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Discrepancy Theory and Applications to Bin Packing Thomas Rothvoss - - PowerPoint PPT Presentation

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Discrepancy Theory and Applications to Bin Packing Thomas Rothvoss Joint work with Becca Hoberg Discrepancy theory Set system S = { S 1 , . . . , S m } , S i [ n ] i S b b b Discrepancy theory 1 Set system S = { S 1 , . . .

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SLIDE 1

Discrepancy Theory and Applications to Bin Packing

Thomas Rothvoss

Joint work with Becca Hoberg

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SLIDE 2

Discrepancy theory

◮ Set system S = {S1, . . . , Sm}, Si ⊆ [n]

i S

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SLIDE 3

Discrepancy theory

◮ Set system S = {S1, . . . , Sm}, Si ⊆ [n] ◮ Coloring χ : [n] → {−1, +1}

i S

b b

−1

b

+1 −1

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SLIDE 4

Discrepancy theory

◮ Set system S = {S1, . . . , Sm}, Si ⊆ [n] ◮ Coloring χ : [n] → {−1, +1} ◮ Discrepancy

disc(S) = min

χ:[n]→{±1} max S∈S

  • i∈S

χ(i)

  • .

i S

b b

−1

b

+1 −1

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SLIDE 5

Discrepancy theory

◮ Set system S = {S1, . . . , Sm}, Si ⊆ [n] ◮ Coloring χ : [n] → {−1, +1} ◮ Discrepancy

disc(S) = min

χ:[n]→{±1} max S∈S

  • i∈S

χ(i)

  • .

i S

b b

−1

b

+1 −1 Known results:

◮ n sets, n elements: disc(S) = O(√n) [Spencer ’85]

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SLIDE 6

Discrepancy theory

◮ Set system S = {S1, . . . , Sm}, Si ⊆ [n] ◮ Coloring χ : [n] → {−1, +1} ◮ Discrepancy

disc(S) = min

χ:[n]→{±1} max S∈S

  • i∈S

χ(i)

  • .

i S

b b

−1

b

+1 −1 Known results:

◮ n sets, n elements: disc(S) = O(√n) [Spencer ’85] ◮ Every element in ≤ t sets: disc(S) < 2t [Beck & Fiala ’81]

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SLIDE 7

Discrepancy theory

◮ Set system S = {S1, . . . , Sm}, Si ⊆ [n] ◮ Coloring χ : [n] → {−1, +1} ◮ Discrepancy

disc(S) = min

χ:[n]→{±1} max S∈S

  • i∈S

χ(i)

  • .

i S

b b

−1

b

+1 −1 Known results:

◮ n sets, n elements: disc(S) = O(√n) [Spencer ’85] ◮ Every element in ≤ t sets: disc(S) < 2t [Beck & Fiala ’81]

Main method: Iteratively find a partial coloring.

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SLIDE 8

Discrepancy algorithm

Theorem (R., FOCS 2014)

For a convex symmetric set K ⊆ Rn with Pr[gaussian ∈ K] ≥ e−Θ(n), one can find a y ∈ K ∩ [−1, 1]n with |{i : yi = ±1}| ≥ Θ(n) in poly-time. K y∗ [−1, 1]n

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SLIDE 9

Discrepancy algorithm

Theorem (R., FOCS 2014)

For a convex symmetric set K ⊆ Rn with Pr[gaussian ∈ K] ≥ e−Θ(n), one can find a y ∈ K ∩ [−1, 1]n with |{i : yi = ±1}| ≥ Θ(n) in poly-time. Algorithm: (1) take a random Gaussian x∗ K x∗ [−1, 1]n

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SLIDE 10

Discrepancy algorithm

Theorem (R., FOCS 2014)

For a convex symmetric set K ⊆ Rn with Pr[gaussian ∈ K] ≥ e−Θ(n), one can find a y ∈ K ∩ [−1, 1]n with |{i : yi = ±1}| ≥ Θ(n) in poly-time. Algorithm: (1) take a random Gaussian x∗ (2) compute y∗ = argmin{x∗ − y2 | y ∈ K ∩ [−1, 1]n} K x∗ y∗ [−1, 1]n

