˆ π‘†πœ π‘ˆ

π‘π‘†πœ βˆ’ 𝑐𝑆

πœβ€²π‘ˆ π‘ˆ π‘†πœ π‘ˆ π‘ˆ

≽ 0

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SLIDE 32

Bounding With Square Terms

  • π‘π‘†πœ βˆ’ 𝑐𝑆

πœβ€²π‘ˆ π‘ˆ π‘†πœ π‘ˆ

π‘π‘†πœ βˆ’ 𝑐𝑆

πœβ€²π‘ˆ π‘ˆ π‘†πœ π‘ˆ π‘ˆ

≽ 0

  • Rearranging, ab π‘†πœπ‘†πœπ‘†πœβ€²π‘ˆ + π‘†πœπ‘†πœπ‘†πœβ€²π‘ˆ

π‘ˆ

β‰Ό 𝑏2π‘†πœπ‘†πœ

π‘ˆ + 𝑐2𝑆 πœβ€²π‘ˆ π‘ˆ π‘†πœ π‘ˆπ‘†πœπ‘†πœβ€²π‘ˆ β‰Ό 𝑏2π‘†πœπ‘†πœ π‘ˆ +

𝑐2 π‘†πœ

π‘ˆπ‘†πœ 𝑆 πœβ€²π‘ˆ π‘ˆ π‘†πœβ€²π‘ˆ

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SLIDE 33

Example

  • What square terms would the following 𝑆𝐼 be

bounded by (ignoring intersection terms)? U V S T 𝑆𝐼

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SLIDE 34

Example Answer

  • Answer: Take the left part and its mirror

image and take the right part and its mirror image 𝜏 πœπ‘ˆ πœβ€² πœβ€²π‘ˆ

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SLIDE 35

Bounding With Square Terms Failure

  • Unfortunately, the coefficients on the square

terms aren’t high enough for this idea to work.

  • We need a more sophisticated analysis.
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SLIDE 36

Part V: Approximate PSD Decomposition

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SLIDE 37

π‘€π‘…π‘€π‘ˆ factorization

  • Definition: Define 𝑀𝑠 = Οƒπœ:|π‘Š

𝜏|=𝑠

𝑙 π‘œ π‘Š 𝜏 βˆ’π‘ 

2 π‘†πœ

and define 𝑅𝑠 = Οƒπœ:|π‘‰πœ|=|π‘Š

𝜐|=𝑠

𝑙 π‘œ π‘Š 𝜐 βˆ’π‘ 

π‘†πœ where we require that π‘Š

𝜏 is the unique

minimum vertex separator of 𝜏 and π‘‰πœ, π‘Š

𝜐 are

the leftmost and rightmost minimum vertex separators of 𝜐. Define 𝑁𝑔𝑏𝑑𝑒 = σ𝑠=0

𝑒 2

𝑀𝑠𝑅𝑠𝑀𝑠

π‘ˆ

  • Claim: 𝑁 β‰ˆ 𝑁𝑔𝑏𝑑𝑒 = σ𝑠=0

𝑒 2

𝑀𝑠𝑅𝑠𝑀𝑠

π‘ˆ

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SLIDE 38

Claim Justification

  • Claim: 𝑁 β‰ˆ 𝑁𝑔𝑏𝑑𝑒 = σ𝑠=0

𝑒 2

𝑀𝑠𝑅𝑠𝑀𝑠

π‘ˆ

  • This follows from the decomposition of each

𝐼 into left, middle, and right parts 𝜏, 𝜐, πœβ€²π‘ˆ and the claim that up to intersection terms,

