[PPT] - The Entropy Rounding Method in Approximation Algorithms Thomas PowerPoint Presentation

SLIDE 1

The Entropy Rounding Method in Approximation Algorithms

Thomas Rothvoß

Department of Mathematics, M.I.T.

Carg` ese 2011

SLIDE 2

A general rounding problem

Problem:

◮ Given: A ∈ Rn×m, fractional solution x ∈ [0, 1]m ◮ Find: y ∈ {0, 1}m : Ax ≈ Ay

SLIDE 3

A general rounding problem

Problem:

◮ Given: A ∈ Rn×m, fractional solution x ∈ [0, 1]m ◮ Find: y ∈ {0, 1}m : Ax ≈ Ay

General strategies:

◮ Randomized rounding:

Flip Pr[yi = 1] = xi (in)dependently

SLIDE 4

A general rounding problem

Problem:

◮ Given: A ∈ Rn×m, fractional solution x ∈ [0, 1]m ◮ Find: y ∈ {0, 1}m : Ax ≈ Ay

General strategies:

◮ Randomized rounding:

Flip Pr[yi = 1] = xi (in)dependently

◮ Use properties of basic solutions:

|supp(x)| ≤ #rows(A)

SLIDE 5

A general rounding problem

Problem:

◮ Given: A ∈ Rn×m, fractional solution x ∈ [0, 1]m ◮ Find: y ∈ {0, 1}m : Ax ≈ Ay

General strategies:

◮ Randomized rounding:

Flip Pr[yi = 1] = xi (in)dependently

◮ Use properties of basic solutions:

|supp(x)| ≤ #rows(A) Try another way:

◮ “Entropy rounding method” based on discrepancy theory

SLIDE 6

A general rounding problem

Problem:

◮ Given: A ∈ Rn×m, fractional solution x ∈ [0, 1]m ◮ Find: y ∈ {0, 1}m : Ax ≈ Ay

General strategies:

◮ Randomized rounding:

Flip Pr[yi = 1] = xi (in)dependently

◮ Use properties of basic solutions:

|supp(x)| ≤ #rows(A) Try another way:

◮ “Entropy rounding method” based on discrepancy theory

Application:

◮ A (OPT + O(log2 OPT))-algorithm for Bin Packing

With Rejection

SLIDE 7

Discrepancy theory

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

i S

SLIDE 8

Discrepancy theory

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

i S

b b b

SLIDE 9

Discrepancy theory

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

disc(S) = min

χ:[n]→{±1} max S∈S |χ(S)|.

where χ(S) =

i∈S χ(i).

i S

b b b

SLIDE 10

Discrepancy theory

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

disc(S) = min

χ:[n]→{±1} max S∈S |χ(S)|.

where χ(S) =

i∈S χ(i).

i S

b b b

Known results:

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

SLIDE 11

Discrepancy theory

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

disc(S) = min

χ:[n]→{±1} max S∈S |χ(S)|.

where χ(S) =

i∈S χ(i).

i S

b b b

Known results:

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

Conjecture: disc(S) ≤ O( √ t)

SLIDE 12

Discrepancy theory

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

disc(S) = min

χ:[n]→{±1} max S∈S |χ(S)|.

where χ(S) =

i∈S χ(i).

i S

b b b

b

Known results:

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

Conjecture: disc(S) ≤ O( √ t) More definitions:

◮ Partial coloring: χ : [n] → {0, −1, +1} ◮ Half coloring: χ : [n] → {0, −1, +1}, |supp(χ)| ≥ n/2

SLIDE 13

Matrix discrepancy

◮ Matrix A

disc(A) = min

χ∈{±1}n Aχ∞

i S A =   1 1 1 1 1 1   set S i disc(S) = disc(A)

SLIDE 14

Matrix discrepancy

◮ Matrix A

disc(A) = min

χ∈{±1}n Aχ∞

i S A =   1 1 1 1 1 1   set S i disc(S) = disc(A)

Theorem (Lov´ asz, Spencer & Vesztergombi ’86)

Finding good colorings ⇒ ∀x ∈ [0, 1]m can find y ∈ {0, 1}m with Ax − Ay∞ small

SLIDE 15

Entropy rounding (simple version)

◮ Input: A ∈ Rn×m, x ∈ [0, 1]m ◮ Assume: ∀ submatrix A′ ⊆ A: half-coloring χ:

A′χ∞ ≤ ∆

◮ Output: y ∈ {0, 1}m: Ax − Ay∞ ≤ O(log m) · ∆

(1) y := x (2) FOR phase k = last bit TO 1 DO

(3) Call yi active if kth bit is 1 (4) Find half-coloring χ : active var → {−1, +1, 0} (5) Update y′ := y + ( 1

2)kχ

(6) REPEAT WHILE ∃ active var.

A =

y = (0.11 0.10 0.11 0.01 )

SLIDE 16

Entropy rounding (simple version)

◮ Input: A ∈ Rn×m, x ∈ [0, 1]m ◮ Assume: ∀ submatrix A′ ⊆ A: half-coloring χ:

A′χ∞ ≤ ∆

◮ Output: y ∈ {0, 1}m: Ax − Ay∞ ≤ O(log m) · ∆

(1) y := x (2) FOR phase k = last bit TO 1 DO

(3) Call yi active if kth bit is 1 (4) Find half-coloring χ : active var → {−1, +1, 0} (5) Update y′ := y + ( 1

2)kχ

(6) REPEAT WHILE ∃ active var.

A =

y = (0.11 0.10 0.11 0.01)

SLIDE 17

Entropy rounding (simple version)

◮ Input: A ∈ Rn×m, x ∈ [0, 1]m ◮ Assume: ∀ submatrix A′ ⊆ A: half-coloring χ:

A′χ∞ ≤ ∆

◮ Output: y ∈ {0, 1}m: Ax − Ay∞ ≤ O(log m) · ∆

(1) y := x (2) FOR phase k = last bit TO 1 DO

(3) Call yi active if kth bit is 1 (4) Find half-coloring χ : active var → {−1, +1, 0} (5) Update y′ := y + ( 1

2)kχ

(6) REPEAT WHILE ∃ active var.

A =

y = (0.11 0.10 0.11 0.01)

χ = ( +1 −1 0 )

SLIDE 18

Entropy rounding (simple version)

◮ Input: A ∈ Rn×m, x ∈ [0, 1]m ◮ Assume: ∀ submatrix A′ ⊆ A: half-coloring χ:

A′χ∞ ≤ ∆

◮ Output: y ∈ {0, 1}m: Ax − Ay∞ ≤ O(log m) · ∆

(1) y := x (2) FOR phase k = last bit TO 1 DO

(3) Call yi active if kth bit is 1 (4) Find half-coloring χ : active var → {−1, +1, 0} (5) Update y′ := y + ( 1

2)kχ

(6) REPEAT WHILE ∃ active var.

