The Entropy Rounding Method in Approximation Algorithms

Transcription

1 The Entropy Rounding Method in Approximation Algorithms Thomas Rothvoß Department of Mathematics, M.I.T. SODA 2012

2 A general LP rounding problem Problem: Given: A R n m, fractional solution x [0,1] m Find: y {0,1} m : Ax Ay

3 A general LP rounding problem Problem: Given: A R n m, fractional solution x [0,1] m Find: y {0,1} m : Ax Ay General strategies: Randomized rounding: Flip Pr[y i = 1] = x i (in)dependently

4 A general LP rounding problem Problem: Given: A R n m, fractional solution x [0,1] m Find: y {0,1} m : Ax Ay General strategies: Randomized rounding: Flip Pr[y i = 1] = x i (in)dependently Use properties of basic solutions: supp(x) #rows(a)

5 A general LP rounding problem Problem: Given: A R n m, fractional solution x [0,1] m Find: y {0,1} m : Ax Ay General strategies: Randomized rounding: Flip Pr[y i = 1] = x i (in)dependently Use properties of basic solutions: supp(x) #rows(a) Try another way: Entropy rounding method based on discrepancy theory

6 A schematic view Input: x [0,1] m, matrix A Bansal algorithm Lovász, Spencer & Entropy. Vesztergombi Theoremrounding Beck s entropy method Output: y {0,1} m with Ax Ay small

7 A schematic view Input: x [0,1] m, matrix A Lovász, Spencer &. Vesztergombi Theorem Bansal algorithm Beck s entropy method Output: y {0,1} m with Ax Ay small

8 A schematic view Input: x [0,1] m, matrix A Lovász, Spencer &. Vesztergombi Theorem Bansal algorithm Beck s entropy method Output: y {0,1} m with Ax Ay small

9 A schematic view Input: x [0,1] m, matrix A Lovász, Spencer &. Vesztergombi Theorem Bansal algorithm Beck s entropy method Output: y {0,1} m with Ax Ay small

10 Discrepancy theory Set system S = {S 1,...,S m },S i [n] i S

11 Discrepancy theory Set system S = {S 1,...,S m },S i [n] Coloring χ : [n] { 1,+1} i S

12 Discrepancy theory Set system S = {S 1,...,S m },S i [n] Coloring χ : [n] { 1,+1} Discrepancy disc(s) = min max χ(s). χ:[n] {±1} S S i S where χ(s) = i S χ(i).

13 Discrepancy theory Set system S = {S 1,...,S m },S i [n] Coloring χ : [n] { 1,+1} Discrepancy disc(s) = min max χ(s). χ:[n] {±1} S S i S where χ(s) = i S χ(i). Known results: n sets, n elements: disc(s) = O( n) [Spencer 85]

14 Discrepancy theory Set system S = {S 1,...,S m },S i [n] Coloring χ : [n] { 1,+1} Discrepancy disc(s) = min max χ(s). χ:[n] {±1} S S i S where χ(s) = i S χ(i). 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)

15 Discrepancy theory Set system S = {S 1,...,S m },S i [n] Coloring χ : [n] { 1,+1} Discrepancy disc(s) = min max χ(s). χ:[n] {±1} S S i S where χ(s) = i S χ(i). 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: Half coloring: χ : [n] {0, 1,+1}, supp(χ) n/2

16 Discrepancy theory Set system S = {S 1,...,S m },S i [n] Coloring χ : [n] { 1,+1} Discrepancy disc(s) = min max χ(s). χ:[n] {±1} S S i S where χ(s) = i S χ(i). 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: Half coloring: χ : [n] {0, 1,+1}, supp(χ) n/2 For matrix A: disc(a) = min χ {±1} n Aχ

17 The LSV-Theorem Theorem (Lovász, Spencer & Vesztergombi 86) Given A R n m,x [0,1] m. Suppose for any A A, coloring χ : A χ. Then y {0,1} m with Ax Ay.

18 The LSV-Theorem Theorem (Lovász, Spencer & Vesztergombi 86) Given A R n m,x [0,1] m. Suppose for any A A, coloring χ : A χ. Then y {0,1} m with Ax Ay. Suffices to find good matrix colorings!

19 The LSV-Theorem Theorem (Lovász, Spencer & Vesztergombi 86) Given A R n m,x [0,1] m. Suppose for any A A, half coloring χ : A χ. Then y {0,1} m with Ax Ay log m. Suffices to find good matrix half colorings!