𝑙 π‘œ |π‘Š(𝐼)|

𝑆𝐼 =

𝑙 π‘œ π‘Š 𝐼 βˆ’π‘ 

2 π‘†πœ

𝑙 π‘œ π‘Š 𝐼 βˆ’π‘ 

π‘†πœ

𝑙 π‘œ π‘Š 𝐼 βˆ’π‘ 

2 π‘†πœβ€²π‘ˆ

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SLIDE 39

Analysis of 𝑅𝑠

  • 𝑅𝑠 = Οƒπœ:|π‘‰πœ|=|π‘Š

𝜐|=𝑠

𝑙 π‘œ π‘Š 𝜐 βˆ’π‘ 

π‘†πœ

  • Probabilistic norm bounds: With high

probability, π‘†πœ is ΰ·¨ 𝑃(π‘œ

π‘Š 𝜐 βˆ’π‘  2

) because 𝑠 is the size of the minimum vertex separator of 𝐼

  • Corollary: If 𝑙 ≀ π‘œ

1 2βˆ’πœ— then with high

probability, 𝑅𝑠 ≽

1 2 𝐽𝑒 as the identity is the

dominant term of 𝑅𝑠

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SLIDE 40

Summary

  • If 𝑙 ≀ π‘œ

1 2βˆ’πœ— then with high probability,

𝑁𝑔𝑏𝑑𝑒 = σ𝑠=0

𝑒 2

𝑀𝑠𝑅𝑠𝑀𝑠

π‘ˆ ≽ 1 2 σ𝑠=0

𝑒 2

𝑀𝑠𝑀𝑠

π‘ˆ

  • The 1

2 σ𝑠=0

𝑒 2

𝑀𝑠𝑀𝑠

π‘ˆ allows us to deal with the

error 𝑁𝑔𝑏𝑑𝑒 βˆ’ 𝑁.

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SLIDE 41

Part VI: Further Work and Open Problems

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SLIDE 42

Further Work

  • The techniques used for planted clique can be

generalized to other planted problems where we are trying to distinguish a planted distribution from a random distribution [HKP+17]

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SLIDE 43

Open Problems

  • Can we prove the full lower bound for planted

clique with the exact constraint that σ𝑗=1

π‘œ

𝑦𝑗 = 𝑙?

  • How close to

π‘œ can we make the lower bound?

  • It turns out that the current machinery

doesn’t work as well for random sparse

  • graphs. What bounds can we prove for

problems such as densest k-subgraph and independent set on sparse graphs?

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SLIDE 44

References

  • [BHK+16] B. Barak, S. B. Hopkins, J. A. Kelner, P. Kothari, A. Moitra, and A. Potechin,

A nearly tight sum-of-squares lower bound for the planted clique problem, FOCS p.428–437, 2016.

  • [HKP+17] S. Hopkins, P. Kothari, A. Potechin, P. Raghavendra, T. Schramm, D. Steurer.

The power of sum-of-squares for detecting hidden structures. FOCS 2017

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SLIDE 45

Appendix I: Handling Intersection Terms

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SLIDE 46

High Level Idea

  • If there are intersections between the left,

middle, and right parts, this creates a new graph 𝐼2.

  • We can decompose 𝐼2 into new left, middle,

and right parts!

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SLIDE 47

𝑉 𝐼 𝑇 = π‘ˆ π‘Š = 𝑉 𝐼2 π‘Š 𝑇2 π‘ˆβ€² 𝜏2 𝜐2 𝜏2

β€²π‘ˆ

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SLIDE 48

Choosing New Separators

  • How do we choose the new separators 𝑇′ and

π‘ˆβ€²?

  • We take 𝑇′ to be the leftmost minimum vertex

separator between 𝑉 and {intersected vertices} βˆͺ 𝑇.

  • Similarly, we take π‘ˆβ€² to be the rightmost

minimum vertex separator between {intersected vertices} βˆͺ π‘ˆ and π‘Š.

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SLIDE 49

Key Idea

  • This decomposition works the same

regardless of what 𝜏2 and 𝜏2

β€²π‘ˆ look like (see

Claim 6.11 of [BHK+16])!

  • Thus, we get a new approximate

decomposition of the form σ𝑠′=0

𝑒 2

𝑀𝑠′𝑅𝑠′

β€² 𝑀𝑠′

  • This can be bounded by

1 2 σ𝑠=0

𝑒 2

𝑀𝑠𝑀𝑠

π‘ˆ as long as

we always have that 𝑅𝑠′

β€²

β‰ͺ 1

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SLIDE 50

Bounding New Middle Parts

  • We need to show that the new middle parts

don’t have norms which are too high.