A =

y = (0.11 0.10 0.11 0.01)

χ = ( +1 −1 0 ) y′ = (1.00 0.10 0.10 0.01)

SLIDE 19

Entropy rounding (simple version)

◮ Input: A ∈ Rn×m, x ∈ [0, 1]m ◮ Assume: ∀ submatrix A′ ⊆ A: half-coloring χ:

A′χ∞ ≤ ∆

◮ Output: y ∈ {0, 1}m: Ax − Ay∞ ≤ O(log m) · ∆

(1) y := x (2) FOR phase k = last bit TO 1 DO

(3) Call yi active if kth bit is 1 (4) Find half-coloring χ : active var → {−1, +1, 0} (5) Update y′ := y + ( 1

2)kχ

(6) REPEAT WHILE ∃ active var.

A =

y = (0.11 0.10 0.11 0.01)

χ = ( +1 −1 0 ) y′ = (1.00 0.10 0.10 0.01) Analysis:

◮ A phase has ≤ log m iterations

SLIDE 20

Entropy rounding (simple version)

◮ Input: A ∈ Rn×m, x ∈ [0, 1]m ◮ Assume: ∀ submatrix A′ ⊆ A: half-coloring χ:

A′χ∞ ≤ ∆

◮ Output: y ∈ {0, 1}m: Ax − Ay∞ ≤ O(log m) · ∆

(1) y := x (2) FOR phase k = last bit TO 1 DO

(3) Call yi active if kth bit is 1 (4) Find half-coloring χ : active var → {−1, +1, 0} (5) Update y′ := y + ( 1

2)kχ

(6) REPEAT WHILE ∃ active var.

A =

y = (0.11 0.10 0.11 0.01)

χ = ( +1 −1 0 ) y′ = (1.00 0.10 0.10 0.01) Analysis:

◮ A phase has ≤ log m iterations ◮ During phase k: Ay′ − Ay∞ = (1 2)kAχ∞ ≤ (1 2)k∆

SLIDE 21

Entropy rounding (simple version)

◮ Input: A ∈ Rn×m, x ∈ [0, 1]m ◮ Assume: ∀ submatrix A′ ⊆ A: half-coloring χ:

A′χ∞ ≤ ∆

◮ Output: y ∈ {0, 1}m: Ax − Ay∞ ≤ O(log m) · ∆

(1) y := x (2) FOR phase k = last bit TO 1 DO

(3) Call yi active if kth bit is 1 (4) Find half-coloring χ : active var → {−1, +1, 0} (5) Update y′ := y + ( 1

2)kχ

(6) REPEAT WHILE ∃ active var.

A =

y = (0.11 0.10 0.11 0.01)

χ = ( +1 −1 0 ) y′ = (1.00 0.10 0.10 0.01) Analysis:

◮ A phase has ≤ log m iterations ◮ During phase k: Ay′ − Ay∞ = (1 2)kAχ∞ ≤ (1 2)k∆ ◮ Triangle inequality:

Ax − Ay∞ ≤

k≥1

log m

t=1

1 2 k · ∆ = O(log m) · ∆

SLIDE 22

Entropy

Definition

For random variable Z, the entropy is H(Z) =

z

Pr[Z = z] · log2

1

Pr[Z = z]

SLIDE 23

Entropy

Definition

For random variable Z, the entropy is H(Z) =

z

Pr[Z = z] · log2

1

Pr[Z = z]

Example: Pr[Z = a] = p and Pr[Z = b] = 1 − p

0.5 1.0 0.5 1.0 H(Z) = p · log(1

p) + (1 − p) · log( 1 1−p)

p 0.5 1.0 0.5 1.0 x · log( 1

x)

x

SLIDE 24

Entropy

Definition

For random variable Z, the entropy is H(Z) =

z

Pr[Z = z] · log2

1

Pr[Z = z]

Properties:

◮ Uniform distribution maximizes entropy: If Z attains k

distinct values: H(Z) ≤ log2(k) (attained if Pr[Z = z] = 1

k ∀z).

SLIDE 25

Entropy

Definition

For random variable Z, the entropy is H(Z) =

z

Pr[Z = z] · log2

1

Pr[Z = z]

Properties:

◮ Uniform distribution maximizes entropy: If Z attains k

distinct values: H(Z) ≤ log2(k) (attained if Pr[Z = z] = 1

k ∀z). ◮ One likely event: ∃z : Pr[Z = z] ≥ (1 2)H(Z)

SLIDE 26

Entropy

Definition

For random variable Z, the entropy is H(Z) =

z

Pr[Z = z] · log2

1

Pr[Z = z]

Properties:

◮ Uniform distribution maximizes entropy: If Z attains k

distinct values: H(Z) ≤ log2(k) (attained if Pr[Z = z] = 1

k ∀z). ◮ One likely event: ∃z : Pr[Z = z] ≥ (1 2)H(Z) ◮ Subadditivity: H(f(Z, Z′)) ≤ H(Z) + H(Z′).

SLIDE 27

Chernov-type bound

Lemma

Let X1, . . . , Xn be indep. RV with Pr[Xi = ±1] = 1

2.

Pr

n
i=1

Xi

≥ λ√n
≤ 2e−λ2/2.

SLIDE 28

Chernov-type bound

Lemma

Let X1, . . . , Xn be indep. RV with Pr[Xi = ±αi] = 1

2.

Pr

n
i=1

Xi

≥ λα2
≤ 2e−λ2/2.

◮ Standard deviation:

Var[
i

Xi] =

i

E[(Xi − E[Xi])2] =

n
i=1

α2

i = α2

SLIDE 29

An isoperimetric inequality

Lemma (Special case of Isoperimetric Ineq – Kleitman’66)

For any X ⊆ {0, 1}n of size |X| ≥ 20.8n and n ≥ 2, there are x, y ∈ X with x − y1 ≥ n/2. y x

SLIDE 30

An isoperimetric inequality

Lemma (Special case of Isoperimetric Ineq – Kleitman’66)

For any X ⊆ {0, 1}n of size |X| ≥ 20.8n and n ≥ 2, there are x, y ∈ X with x − y1 ≥ n/2. y x

◮ Proof with weaker constant:

ball of radius n/10

around 0

≤
0≤q<n/10

n q

≤

en n/10 n/10 < 20.8n

SLIDE 31

Theorem [Beck’s entropy method]

H

χi∈{±1}

Aχ

2∆

≤ m

5 ⇒ ∃half-coloring χ0 : Aχ0∞ ≤ ∆.