20 A schematic view Input: x [0,1] m, matrix A Lovász, Spencer &. Vesztergombi Theorem Bansal algorithm Beck s entropy method Output: y {0,1} m with Ax Ay small

21 Entropy Definition For random variable Z, the entropy is H(Z) = ( ) 1 Pr[Z = z] log 2 Pr[Z = z] z

22 Entropy Definition For random variable Z, the entropy is H(Z) = ( ) 1 Pr[Z = z] log 2 Pr[Z = z] z Properties: One likely event: z : Pr[Z = z] ( 1 2 )H(Z)

23 Entropy Definition For random variable Z, the entropy is H(Z) = ( ) 1 Pr[Z = z] log 2 Pr[Z = z] z Properties: One likely event: z : Pr[Z = z] ( 1 2 )H(Z) Subadditivity: H(f(Z,Z )) H(Z)+H(Z ).

24 Theorem [Beck s entropy method] ) m 5 half-coloring χ0 : Aχ 0. H χ i {±1} ( Aχ A = m ( ) A 2 χ A 1 χ {±1} m

25 Theorem [Beck s entropy method] ) m 5 half-coloring χ0 : Aχ 0. H χ i {±1} ( Aχ A = m ( ) A 2 χ A 1 χ {±1} m

26 Theorem [Beck s entropy method] ) m 5 half-coloring χ0 : Aχ 0. H χ i {±1} ( Aχ A = m ( ) A 2 χ A 1 χ {±1} m

27 Theorem [Beck s entropy method] ) m 5 half-coloring χ0 : Aχ 0. H χ i {±1} ( Aχ A = {±1} m m ( ) A 2 χ A 1 χ

28 Theorem [Beck s entropy method] ) m 5 half-coloring χ0 : Aχ 0. H χ i {±1} ( Aχ A = {±1} m m ( ) A 2 χ A 1 χ

29 Theorem [Beck s entropy method] ) m 5 half-coloring χ0 : Aχ 0. H χ i {±1} ( Aχ A 2 χ A = {±1} m m ( ) A 1 χ

30 Theorem [Beck s entropy method] ) m 5 half-coloring χ0 : Aχ 0. H χ i {±1} ( Aχ A 2 χ A = {±1} m m ( ) A 1 χ

31 Theorem [Beck s entropy method] ) m 5 half-coloring χ0 : Aχ 0. H χ i {±1} ( Aχ A 2 χ A = {±1} m m ( ) A 1 χ

32 Theorem [Beck s entropy method] ) m 5 half-coloring χ0 : Aχ 0. H χ i {±1} ( Aχ A 2 χ A = {±1} m m ( ) A 1 χ

33 Theorem [Beck s entropy method] ) m 5 half-coloring χ0 : Aχ 0. H χ i {±1} ( Aχ cell : Pr[Aχ cell] ( 1 2 )m/5. A 2 χ A = {±1} m m ( ) A 1 χ

34 Theorem [Beck s entropy method] ) m 5 half-coloring χ0 : Aχ 0. H χ i {±1} ( Aχ cell : Pr[Aχ cell] ( 1 2 )m/5. At least 2 m ( 1 2 )m/5 = 2 0.8m colorings χ have Aχ cell A 2 χ A = m ( ) A 1 χ {±1} m

35 Theorem [Beck s entropy method] ) m 5 half-coloring χ0 : Aχ 0. H χ i {±1} ( Aχ cell : Pr[Aχ cell] ( 1 2 )m/5. At least 2 m ( 1 2 )m/5 = 2 0.8m colorings χ have Aχ cell Pick χ 1,χ 2 differing in half of entries A = {±1} m χ 2 m ( ) χ 1 A 2 χ Aχ 1 Aχ 2 A 1 χ

36 Theorem [Beck s entropy method] ) m 5 half-coloring χ0 : Aχ 0. H χ i {±1} ( Aχ cell : Pr[Aχ cell] ( 1 2 )m/5. At least 2 m ( 1 2 )m/5 = 2 0.8m colorings χ have Aχ cell Pick χ 1,χ 2 differing in half of entries Define χ 0 (i) := 1 2 (χ1 (i) χ 2 (i)) {0,±1}. A = {±1} m χ 2 m ( ) χ 1 A 2 χ Aχ 1 Aχ 2 A 1 χ