  • This is done with the intersection tradeoff

lemma (Lemma 7.12 of [BHK+16])

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SLIDE 51

Appendix II: Technical Mines

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SLIDE 52

Approximate Decomposition Mine

  • Claim: If 𝑠 is the size of the minimum vertex

separator of 𝐼, 𝑆𝐼 β‰ˆ π‘†πœπ‘†πœπ‘†πœβ€²π‘ˆ

  • There are subtle issues related to the ordering
  • f 𝑇 and π‘ˆ, the leftmost and rightmost

minimum vertex separators of 𝐼

  • How these issues should be handled depends
  • n whether we require matrix indices to be in

ascending order.

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SLIDE 53

Approximate Decomposition Mine

  • If we require matrix indices to be in ascending
  • rder, what we actually have is 𝑆𝐼 β‰ˆ

Οƒπœ,𝜐,πœβ€²π‘ˆ:𝐼=𝜏βˆͺ𝜐βˆͺπœβ€²π‘ˆ π‘†πœπ‘†πœπ‘†πœβ€²π‘ˆ where 𝜏 βˆͺ 𝜐 βˆͺ πœβ€²π‘ˆ is the graph formed by gluing 𝜏, 𝜐, πœβ€²π‘ˆ together.

  • In fact, this equation is precisely what is

needed for the approximate PSD decomposition 𝑁 β‰ˆ 𝑁𝑔𝑏𝑑𝑒 = σ𝑠=0

𝑒 2

𝑀𝑠𝑅𝑠𝑀𝑠

π‘ˆ.

slide-54
SLIDE 54

Approximate Decomposition Mine

  • Remark: [BHK+16] navigates this issue by

keeping everything in terms of the individual ribbons (Fourier characters for a given matrix entry) until it is time to use the matrix norm bounds (see Definition 6.1 and subsection 6.4

  • f [BHK+16])
slide-55
SLIDE 55

Approximate Decomposition Mine

  • If we do not require matrix indices to be in

ascending order, we actually have the following two equations

1. 𝑆𝐼 β‰ˆ 𝐡𝑣𝑒 𝜏, 𝜐, πœβ€²π‘ˆ π‘†πœπ‘†πœπ‘†πœβ€²π‘ˆwhere 𝐡𝑣𝑒 𝜏, 𝜐, πœβ€²π‘ˆ is the number of different ways to decompose 𝐼 into 𝜏, 𝜐, πœβ€²π‘ˆ. 2. 𝑆𝐼 β‰ˆ

1 𝑑𝐼 !2 Οƒπœ,𝜐,πœβ€²π‘ˆ:𝐼=𝜏βˆͺ𝜐βˆͺπœβ€²π‘ˆ π‘†πœπ‘†πœπ‘†πœβ€²π‘ˆ where

𝜏 βˆͺ 𝜐 βˆͺ πœβ€²π‘ˆ is the graph formed by gluing 𝜏, 𝜐, πœβ€²π‘ˆ together.

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SLIDE 56

Truncation Mine

  • Definition: Define 𝑀𝑠 = Οƒπœ:|π‘Š

𝜏|=𝑠

𝑙 π‘œ π‘Š 𝜏 βˆ’π‘ 

2 π‘†πœ

and define 𝑅𝑠 = Οƒπœ:|π‘‰πœ|=|π‘Š

𝜐|=𝑠

𝑙 π‘œ π‘Š 𝜐 βˆ’π‘ 

π‘†πœ where we require that π‘Š

𝜏 is the unique

minimum vertex separator of 𝜏 and π‘‰πœ, π‘Š

𝜐 are

the leftmost and rightmost minimum vertex separators of 𝜐. Define 𝑁𝑔𝑏𝑑𝑒 = σ𝑠=0

𝑒 2

𝑀𝑠𝑅𝑠𝑀𝑠

π‘ˆ

  • Actually, we need to truncate 𝑀𝑠 and 𝑆𝑠 by only

taking 𝜏, 𝜐 with at most 𝐸 vertices

slide-57
SLIDE 57

Truncation Mine

  • Warning: There is a mismatch between 𝐼

which have at most 𝐸 vertices and triples 𝜏, 𝜐, πœβ€²π‘ˆ which each have at most 𝐸 vertices.

  • This truncation error turns out to have very

small total norm, see Lemma 7.4 of [BHK+16]