A =

m

{±1}m

A1χ A2χ 2∆ 2∆

SLIDE 32

Theorem [Beck’s entropy method]

H

χi∈{±1}

Aχ

2∆

≤ m

5 ⇒ ∃half-coloring χ0 : Aχ0∞ ≤ ∆.

b

A =

m

{±1}m

b

A1χ A2χ 2∆ 2∆

SLIDE 33

Theorem [Beck’s entropy method]

H

χi∈{±1}

Aχ

2∆

≤ m

5 ⇒ ∃half-coloring χ0 : Aχ0∞ ≤ ∆.

b b

A =

m

{±1}m

b b

A1χ A2χ 2∆ 2∆

SLIDE 34

Theorem [Beck’s entropy method]

H

χi∈{±1}

Aχ

2∆

≤ m

5 ⇒ ∃half-coloring χ0 : Aχ0∞ ≤ ∆.

b b b

A =

m

{±1}m

b b b

A1χ A2χ 2∆ 2∆

SLIDE 35

Theorem [Beck’s entropy method]

H

χi∈{±1}

Aχ

2∆

≤ m

5 ⇒ ∃half-coloring χ0 : Aχ0∞ ≤ ∆.

b b b b

A =

m

{±1}m

b b b b

A1χ A2χ 2∆ 2∆

SLIDE 36

Theorem [Beck’s entropy method]

H

χi∈{±1}

Aχ

2∆

≤ m

5 ⇒ ∃half-coloring χ0 : Aχ0∞ ≤ ∆.

b b b b b

A =

m

{±1}m

b b b b b

A1χ A2χ 2∆ 2∆

SLIDE 37

Theorem [Beck’s entropy method]

H

χi∈{±1}

Aχ

2∆

≤ m

5 ⇒ ∃half-coloring χ0 : Aχ0∞ ≤ ∆.

b b b b b b

A =

m

{±1}m

b b b b b b

A1χ A2χ 2∆ 2∆

SLIDE 38

Theorem [Beck’s entropy method]

H

χi∈{±1}

Aχ

2∆

≤ m

5 ⇒ ∃half-coloring χ0 : Aχ0∞ ≤ ∆.

b b b b b b b

A =

m

{±1}m

b b b b b b b

A1χ A2χ 2∆ 2∆

SLIDE 39

Theorem [Beck’s entropy method]

H

χi∈{±1}

Aχ

2∆

≤ m

5 ⇒ ∃half-coloring χ0 : Aχ0∞ ≤ ∆.

b b b b b b b b

A =

m

{±1}m

b b b b b b b b

A1χ A2χ 2∆ 2∆

SLIDE 40

Theorem [Beck’s entropy method]

H

χi∈{±1}

Aχ

2∆

≤ m

5 ⇒ ∃half-coloring χ0 : Aχ0∞ ≤ ∆. ◮ ∃cell : Pr[Aχ ∈ cell] ≥ (1 2)m/5.

b b b b b b b b

A =

m

{±1}m

b b b b b b b b

A1χ A2χ 2∆ 2∆

SLIDE 41

Theorem [Beck’s entropy method]

H

χi∈{±1}

Aχ

2∆

≤ m

5 ⇒ ∃half-coloring χ0 : Aχ0∞ ≤ ∆. ◮ ∃cell : Pr[Aχ ∈ cell] ≥ (1 2)m/5. ◮ At least 2m · (1 2)m/5 = 20.8m colorings χ have Aχ ∈ cell

b b b

A =

m

{±1}m

b b b

A1χ A2χ 2∆ 2∆

SLIDE 42

Theorem [Beck’s entropy method]

H

χi∈{±1}

Aχ

2∆

≤ m

5 ⇒ ∃half-coloring χ0 : Aχ0∞ ≤ ∆. ◮ ∃cell : Pr[Aχ ∈ cell] ≥ (1 2)m/5. ◮ At least 2m · (1 2)m/5 = 20.8m colorings χ have Aχ ∈ cell ◮ Pick χ1, χ2 differing in half of entries

b b

A =

m

χ1 χ2

{±1}m

b b

A1χ A2χ 2∆ 2∆ Aχ1 Aχ2

SLIDE 43

Theorem [Beck’s entropy method]

H

χi∈{±1}

Aχ

2∆

≤ m

5 ⇒ ∃half-coloring χ0 : Aχ0∞ ≤ ∆. ◮ ∃cell : Pr[Aχ ∈ cell] ≥ (1 2)m/5. ◮ At least 2m · (1 2)m/5 = 20.8m colorings χ have Aχ ∈ cell ◮ Pick χ1, χ2 differing in half of entries ◮ Define χ0(i) := 1 2(χ1(i) − χ2(i)) ∈ {0, ±1}.

b b

A =

m

χ1 χ2

{±1}m

b b

A1χ A2χ 2∆ 2∆ Aχ1 Aχ2

SLIDE 44

Theorem [Beck’s entropy method]

H

χi∈{±1}

Aχ

2∆

≤ m

5 ⇒ ∃half-coloring χ0 : Aχ0∞ ≤ ∆. ◮ ∃cell : Pr[Aχ ∈ cell] ≥ (1 2)m/5. ◮ At least 2m · (1 2)m/5 = 20.8m colorings χ have Aχ ∈ cell ◮ Pick χ1, χ2 differing in half of entries ◮ Define χ0(i) := 1 2(χ1(i) − χ2(i)) ∈ {0, ±1}. ◮ Then Aχ0∞ ≤ 1 2Aχ1 − Aχ2∞ ≤ ∆.

b b

A =

m

χ1 χ2

{±1}m

b b

A1χ A2χ 2∆ 2∆ Aχ1 Aχ2

SLIDE 45

A slight generalization

Theorem

For any auxiliary function f(χ) with Aχ − f(χ)∞ ≤ ∆: H

χi∈{±1}

(f(χ)) ≤ m

5 ⇒ ∃half-coloring χ0 : Aχ0∞ ≤ ∆.

b b b b b b b b

A =

m

{±1}m

b b b b b b b b

A1χ A2χ 2∆ 2∆ Aχ f(χ)

SLIDE 46

A bound on the entropy

Lemma

For α ∈ Rm, ∆ > 0: H

χi∈{±1}

αT χ 2∆ ≤ G ∆ α2

=:λ

:=

9e−λ2/5

if λ ≥ 2 log2(32 + 64/λ) if λ < 2

b

2 e−Ω(λ2) λ O(log( 1

λ)) + O(1)

bound on H( αT χ

2∆

)

SLIDE 47

Proof – Case λ ≥ 2

Z = 0 Z = −1 Z = 1 ∆ −∆ 3∆ −3∆ 0.5 1.0 0.5 1.0 x · log( 1

x)

x

◮ Recall: ∆ = λ · α2 with λ ≥ 2 and Z =

αT χ

2∆

H(Z) =
i∈Z

Pr[Z = i] · log

1

Pr[Z = i]

SLIDE 48

Proof – Case λ ≥ 2

Z = 0 Z = −1 Z = 1 ∆ −∆ 3∆ −3∆ Pr[Z = 0] ≥ 1 − e−Ω(λ2) 0.5 1.0 0.5 1.0 x · log( 1

x)

x

◮ Recall: ∆ = λ · α2 with λ ≥ 2 and Z =

αT χ

2∆

◮ Pr[Z = 0] · log(

1 Pr[Z=0]) ≤ e−Ω(λ2)

H(Z) =

i∈Z

Pr[Z = i] · log

1

Pr[Z = i]