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SLIDE 11

Analysis

K [−1, 1]n

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SLIDE 12

Analysis

◮ W.h.p. x∗ − y∗2 ≥ 1 5

√n K x∗ y∗ [−1, 1]n

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SLIDE 13

Analysis

◮ W.h.p. x∗ − y∗2 ≥ 1 5

√n

◮ Fact: For any set Q: Pr[gaussian ∈ Q] ≥ e−o(n) ⇒

E[dist(gaussian, Q)] ≤ o(√n). K x∗ y∗ [−1, 1]n

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SLIDE 14

Analysis

◮ W.h.p. x∗ − y∗2 ≥ 1 5

√n

◮ Fact: For any set Q: Pr[gaussian ∈ Q] ≥ e−o(n) ⇒

E[dist(gaussian, Q)] ≤ o(√n).

◮ Key observation:

y∗ − x∗2 = min{y − x∗2 | y ∈ K and |yi| ≤ 1 ∀ i} K x∗ y∗ [−1, 1]n

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SLIDE 15

Analysis

◮ W.h.p. x∗ − y∗2 ≥ 1 5

√n

◮ Fact: For any set Q: Pr[gaussian ∈ Q] ≥ e−o(n) ⇒

E[dist(gaussian, Q)] ≤ o(√n).

◮ Key observation:

y∗ −x∗2 = min{y −x∗2 | y ∈ K and |yi| ≤ 1 ∀ tight i} K ∩ STRIP K x∗ y∗ [−1, 1]n

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SLIDE 16

Analysis

◮ W.h.p. x∗ − y∗2 ≥ 1 5

√n

◮ Fact: For any set Q: Pr[gaussian ∈ Q] ≥ e−o(n) ⇒

E[dist(gaussian, Q)] ≤ o(√n).

◮ Key observation:

y∗ −x∗2 = min{y −x∗2 | y ∈ K and |yi| ≤ 1 ∀ tight i}

◮ Strip of o(n) coord.: Pr[gaussian ∈ K ∩ STRIP] ≥ e−Ω(n).

K ∩ STRIP K x∗ y∗ [−1, 1]n

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SLIDE 17

Analysis

◮ W.h.p. x∗ − y∗2 ≥ 1 5

√n

◮ Fact: For any set Q: Pr[gaussian ∈ Q] ≥ e−o(n) ⇒

E[dist(gaussian, Q)] ≤ o(√n).

◮ Key observation:

y∗ −x∗2 = min{y −x∗2 | y ∈ K and |yi| ≤ 1 ∀ tight i}

◮ Strip of o(n) coord.: Pr[gaussian ∈ K ∩ STRIP] ≥ e−Ω(n). ◮ Then E[dist(gaussian, K ∩ STRIP)] ≤ o(√n).

Contradiction! K ∩ STRIP K x∗ y∗ [−1, 1]n

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SLIDE 18

Application to Bin Packing

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SLIDE 19

Bin Packing

Input: Items with sizes s1, . . . , sn ∈ [0, 1] Goal: Pack items into minimum number of bins of size 1. bin 1 bin 2 input si 1 1

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SLIDE 20

Bin Packing

Input: Items with sizes s1, . . . , sn ∈ [0, 1] Goal: Pack items into minimum number of bins of size 1. bin 1 bin 2 input si 1 1

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SLIDE 21

Bin Packing

Input: Items with sizes s1, . . . , sn ∈ [0, 1] Goal: Pack items into minimum number of bins of size 1. bin 1 bin 2 input si 1 1

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SLIDE 22

Bin Packing

Input: Items with sizes s1, . . . , sn ∈ [0, 1] Goal: Pack items into minimum number of bins of size 1. bin 1 bin 2 input si 1 1

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SLIDE 23

Bin Packing

Input: Items with sizes s1, . . . , sn ∈ [0, 1] Goal: Pack items into minimum number of bins of size 1. bin 1 bin 2 input si 1 1

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SLIDE 24

Bin Packing

Input: Items with sizes s1, . . . , sn ∈ [0, 1] Goal: Pack items into minimum number of bins of size 1. bin 1 bin 2 input si 1 1

◮ NP-hard to distinguish OPT ≤ 2 or OPT ≥ 3

[Garey & Johnson ’79]

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SLIDE 25

Bin Packing

Input: Items with sizes s1, . . . , sn ∈ [0, 1] Goal: Pack items into minimum number of bins of size 1. bin 1 bin 2 input si 1 1