SLIDE 49

Proof – Case λ ≥ 2

Z = 0 Z = −1 Z = 1 ∆ −∆ 3∆ −3∆ Pr[Z = i] ≤ Pr[αT χ is ≥ iλ times standard dev] ≤ e−Ω(i2λ2) 0.5 1.0 0.5 1.0 x · log( 1

x)

x

◮ Recall: ∆ = λ · α2 with λ ≥ 2 and Z =

αT χ

2∆

◮ Pr[Z = 0] · log(

1 Pr[Z=0]) ≤ e−Ω(λ2) ◮ Pr[Z = i] · log( 1 Pr[Z=i]) ≤ e−Ω(i2λ2) · log(ei2λ2)

H(Z) =

i∈Z

Pr[Z = i] · log

1

Pr[Z = i]

SLIDE 50

Proof – Case λ ≥ 2

Z = 0 Z = −1 Z = 1 ∆ −∆ 3∆ −3∆ Pr[Z = i] ≤ Pr[αT χ is ≥ iλ times standard dev] ≤ e−Ω(i2λ2) 0.5 1.0 0.5 1.0 x · log( 1

x)

x

◮ Recall: ∆ = λ · α2 with λ ≥ 2 and Z =

αT χ

2∆

◮ Pr[Z = 0] · log(

1 Pr[Z=0]) ≤ e−Ω(λ2) ◮ Pr[Z = i] · log( 1 Pr[Z=i]) ≤ e−Ω(i2λ2) · log(ei2λ2)

H(Z) =

i∈Z

Pr[Z = i] · log

1

Pr[Z = i]

≤ e−Ω(λ2)

SLIDE 51

Proof – Case λ < 2

α2 −α2 3α2 −3α2 2∆ Subadditivity: H(Z) ≤

SLIDE 52

Proof – Case λ < 2

block α2 −α2 3α2 −3α2 2∆ Subadditivity: H(Z) ≤ H(which block of length 2α2) + ≤ O(1) +

SLIDE 53

Proof – Case λ < 2

block α2 −α2 3α2 −3α2 2∆

2α2 2∆

= 1

λ intervals

Subadditivity: H(Z) ≤ H(which block of length 2α2) + H(index) ≤ O(1) + O(log 1

λ)

SLIDE 54

Entropy rounding (extended version)

Algorithm:

◮ Input: A ∈ Rn×m, x ∈ [0, 1]m

(1) y := x (2) FOR phase k = last bit TO 1 DO

(3) Call yi active if kth bit is 1 (4) Find half-coloring χ : active var → {−1, +1, 0} (5) Update y′ := y + ( 1

2)kχ

(6) REPEAT WHILE ∃ active var.

SLIDE 55

Entropy rounding (extended version)

Algorithm:

◮ Input: A ∈ [−1, 1]n×m, x ∈ [0, 1]m ◮ Row weights w(i) ( i w(i) = 1; w(i) ≥ 0)

(1) y := x (2) FOR phase k = last bit TO 1 DO

(3) Call yi active if kth bit is 1 (4) Find half-coloring χ : active var → {−1, +1, 0} with |Aiχ| ≤ ∆i s.t. G(

∆i √#active var ) ≤ w(i) · #active var 5

(5) Update y′ := y + ( 1

2)kχ

(6) REPEAT WHILE ∃ active var.

SLIDE 56

Entropy rounding (extended version)

Algorithm:

◮ Input: A ∈ [−1, 1]n×m, x ∈ [0, 1]m ◮ Row weights w(i) ( i w(i) = 1; w(i) ≥ 0)

(1) y := x (2) FOR phase k = last bit TO 1 DO

(3) Call yi active if kth bit is 1 (4) Find half-coloring χ : active var → {−1, +1, 0} with |Aiχ| ≤ ∆i s.t. G(

∆i √#active var ) ≤ w(i) · #active var 5

(5) Update y′ := y + ( 1

2)kχ

(6) REPEAT WHILE ∃ active var.

◮ In each step:

H

Aiχ

2∆i

i

Subadd. ≤

n

i=1

H Aiχ 2∆i

≤

n

i=1

G

∆i

√#act. var

≤ #act.var.

5

◮ Use α ∈ [−1, 1]m′ ⇒ α2 ≤

√ m′

SLIDE 57

Entropy rounding (extended version) (2)

◮ Consider row i while rounding bit k:

∆i

b

Θ

1

w(i)

v := #active var.

∼ (1

2)v·w(i) · √v

∼

ln(

1 v·w(i)) · √v

Θ

1

w(i)

SLIDE 58

Entropy rounding (extended version) (2)

◮ Consider row i while rounding bit k:

∆i

b

Θ

1

w(i)

v := #active var.

∼ (1

2)v·w(i) · √v

∼

ln(

1 v·w(i)) · √v

Θ

1

w(i)

it 1

SLIDE 59

Entropy rounding (extended version) (2)

◮ Consider row i while rounding bit k:

∆i

b

Θ

1

w(i)

v := #active var.

∼ (1

2)v·w(i) · √v

∼

ln(

1 v·w(i)) · √v

Θ

1

w(i)

it 1

it 2 v halves

SLIDE 60

Entropy rounding (extended version) (2)

◮ Consider row i while rounding bit k:

∆i

b

Θ

1

w(i)

v := #active var.

∼ (1

2)v·w(i) · √v

∼

ln(

1 v·w(i)) · √v

Θ

1

w(i)

it 1

it 2 it . . . v halves

SLIDE 61

Entropy rounding (extended version) (2)

◮ Consider row i while rounding bit k:

∆i

b

Θ

1

w(i)

v := #active var.

∼ (1

2)v·w(i) · √v

∼

ln(

1 v·w(i)) · √v

Θ

1

w(i)

it 1

it 2 it . . . v halves

SLIDE 62

Entropy rounding (extended version) (2)

◮ Consider row i while rounding bit k:

∆i

b

Θ

1

w(i)

v := #active var.

∼ (1

2)v·w(i) · √v

∼

ln(

1 v·w(i)) · √v

Θ

1

w(i)

it 1

it 2 it . . . v halves

SLIDE 63

Entropy rounding (extended version) (2)

◮ Consider row i while rounding bit k:

∆i

b

Θ

1

w(i)

v := #active var.

∼ (1

2)v·w(i) · √v

∼

ln(

1 v·w(i)) · √v

Θ

1

w(i)

it 1

it 2 it . . . v halves

SLIDE 64

Entropy rounding (extended version) (2)

◮ Consider row i while rounding bit k:

∆i

b

Θ

1

w(i)

v := #active var.

∼ (1

2)v·w(i) · √v

∼

ln(

1 v·w(i)) · √v

Θ

1

w(i)

it 1

it 2 it . . . it log m v halves

SLIDE 65

Entropy rounding (extended version) (2)

◮ Consider row i while rounding bit k:

∆i

b

Θ

1

w(i)

v := #active var.

∼ (1

2)v·w(i) · √v

∼

ln(

1 v·w(i)) · √v

Θ

1

w(i)

it 1

it 2 it . . . it log m v halves

◮ By convergence: |Aix − Aiy| ≤ O

1

w(i)

SLIDE 66

Example: Discrepancy of set systems

◮ Given: Set system S with n sets and n elements

i S A =   1 1 1 1 1 1   set S i

SLIDE 67

Example: Discrepancy of set systems

◮ Given: Set system S with n sets and n elements

i S A =   1 1 1 1 1 1   set S i

◮ Pick x := (1 2, . . . , 1 2); weight w(i) := 1 n for each row

SLIDE 68

Example: Discrepancy of set systems

◮ Given: Set system S with n sets and n elements

i S A =   1 1 1 1 1 1   set S i

◮ Pick x := (1 2, . . . , 1 2); weight w(i) := 1 n for each row ◮ There is y ∈ {0, 1}n : Ax − Ay∞ = O(

1

1/n) = O(√n).

SLIDE 69

Example: Discrepancy of set systems

◮ Given: Set system S with n sets and n elements

i S A =   1 1 1 1 1 1   set S i

◮ Pick x := (1 2, . . . , 1 2); weight w(i) := 1 n for each row ◮ There is y ∈ {0, 1}n : Ax − Ay∞ = O(

1

1/n) = O(√n). ◮ Coloring χ(i) =

+1

yi = 1 −1 yi = 0 has discrepancy O(√n).

◮ “6 Standard deviations suffice”-Thm [Spencer ’85]

SLIDE 70

Summarizing

Theorem

Input:

◮ matrix A ∈ [−1, 1]n×m (∀A′ ⊆ A: there are

f : −∆ ≤ A′χ − f(χ) ≤ ∆ and H(f(χ)) ≤ #cols(A′)

10

)

◮ vector x ∈ [0, 1]m ◮ row weights w(i) ( i w(i) = 1)

There is a y ∈ {0, 1}m with

◮ Bounded difference:

◮ |Aix − Aiy| ≤ O(log(m)) · ∆i ◮ |Aix − Aiy| ≤ O(

1/w(i))

SLIDE 71

Summarizing

Theorem

Input:

◮ matrix A ∈ [−1, 1]n×m (∀A′ ⊆ A: there are

f : −∆ ≤ A′χ − f(χ) ≤ ∆ and H(f(χ)) ≤ #cols(A′)

10

)

◮ vector x ∈ [0, 1]m ◮ row weights w(i) ( i w(i) = 1)

There is a random variable y ∈ {0, 1}m with

◮ Bounded difference:

◮ |Aix − Aiy| ≤ O(log(m)) · ∆i ◮ |Aix − Aiy| ≤ O(

1/w(i))

◮ Preserved expectation: E[yi] = xi. ◮ Randomness: y = x + k≥1

log m

t=1 (1 2)k · χ(k,t)

SLIDE 72

Summarizing

Theorem

Input:

◮ matrix A ∈ [−1, 1]n×m (∀A′ ⊆ A: there are

f : −∆ ≤ A′χ − f(χ) ≤ ∆ and H(f(χ)) ≤ #cols(A′)

10

)

◮ vector x ∈ [0, 1]m ◮ row weights w(i) ( i w(i) = 1)

There is a random variable y ∈ {0, 1}m with

◮ Bounded difference:

◮ |Aix − Aiy| ≤ O(log(m)) · ∆i ◮ |Aix − Aiy| ≤ O(

1/w(i))

◮ Preserved expectation: E[yi] = xi. ◮ Randomness: y = x + k≥1

log m

t=1 (1 2)k · rand. ± 1 · χ(k,t)

SLIDE 73

Summarizing

Theorem

Input:

◮ matrix A ∈ [−1, 1]n×m (∀A′ ⊆ A: there are

f : −∆ ≤ A′χ − f(χ) ≤ ∆ and H(f(χ)) ≤ #cols(A′)

10

)

◮ vector x ∈ [0, 1]m ◮ row weights w(i) ( i w(i) = 1)

There is a random variable y ∈ {0, 1}m with

◮ Bounded difference:

◮ |Aix − Aiy| ≤ O(log(m)) · ∆i ◮ |Aix − Aiy| ≤ O(

1/w(i))

◮ Preserved expectation: E[yi] = xi. ◮ Randomness: y = x + k≥1

log m

t=1 (1 2)k · rand. ± 1 · χ(k,t) ◮ Can be computed by SDP in poly-time using [Bansal ’10]

SLIDE 74

Bin Packing With Rejection

Input:

◮ Items i ∈ {1, . . . , n} with size si ∈ [0, 1], and rejection

penalty πi ∈ [0, 1] Goal: Pack or reject. Minimize # bins + rejection cost. πi 0.9 0.7 0.4 0.6 bin 1 bin 2 input si 1 1

SLIDE 75

Bin Packing With Rejection

Input:

◮ Items i ∈ {1, . . . , n} with size si ∈ [0, 1], and rejection

penalty πi ∈ [0, 1] Goal: Pack or reject. Minimize # bins + rejection cost. πi 0.7 0.4 0.6 bin 1 bin 2 input si 1 1

SLIDE 76

Bin Packing With Rejection

Input:

◮ Items i ∈ {1, . . . , n} with size si ∈ [0, 1], and rejection

penalty πi ∈ [0, 1] Goal: Pack or reject. Minimize # bins + rejection cost. πi 0.7 0.4 0.6 bin 1 bin 2 input si 1 1

×

SLIDE 77

Bin Packing With Rejection

Input:

◮ Items i ∈ {1, . . . , n} with size si ∈ [0, 1], and rejection

penalty πi ∈ [0, 1] Goal: Pack or reject. Minimize # bins + rejection cost. πi 0.7 0.6 bin 1 bin 2 input si 1 1

×

SLIDE 78

Bin Packing With Rejection

Input:

◮ Items i ∈ {1, . . . , n} with size si ∈ [0, 1], and rejection

penalty πi ∈ [0, 1] Goal: Pack or reject. Minimize # bins + rejection cost. πi 0.7 bin 1 bin 2 input si 1 1

×

SLIDE 79

Known results

Bin Packing With Rejection:

◮ APTAS [Epstein ’06] ◮ Faster APTAS [Bein, Correa & Han ’08] ◮ AFPTAS (APX ≤ OPT + OP T (log OP T)1−o(1) )

[Epstein & Levin ’10]

SLIDE 80

Known results

Bin Packing With Rejection:

◮ APTAS [Epstein ’06] ◮ Faster APTAS [Bein, Correa & Han ’08] ◮ AFPTAS (APX ≤ OPT + OP T (log OP T)1−o(1) )

[Epstein & Levin ’10] Bin Packing:

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

SLIDE 81

Known results

Bin Packing With Rejection:

◮ APTAS [Epstein ’06] ◮ Faster APTAS [Bein, Correa & Han ’08] ◮ AFPTAS (APX ≤ OPT + OP T (log OP T)1−o(1) )

[Epstein & Levin ’10] Bin Packing:

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

Theorem

There is a randomized approximation algorithm for Bin Packing With Rejection with APX ≤ OPT + O(log2 OPT) (with high probability).