◮ NP-hard to distinguish OPT ≤ 2 or OPT ≥ 3

[Garey & Johnson ’79]

◮ First Fit Decreasing [Johnson ’73]: APX ≤ 11 9 OPT + 4

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SLIDE 26

Bin Packing

Input: Items with sizes s1, . . . , sn ∈ [0, 1] Goal: Pack items into minimum number of bins of size 1. bin 1 bin 2 input si 1 1

◮ NP-hard to distinguish OPT ≤ 2 or OPT ≥ 3

[Garey & Johnson ’79]

◮ First Fit Decreasing [Johnson ’73]: APX ≤ 11 9 OPT + 4 ◮ [de la Vega & L¨

ucker ’81] : APX ≤ (1 + ε)OPT + O(1/ε2) in time O(n) · f(ε)

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SLIDE 27

Bin Packing

Input: Items with sizes s1, . . . , sn ∈ [0, 1] Goal: Pack items into minimum number of bins of size 1. bin 1 bin 2 input si 1 1

◮ NP-hard to distinguish OPT ≤ 2 or OPT ≥ 3

[Garey & Johnson ’79]

◮ First Fit Decreasing [Johnson ’73]: APX ≤ 11 9 OPT + 4 ◮ [de la Vega & L¨

ucker ’81] : APX ≤ (1 + ε)OPT + O(1/ε2) in time O(n) · f(ε)

◮ [Karmarkar & Karp ’82]: APX ≤ OPT + O(log2 OPT) in

poly-time

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SLIDE 28

The Gilmore Gomory LP relaxation

◮ bi = #items with size si ◮ Feasible patterns:

P =

  • p ∈ Zn

≥0 | n

  • i=1

sipi ≤ 1

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SLIDE 29

The Gilmore Gomory LP relaxation

◮ bi = #items with size si ◮ Feasible patterns:

P =

  • p ∈ Zn

≥0 | n

  • i=1

sipi ≤ 1

  • ◮ Gilmore Gomory LP relaxation:

min

  • p∈P

xp

  • p∈P

pi · xp ≥ bi ∀i ∈ [n] xp ≥ ∀p ∈ P

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SLIDE 30

The Gilmore Gomory LP relaxation

◮ bi = #items with size si ◮ Feasible patterns:

P =

  • p ∈ Zn

≥0 | n

  • i=1

sipi ≤ 1

  • ◮ Gilmore Gomory LP relaxation:

min 1Tx

  • p∈P

Ax ≥ b xp ≥ ∀p ∈ P

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SLIDE 31

The Gilmore Gomory LP relaxation

◮ bi = #items with size si ◮ Feasible patterns:

P =

  • p ∈ Zn

≥0 | n

  • i=1

sipi ≤ 1

  • ◮ Gilmore Gomory LP relaxation:

min 1Tx

  • p∈P

Ax ≥ b xp ≥ ∀p ∈ P

◮ Can find x with 1Tx ≤ OPTf + δ in time poly(b1, 1 δ)

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SLIDE 32

The Gilmore Gomory LP - Example

input si 1 0.44 0.4 0.3 0.26

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SLIDE 33

The Gilmore Gomory LP - Example

input si 1 0.44 0.4 0.3 0.26 min 1Tx     2 1 1 1 1 2 1 1 1 1 3 1 1 1 1 1 3 1 1 1 1 1     x ≥     1 1 1 1     x ≥

A

b

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SLIDE 34

The Gilmore Gomory LP - Example

input si 1 0.44 0.4 0.3 0.26 min 1Tx     2 1 1 1 1 2 1 1 1 1 3 1 1 1 1 1 3 1 1 1 1 1     x ≥     1 1 1 1     x ≥

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SLIDE 35

The Gilmore Gomory LP - Example

input si 1 0.44 0.4 0.3 0.26 min 1Tx     2 1 1 1 1 2 1 1 1 1 3 1 1 1 1 1 3 1 1 1 1 1     x ≥     1 1 1 1     x ≥ 1/2× 1/2× 1/2×

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SLIDE 36

Karmarkar-Karp’s Grouping

input:

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SLIDE 37

Karmarkar-Karp’s Grouping

input: si ∈ [2, 3]

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SLIDE 38

Karmarkar-Karp’s Grouping

input: si ∈ [2, 3] new input: 3× 4× 4×

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SLIDE 39

Karmarkar-Karp’s Grouping

input: si ∈ [2, 3] new input: 3× 4× 4×

◮ increases OPT by O(log n)

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SLIDE 40

Karmarkar-Karp’s Grouping

input: si ∈ [2, 3] new input: 3× 4× 4×

◮ increases OPT by O(log n)

A

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SLIDE 41

Karmarkar-Karp’s Grouping

input: si ∈ [2, 3] new input: 3× 4× 4×

◮ increases OPT by O(log n)

A

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SLIDE 42

Karmarkar-Karp’s Grouping

input: si ∈ [2, 3] new input: 3× 4× 4×

◮ increases OPT by O(log n)

A

A

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SLIDE 43

Karmarkar-Karp’s Grouping

input: si ∈ [2, 3] new input: 3× 4× 4×

◮ increases OPT by O(log n) ◮ row sum · si ≥ 2

A

A

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SLIDE 44

Karmarkar-Karp’s Grouping

input: si ∈ [2, 3] new input: 3× 4× 4×

◮ increases OPT by O(log n) ◮ row sum · si ≥ 2

← → column sum (w.r.t si) ≤ 1

A

A

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SLIDE 45

Karmarkar-Karp’s Grouping

input: si ∈ [2, 3] new input: 3× 4× 4×

◮ increases OPT by O(log n) ◮ row sum · si ≥ 2

← → column sum (w.r.t si) ≤ 1

◮ # constraints ≤ 1 2support(x)

A

A

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SLIDE 46

Karmarkar-Karp algo (2)

b

x

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SLIDE 47

Karmarkar-Karp algo (2)

◮ After grouping: # constraints ≤ 1 2|supp(x)|

A

b

x

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SLIDE 48

Karmarkar-Karp algo (2)

◮ After grouping: # constraints ≤ 1 2|supp(x)|

A

Ax = b

b

x

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SLIDE 49

Karmarkar-Karp algo (2)

◮ After grouping: # constraints ≤ 1 2|supp(x)| ◮ Move to basic solution x′ with Ax′ = Ax

A

Ax = b

b

x

b x′

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SLIDE 50

Karmarkar-Karp algo (2)

◮ After grouping: # constraints ≤ 1 2|supp(x)| ◮ Move to basic solution x′ with Ax′ = Ax ◮ |supp(x′ − ⌊x′⌋)| ≤ 1 2|supp(x)|

A

Ax = b

b

x

b x′

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SLIDE 51

Karmarkar-Karp algo (2)

◮ After grouping: # constraints ≤ 1 2|supp(x)| ◮ Move to basic solution x′ with Ax′ = Ax ◮ |supp(x′ − ⌊x′⌋)| ≤ 1 2|supp(x)| ◮ Repeat O(log n) times → O(log2n)

A

Ax = b

b

x

b x′

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SLIDE 52

Applying Discrepancy to Bin Packing

Ax = Ax0

b

x0

b x

Karmarkar & Karp

b

x0

b x

Hoberg & R.

Theorem (R. FOCS ’13, Hoberg-R. SODA ’17)

One can find a packing with OP T + O(log n) bins in poly-time.

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SLIDE 53

Beating Karmarkar & Karp

Assumptions:

◮ Fractional solution x ∈ [0, 1]n, matrix A is O(n) × n

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SLIDE 54

Beating Karmarkar & Karp

Assumptions:

◮ Fractional solution x ∈ [0, 1]n, matrix A is O(n) × n ◮ Item sizes in [ 1 k, 2 k]

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SLIDE 55

Beating Karmarkar & Karp

Assumptions:

◮ Fractional solution x ∈ [0, 1]n, matrix A is O(n) × n ◮ Item sizes in [ 1 k, 2 k] ◮ OPTf increases whenever glue 2 items

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SLIDE 56

Beating Karmarkar & Karp

Assumptions:

◮ Fractional solution x ∈ [0, 1]n, matrix A is O(n) × n ◮ Item sizes in [ 1 k, 2 k] ◮ OPTf increases whenever glue 2 items ⇒ Ai1 ≥ k1/2

and A∞ ≤ k1/4.