SLIDE 82

The column-based LP

Set Cover formulation:

◮ Bins: Sets S ⊆ [n] with i∈S si ≤ 1 of cost c(S) = 1 ◮ Rejections: Sets S = {i} of cost c(S) = πi

LP: min

S∈S

c(S) · xS

S∈S

1S · xS ≥ 1 xS ≥ ∀S ∈ S

SLIDE 83

The column-based LP - Example

input si 1 0.44 0.4 0.3 0.26 πi 0.9 0.7 0.4 0.6

SLIDE 84

The column-based LP - Example

input si 1 0.44 0.4 0.3 0.26 πi 0.9 0.7 0.4 0.6 min

1

1 1 1 1 1 1 1 1 1 1 1 | .9 .7 .4 .6

x

    1 1 1 1 1 | 1 1 1 1 1 1 | 0 1 1 1 1 1 1 1 | 0 1 1 1 1 1 1 1 | 0 1     x ≥     1 1 1 1     x ≥

SLIDE 85

The column-based LP - Example

input si 1 0.44 0.4 0.3 0.26 πi 0.9 0.7 0.4 0.6 min

1

1 1 1 1 1 1 1 1 1 1 1 | .9 .7 .4 .6

x

    1 1 1 1 1 | 1 1 1 1 1 1 | 0 1 1 1 1 1 1 1 | 0 1 1 1 1 1 1 1 | 0 1     x ≥     1 1 1 1     x ≥ 1/2× 1/2× 1/2×

SLIDE 86

Massaging the LP

◮ Sort items s1 ≥ . . . ≥ sn

SLIDE 87

Massaging the LP

◮ Sort items s1 ≥ . . . ≥ sn ◮ Call items large if size at least ε := log OP Tf OP Tf

(otherwise small

SLIDE 88

Massaging the LP

◮ Sort items s1 ≥ . . . ≥ sn ◮ Call items large if size at least ε := log OP Tf OP Tf

(otherwise small       1 1 1 1 1 1 1 1 1 1 1 1 1 1 1      

SLIDE 89

Massaging the LP

◮ Sort items s1 ≥ . . . ≥ sn ◮ Call items large if size at least ε := log OP Tf OP Tf

(otherwise small

◮ Add up rows 1, . . . , i of constraint matrix to obtain row i

for matrix A (for large i’s). “1 slot per item” ⇒ “i slots for largest i items”       1 1 1 1 1 1 1 1 1 1 1 1 1 1 1       → A =       1 1 1 2 1 1 1 1 2 2 2 1 1 1      

SLIDE 90

Massaging the LP

◮ Sort items s1 ≥ . . . ≥ sn ◮ Call items large if size at least ε := log OP Tf OP Tf

(otherwise small

◮ Add up rows 1, . . . , i of constraint matrix to obtain row i

for matrix A (for large i’s). “1 slot per item” ⇒ “i slots for largest i items”

◮ Append row vector ( i∈S:si<ε si)S

      1 1 1 1 1 1 1 1 1 1 1 1 1 1 1       → A =       1 1 1 2 1 1 1 1 2 2 2 1 1 1 space for small items      

SLIDE 91

Massaging the LP

◮ Sort items s1 ≥ . . . ≥ sn ◮ Call items large if size at least ε := log OP Tf OP Tf

(otherwise small

◮ Add up rows 1, . . . , i of constraint matrix to obtain row i

for matrix A (for large i’s). “1 slot per item” ⇒ “i slots for largest i items”

◮ Append row vector ( i∈S:si<ε si)S ◮ Append objective function c

      1 1 1 1 1 1 1 1 1 1 1 1 1 1 1       → A =       1 1 1 2 1 1 1 1 2 2 2 1 1 1 space for small items

bjective function

     

SLIDE 92

Entropy bound for monotone matrices

Theorem

Let A be column-monotone matrix, max entry ≤ ∆, sum of last row = σ. There are auxiliary RV f: Aχ − f(χ)∞ = O(∆) and Hχ(f(χ)) ≤ O( σ

∆).

σ = A =             1 1 1 1 2 1 1 2 2 1 2 2 1 2 1 2 1 2 2 2 1 3 2             ≤ ∆ Aiχ i

SLIDE 93

Entropy bound for monotone matrices

Theorem

Let A be column-monotone matrix, max entry ≤ ∆, sum of last row = σ. There are auxiliary RV f: Aχ − f(χ)∞ = O(∆) and Hχ(f(χ)) ≤ O( σ

∆).

χ = (+1,-1,-1,+1) A =             1 1 1 1 2 1 1 2 2 1 2 2 1 2 1 2 1 2 2 2 1 3 2             Aiχ i

◮ Idea: Describe random walk A1χ, . . . , Aσχ

O(∆)-approximately

SLIDE 94

Entropy bound for monotone matrices

χ = (+1,-1,-1,+1) A =             1 1 1 1 2 1 1 2 2 1 2 2 1 2 1 2 1 2 2 2 1 3 2             Aiχ i σ

◮ Idea: Describe random walk A1χ, . . . , Aσχ

O(∆)-approximately

SLIDE 95

Entropy bound for monotone matrices

χ = (+1,-1,-1,+1) A =             1 1 1 1 2 1 1 2 2 1 2 2 1 2 1 2 1 2 2 2 1 3 2             Aiχ i σ D k 2k∆

◮ Idea: Describe random walk A1χ, . . . , Aσχ

O(∆)-approximately

◮ For each interval D of length 2k · ∆:

gD(χ) := covered distance in D rounded to

∆ 1.1k

SLIDE 96

Entropy bound for monotone matrices

χ = (+1,-1,-1,+1) A =             1 1 1 1 2 1 1 2 2 1 2 2 1 2 1 2 1 2 2 2 1 3 2             Aiχ i σ D k 2k∆ gD(χ)

◮ Idea: Describe random walk A1χ, . . . , Aσχ

O(∆)-approximately

◮ For each interval D of length 2k · ∆:

gD(χ) := covered distance in D rounded to

∆ 1.1k

SLIDE 97

Entropy bound for monotone matrices

χ = (+1,-1,-1,+1) A =             1 1 1 1 2 1 1 2 2 1 2 2 1 2 1 2 1 2 2 2 1 3 2             Aiχ i σ D k 2k∆ gD(χ)