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SLIDE 57

Beating Karmarkar & Karp

Assumptions:

◮ Fractional solution x ∈ [0, 1]n, matrix A is O(n) × n ◮ Item sizes in [ 1 k, 2 k] ◮ OPTf increases whenever glue 2 items ⇒ Ai1 ≥ k1/2

and A∞ ≤ k1/4. Goal: Replace x with y so that |frac(y)| ≤ n

2.

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SLIDE 58

Beating Karmarkar & Karp

Assumptions:

◮ Fractional solution x ∈ [0, 1]n, matrix A is O(n) × n ◮ Item sizes in [ 1 k, 2 k] ◮ OPTf increases whenever glue 2 items ⇒ Ai1 ≥ k1/2

and A∞ ≤ k1/4. Goal: Replace x with y so that |frac(y)| ≤ n

2.

Karmarkar-Karp: Loose O(k) items = O(1) bins

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SLIDE 59

Beating Karmarkar & Karp

Assumptions:

◮ Fractional solution x ∈ [0, 1]n, matrix A is O(n) × n ◮ Item sizes in [ 1 k, 2 k] ◮ OPTf increases whenever glue 2 items ⇒ Ai1 ≥ k1/2

and A∞ ≤ k1/4. Goal: Replace x with y so that |frac(y)| ≤ n

2.

Karmarkar-Karp: Loose O(k) items = O(1) bins We: Loose O(k15/16) items = O(k−1/16) bins

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SLIDE 60

Beating Karmarkar & Karp

Assumptions:

◮ Fractional solution x ∈ [0, 1]n, matrix A is O(n) × n ◮ Item sizes in [ 1 k, 2 k] ◮ OPTf increases whenever glue 2 items ⇒ Ai1 ≥ k1/2

and A∞ ≤ k1/4. Goal: Replace x with y so that |frac(y)| ≤ n

2.

Karmarkar-Karp: Loose O(k) items = O(1) bins We: Loose O(k15/16) items = O(k−1/16) bins

Lemma (Lovett-Meka)

For x ∈ [0, 1]n, vectors vi, parameters λi ≥ 0 with

m

  • i=1

exp(−λ2

i /16) ≤ n

16 can find partial coloring y ∈ [0, 1]n with at least half the entries in {0, 1} and | vi, x − y | ≤ λivi2.

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SLIDE 61

Applying the Partial Coloring Lemma

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

A

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SLIDE 62

Applying the Partial Coloring Lemma

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

A

◮ Given: x. Find: y with |( j≤i Aj)(x − y)| small

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SLIDE 63

Applying the Partial Coloring Lemma

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

I

◮ Given: x. Find: y with |( j≤i Aj)(x − y)| small

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SLIDE 64

Applying the Partial Coloring Lemma

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

I + 1 0 2 1 1 1 1 1 ) vI = (

◮ Given: x. Find: y with |( j≤i Aj)(x − y)| small ◮ For interval I ⊆ [n]: vI := i∈I Ai

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SLIDE 65

Applying the Partial Coloring Lemma

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

. . . . . . ← level I vI1 = ck17/16

◮ Given: x. Find: y with |( j≤i Aj)(x − y)| small ◮ For interval I ⊆ [n]: vI := i∈I Ai

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SLIDE 66

Applying the Partial Coloring Lemma

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

. . . . . . ← level . . . . . . 1 I vI1 = 2−1ck17/16

◮ Given: x. Find: y with |( j≤i Aj)(x − y)| small ◮ For interval I ⊆ [n]: vI := i∈I Ai

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SLIDE 67

Applying the Partial Coloring Lemma

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

. . . . . . ← level . . . . . . 1 . . . . . . 2 I vI1 = 2−2ck17/16

◮ Given: x. Find: y with |( j≤i Aj)(x − y)| small ◮ For interval I ⊆ [n]: vI := i∈I Ai

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SLIDE 68

Applying the Partial Coloring Lemma

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

. . . . . . ← level . . . . . . 1 . . . . . . 2 . . . . . . 3 I vI1 = 2−3ck17/16

◮ Given: x. Find: y with |( j≤i Aj)(x − y)| small ◮ For interval I ⊆ [n]: vI := i∈I Ai