◮ Idea: Describe random walk A1χ, . . . , Aσχ

O(∆)-approximately

◮ For each interval D of length 2k · ∆:

gD(χ) := covered distance in D rounded to

∆ 1.1k ◮ Std-Dev for gD:

max dep. · |D| ≤

√ ∆ · 2k∆ = 2k/2 · ∆

SLIDE 98

Entropy bound for monotone matrices

χ = (+1,-1,-1,+1) A =             1 1 1 1 2 1 1 2 2 1 2 2 1 2 1 2 1 2 2 2 1 3 2             Aiχ i σ D k 2k∆ gD(χ)

◮ Idea: Describe random walk A1χ, . . . , Aσχ

O(∆)-approximately

◮ For each interval D of length 2k · ∆:

gD(χ) := covered distance in D rounded to

∆ 1.1k ◮ Std-Dev for gD:

max dep. · |D| ≤

√ ∆ · 2k∆ = 2k/2 · ∆

◮ H(gD) ≤ G(∆/1.1k 2k/2∆ ) ≤ G(2−k) = O(log 2k) = O(k)

SLIDE 99

Entropy bound for monotone matrices

χ = (+1,-1,-1,+1) A =             1 1 1 1 2 1 1 2 2 1 2 2 1 2 1 2 1 2 2 2 1 3 2             Aiχ i σ D k 2k∆ gD(χ)

◮ Idea: Describe random walk A1χ, . . . , Aσχ

O(∆)-approximately

◮ For each interval D of length 2k · ∆:

gD(χ) := covered distance in D rounded to

∆ 1.1k ◮ Std-Dev for gD:

max dep. · |D| ≤

√ ∆ · 2k∆ = 2k/2 · ∆

◮ H(gD) ≤ G(∆/1.1k 2k/2∆ ) ≤ G(2−k) = O(log 2k) = O(k) ◮ Total entropy of g: k≥1 σ 2k∆ · O(k) = O( σ ∆).

SLIDE 100

Entropy bound for monotone matrices

χ = (+1,-1,-1,+1) A =             1 1 1 1 2 1 1 2 2 1 2 2 1 2 1 2 1 2 2 2 1 3 2             Aiχ i

◮ We know Aiχ up to an error of:

SLIDE 101

Entropy bound for monotone matrices

χ = (+1,-1,-1,+1) A =             1 1 1 1 2 1 1 2 2 1 2 2 1 2 1 2 1 2 2 2 1 3 2             Aiχ i

◮ We know Aiχ up to an error of:

SLIDE 102

Entropy bound for monotone matrices

χ = (+1,-1,-1,+1) A =             1 1 1 1 2 1 1 2 2 1 2 2 1 2 1 2 1 2 2 2 1 3 2             Aiχ i

◮ We know Aiχ up to an error of: k≥1 ∆ 1.1k = O(∆).

(formally fi(χ) :=

D: ˙ D=[i] gD(χ))

SLIDE 103

Approximation algorithm for BPWR

◮ Apply rounding theorem to fractional sol. x

A =       1 1 1 2 1 1 1 1 2 2 2 1 1 1 space for small items

bjective function

      weight 1/2 weight 1/2

SLIDE 104

Approximation algorithm for BPWR

◮ Apply rounding theorem to fractional sol. x ◮ Pick ∆i := Θ( 1 si )

A =       1 1 1 2 1 1 1 1 2 2 2 1 1 1 space for small items

bjective function

      i ∆i := Θ( 1

si )

weight 1/2 weight 1/2

SLIDE 105

Approximation algorithm for BPWR

◮ Apply rounding theorem to fractional sol. x ◮ Pick ∆i := Θ( 1 si ) ◮ Assume for 1 sec that 1 2k ≤ si ≤ 1 k (same size class)

O( σ ∆) ≤ 1 20

S active

|S|· 1 k ≤ 1 10

S active
i∈S

si

≤1

≤ # active var. 10 A =       1 1 1 2 1 1 1 1 2 2 2 1 1 1 space for small items

bjective function

      i ∆i := Θ( 1

si )

weight 1/2 weight 1/2

SLIDE 106

Approximation algorithm for BPWR (2)

Obtain y ∈ {0, 1}m:

◮ |Aiy − Aix| ≤ O(log m) · 1 si ◮ |cT x − cT y| ≤ O(1) ◮ space for small items in x and y differs by O(1)

SLIDE 107

Approximation algorithm for BPWR (2)

Obtain y ∈ {0, 1}m: → repair to feasible solution

◮ Aiy ≥ Aix = i (y reserves ≥ i slots for largest i items)

Buy O(log m) · 1

si slots for each size class → O(log m · log 1 ε) ◮ |cT x − cT y| ≤ O(1) ◮ space for small items in x and y differs by O(1)

Input: (large items) y :

SLIDE 108

Approximation algorithm for BPWR (2)

Obtain y ∈ {0, 1}m: → repair to feasible solution

◮ Aiy ≥ Aix = i (y reserves ≥ i slots for largest i items)

Buy O(log m) · 1

si slots for each size class → O(log m · log 1 ε) ◮ |cT x − cT y| ≤ O(1) ◮ space for small items in x and y differs by O(1)

Input: (large items) y :

SLIDE 109

Approximation algorithm for BPWR (2)

Obtain y ∈ {0, 1}m: → repair to feasible solution

◮ Aiy ≥ Aix = i (y reserves ≥ i slots for largest i items)

Buy O(log m) · 1

si slots for each size class → O(log m · log 1 ε) ◮ |cT x − cT y| ≤ O(1) ◮ space for small items in x and y differs by O(1)

Input: (large items) y :

SLIDE 110

Approximation algorithm for BPWR (2)

Obtain y ∈ {0, 1}m: → repair to feasible solution

◮ Aiy ≥ Aix = i (y reserves ≥ i slots for largest i items)

Buy O(log m) · 1

si slots for each size class → O(log m · log 1 ε) ◮ |cT x − cT y| ≤ O(1) ◮ space for small items in x and y differs by O(1)

Input: (large items) y :

SLIDE 111

Approximation algorithm for BPWR (2)

Obtain y ∈ {0, 1}m: → repair to feasible solution

◮ Aiy ≥ Aix = i (y reserves ≥ i slots for largest i items)

Buy O(log m) · 1

si slots for each size class → O(log m · log 1 ε) ◮ |cT x − cT y| ≤ O(1) ◮ space for small items in x and y differs by O(1)

Input: (large items) y :

SLIDE 112

Approximation algorithm for BPWR (2)

Obtain y ∈ {0, 1}m: → repair to feasible solution

◮ Aiy ≥ Aix = i (y reserves ≥ i slots for largest i items)

Buy O(log m) · 1

si slots for each size class → O(log m · log 1 ε) ◮ |cT x − cT y| ≤ O(1) ◮ space for small items in x and y differs by O(1)

SLIDE 113

Approximation algorithm for BPWR (2)

Obtain y ∈ {0, 1}m: → repair to feasible solution

◮ Aiy ≥ Aix = i (y reserves ≥ i slots for largest i items)

Buy O(log m) · 1

si slots for each size class → O(log m · log 1 ε) ◮ |cT x − cT y| ≤ O(1) ◮ y reserves enough space for small items → O(1)

bin 1 bin 2 bin 3 1 large items:

SLIDE 114

Approximation algorithm for BPWR (2)

Obtain y ∈ {0, 1}m: → repair to feasible solution

◮ Aiy ≥ Aix = i (y reserves ≥ i slots for largest i items)

Buy O(log m) · 1

si slots for each size class → O(log m · log 1 ε) ◮ |cT x − cT y| ≤ O(1) ◮ y reserves enough space for small items → O(1)

bin 1 bin 2 bin 3 1 large items: small items:

SLIDE 115

Approximation algorithm for BPWR (2)

Obtain y ∈ {0, 1}m: → repair to feasible solution

◮ Aiy ≥ Aix = i (y reserves ≥ i slots for largest i items)

Buy O(log m) · 1

si slots for each size class → O(log m · log 1 ε) ◮ |cT x − cT y| ≤ O(1) ◮ y reserves enough space for small items → O(1)

bin 1 bin 2 bin 3 1 large items: small items:

SLIDE 116

Approximation algorithm for BPWR (2)

Obtain y ∈ {0, 1}m: → repair to feasible solution

◮ Aiy ≥ Aix = i (y reserves ≥ i slots for largest i items)

Buy O(log m) · 1

si slots for each size class → O(log m · log 1 ε) ◮ |cT x − cT y| ≤ O(1) ◮ y reserves enough space for small items → O(1)

bin 1 bin 2 bin 3 1 large items: small items:

SLIDE 117

Approximation algorithm for BPWR (2)

Obtain y ∈ {0, 1}m: → repair to feasible solution

◮ Aiy ≥ Aix = i (y reserves ≥ i slots for largest i items)

Buy O(log m) · 1

si slots for each size class → O(log m · log 1 ε) ◮ |cT x − cT y| ≤ O(1) ◮ y reserves enough space for small items → O(1)

bin 1 bin 2 bin 3 1 large items: small items:

SLIDE 118

Approximation algorithm for BPWR (2)

Obtain y ∈ {0, 1}m: → repair to feasible solution

◮ Aiy ≥ Aix = i (y reserves ≥ i slots for largest i items)

Buy O(log m) · 1

si slots for each size class → O(log m · log 1 ε) ◮ |cT x − cT y| ≤ O(1) ◮ y reserves enough space for small items → O(1)

bin 1 bin 2 bin 3 1 large items: small items:

SLIDE 119

Approximation algorithm for BPWR (2)

Obtain y ∈ {0, 1}m: → repair to feasible solution

◮ Aiy ≥ Aix = i (y reserves ≥ i slots for largest i items)

Buy O(log m) · 1

si slots for each size class → O(log m · log 1 ε) ◮ |cT x − cT y| ≤ O(1) ◮ y reserves enough space for small items → O(1)

assign small items integrally → O(ε) · OPTf = O(log OPTf) bin 1 bin 2 bin 3 1 large items: small items:

SLIDE 120

Approximation algorithm for BPWR (2)

Obtain y ∈ {0, 1}m: → repair to feasible solution

◮ Aiy ≥ Aix = i (y reserves ≥ i slots for largest i items)

Buy O(log m) · 1

si slots for each size class → O(log m · log 1 ε) ◮ |cT x − cT y| ≤ O(1) ◮ y reserves enough space for small items → O(1)

assign small items integrally → O(ε) · OPTf = O(log OPTf) Doing the math:

◮ may assume that xS ≤ 1 − ε (o.w. buy them apriori)

SLIDE 121

Approximation algorithm for BPWR (2)

Obtain y ∈ {0, 1}m: → repair to feasible solution

◮ Aiy ≥ Aix = i (y reserves ≥ i slots for largest i items)

Buy O(log m) · 1

si slots for each size class → O(log m · log 1 ε) ◮ |cT x − cT y| ≤ O(1) ◮ y reserves enough space for small items → O(1)

assign small items integrally → O(ε) · OPTf = O(log OPTf) Doing the math:

◮ may assume that xS ≤ 1 − ε (o.w. buy them apriori) ◮ Can assume that m ≤ #large items + 2

SLIDE 122

Approximation algorithm for BPWR (2)

Obtain y ∈ {0, 1}m: → repair to feasible solution

◮ Aiy ≥ Aix = i (y reserves ≥ i slots for largest i items)

Buy O(log m) · 1

si slots for each size class → O(log m · log 1 ε) ◮ |cT x − cT y| ≤ O(1) ◮ y reserves enough space for small items → O(1)

assign small items integrally → O(ε) · OPTf = O(log OPTf) Doing the math:

◮ may assume that xS ≤ 1 − ε (o.w. buy them apriori) ◮ Can assume that m ≤ #large items + 2 ◮ OPTf ≥ ε2 · #large items ⇒ log m, log 1 ε = O(log OPTf)

SLIDE 123

Approximation algorithm for BPWR (2)

Obtain y ∈ {0, 1}m: → repair to feasible solution

◮ Aiy ≥ Aix = i (y reserves ≥ i slots for largest i items)

Buy O(log m) · 1

si slots for each size class → O(log m · log 1 ε) ◮ |cT x − cT y| ≤ O(1) ◮ y reserves enough space for small items → O(1)

assign small items integrally → O(ε) · OPTf = O(log OPTf) Doing the math:

◮ may assume that xS ≤ 1 − ε (o.w. buy them apriori) ◮ Can assume that m ≤ #large items + 2 ◮ OPTf ≥ ε2 · #large items ⇒ log m, log 1 ε = O(log OPTf) ◮ APX − OPTf ≤ O(log2 OPTf)

SLIDE 124

Open problem I

Method works pretty well for other Bin Packing variants. But:

Open question I

Are there other applications?

SLIDE 125

Open problem II

Bin Packing: min

S∈S xS

S∈S 1S · xS

≥ 1 xS ≥ ∀S ∈ S

Modified Integer Roundup Conjecture

OPT ≤ ⌈OPTf⌉ + 1

◮ True, if # of different item sizes ≤ 7 [Seb˝

, Shmonin ’09]

◮ Best known general bound: OPT ≤ OPTf + O(log2 n) ◮ WARNING: No o(log2 n) bound possible by just “selecting

patterns from an initial fractional solution and rounding up items” [Eisenbrand, P´ alv¨

lgyi, R. ’11]

SLIDE 126

Open problem II

Bin Packing: min

S∈S xS

S∈S 1S · xS

≥ 1 xS ≥ ∀S ∈ S

Modified Integer Roundup Conjecture

OPT ≤ ⌈OPTf⌉ + 1

◮ True, if # of different item sizes ≤ 7 [Seb˝

, Shmonin ’09]

◮ Best known general bound: OPT ≤ OPTf + O(log2 n) ◮ WARNING: No o(log2 n) bound possible by just “selecting

patterns from an initial fractional solution and rounding up items” [Eisenbrand, P´ alv¨

lgyi, R. ’11]