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SLIDE 69

Applying the Partial Coloring Lemma

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

. . . . . . ← level . . . . . . 1 . . . . . . 2 . . . . . . 3 I vI1 = 2−3ck17/16

◮ Given: x. Find: y with |( j≤i Aj)(x − y)| small ◮ For interval I ⊆ [n]: vI := i∈I Ai , λI := level(I)

slide-70
SLIDE 70

Applying the Partial Coloring Lemma

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

n columns . . . . . . ← level . . . . . . 1 . . . . . . 2 . . . . . . 3

◮ Run Partial coloring with vI := i∈I Ai and λI := level(I)

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SLIDE 71

Applying the Partial Coloring Lemma

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

n columns . . . . . . ← level . . . . . . 1 . . . . . . 2 . . . . . . 3

◮ Run Partial coloring with vI := i∈I Ai and λI := level(I)

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SLIDE 72

Applying the Partial Coloring Lemma

sum ≤ k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

n columns . . . . . . ← level . . . . . . 1 . . . . . . 2 . . . . . . 3

◮ Run Partial coloring with vI := i∈I Ai and λI := level(I)

  • I

e−λ2

I/16

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SLIDE 73

Applying the Partial Coloring Lemma

sum ≤ k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

n columns . . . . . . ← level . . . . . . 1 . . . . . . 2 . . . . . . 3

◮ Run Partial coloring with vI := i∈I Ai and λI := level(I)

  • I

e−λ2

I/16 ≤

  • ℓ≥0

k · n ck17/162−ℓ · e−ℓ2/16

slide-74
SLIDE 74

Applying the Partial Coloring Lemma

i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

n columns . . . . . . ← level . . . . . . 1 . . . . . . 2 . . . . . . 3

◮ Bound error for item i:

|(

  • j≤i

Aj)(x − y)| ≤

slide-75
SLIDE 75

Applying the Partial Coloring Lemma

i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

n columns . . . . . . ← level . . . . . . 1 . . . . . . 2 . . . . . . 3 . . .

◮ Bound error for item i:

|(

  • j≤i

Aj)(x − y)| ≤

slide-76
SLIDE 76

Applying the Partial Coloring Lemma

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

n columns . . . . . . ← level . . . . . . 1 . . . . . . 2 . . . . . . 3 . . .

◮ Bound error for item i:

|(

  • j≤i

Aj)(x − y)| ≤

  • ℓ≥0

ℓ · vI on level ℓ2

slide-77
SLIDE 77

Applying the Partial Coloring Lemma

vI1 = Θ(k17/16) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

n columns . . . . . . ← level . . . . . . 1 . . . . . . 2 . . . . . . 3 I

◮ Bound error for item i:

|(

  • j≤i

Aj)(x − y)| ≤

  • ℓ≥0

ℓ · vI on level ℓ2 ≤ O(1) · vI2

slide-78
SLIDE 78

Applying the Partial Coloring Lemma

vI1 = Θ(k17/16) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

n columns . . . . . . ← level . . . . . . 1 . . . . . . 2 . . . . . . 3 I

◮ Bound error for item i:

|(

  • j≤i

Aj)(x − y)| ≤

  • ℓ≥0

ℓ · vI on level ℓ2 ≤ O(1) · vI2

H¨

  • lder

≤ vI1 ·

  • k1/4

k1/2

slide-79
SLIDE 79

Applying the Partial Coloring Lemma

vI1 = Θ(k17/16) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

n columns . . . . . . ← level . . . . . . 1 . . . . . . 2 . . . . . . 3 I

◮ Bound error for item i:

|(

  • j≤i

Aj)(x − y)| ≤

  • ℓ≥0

ℓ · vI on level ℓ2 ≤ O(1) · vI2

H¨

  • lder

≤ vI1 ·

  • k1/4

k1/2 ≤ O(k15/16)

slide-80
SLIDE 80

The end

Open problems:

slide-81
SLIDE 81

The end

Open problems:

  • 1. Close the gap

1 ≤ additive integrality gap ≤ O(log OPT)

slide-82
SLIDE 82

The end

Open problems:

  • 1. Close the gap

1 ≤ additive integrality gap ≤ O(log OPT)

  • 2. Find more applications in LP rounding.
slide-83
SLIDE 83

The end

Open problems:

  • 1. Close the gap

1 ≤ additive integrality gap ≤ O(log OPT)

  • 2. Find more applications in LP rounding.

Thanks for your attention