37 Theorem [Beck s entropy method] ) m 5 half-coloring χ0 : Aχ 0. H χ i {±1} ( Aχ cell : Pr[Aχ cell] ( 1 2 )m/5. At least 2 m ( 1 2 )m/5 = 2 0.8m colorings χ have Aχ cell Pick χ 1,χ 2 differing in half of entries Define χ 0 (i) := 1 2 (χ1 (i) χ 2 (i)) {0,±1}. Then Aχ Aχ1 Aχ 2. A 2 χ A = {±1} m χ 2 m ( ) χ 1 Aχ 1 Aχ 2 A 1 χ

38 A bound on the entropy Let α = (α 1,...,α m ) R m. j Consider Z := χ(j) α j λ α 2 with χ(j) {±1} at random

39 A bound on the entropy Let α = (α 1,...,α m ) R m. j Consider Z := χ(j) α j λ α 2 with χ(j) {±1} at random 1 Pr[Z = 0] 0 1 λ

40 A bound on the entropy Let α = (α 1,...,α m ) R m. j Consider Z := χ(j) α j λ α 2 with χ(j) {±1} at random 1 Pr[Z = 0] 0 H(Z) 1 λ e Ω(λ2 ) 1 λ

41 A bound on the entropy Let α = (α 1,...,α m ) R m. j Consider Z := χ(j) α j λ α 2 with χ(j) {±1} at random 1 Pr[Z = 0] 0 H(Z) 1 λ O(log( 1 λ )) e Ω(λ2 ) 1 λ

42 A bound on the entropy Let α = (α 1,...,α m ) R m. j Consider Z := χ(j) α j λ α 2 with χ(j) {±1} at random 1 Pr[Z = 0] 0 H(Z) 1 λ O(log( 1 λ )) randomized rounding e Ω(λ2 ) 1 λ

43 A schematic view Input: x [0,1] m, matrix A Lovász, Spencer &. Vesztergombi Theorem Bansal algorithm Beck s entropy method Output: y {0,1} m with Ax Ay small

44 A general rounding theorem Theorem Input: matrix A [ 1,1] n m, i > 0 satisfying entropy condition for all submatrices vector x [0,1] m row weights w(i) ( iw(i) = 1) There is a random variable y {0,1} m with Bounded difference: Ai x A i y O(log(m)) i Ai x A i y O( 1/w(i)) Preserved expectation: E[y i ] = x i.

45 Bin Packing With Rejection Input: Items i {1,...,n} with size s i [0,1], and rejection penalty π i [0,1] Goal: Pack or reject. Minimize # bins + rejection cost. 1 0 bin 1 bin 2 1 s i input π i

46 Bin Packing With Rejection Input: Items i {1,...,n} with size s i [0,1], and rejection penalty π i [0,1] Goal: Pack or reject. Minimize # bins + rejection cost. 1 0 bin 1 bin 2 1 s i input π i

47 Bin Packing With Rejection Input: Items i {1,...,n} with size s i [0,1], and rejection penalty π i [0,1] Goal: Pack or reject. Minimize # bins + rejection cost s i bin 1 bin 2 π i input

48 Bin Packing With Rejection Input: Items i {1,...,n} with size s i [0,1], and rejection penalty π i [0,1] Goal: Pack or reject. Minimize # bins + rejection cost s i bin 1 bin 2 π i input

49 Bin Packing With Rejection Input: Items i {1,...,n} with size s i [0,1], and rejection penalty π i [0,1] Goal: Pack or reject. Minimize # bins + rejection cost s i bin 1 bin 2 π i 0.7 input

50 The column-based LP Set Cover formulation: Bins: Sets S [n] with i S s i 1 of cost c(s) = 1 Rejections: Sets S = {i} of cost c(s) = π i LP: min c(s) x S S S 1 S x S 1 S S x S 0 S S

51 The column-based LP - Example 1 s i π i input

52 The column-based LP - Example 1 s i π i input min ( ) x x x 0

53 The column-based LP - Example 1 s i π i input min ( ) x x x 0 1/2 1/2 1/2

54 Our results Theorem There is a randomized poly-time approximation algorithm for Bin Packing With Rejection with (with high probability). APX OPT f +O(log 2 OPT f ) Previously best known: APX OPT f + [Epstein & Levin 10] OPT f (logopt f ) 1 o(1) As good as classical Bin Packing [Karmarkar & Karp 82]

55 The end Open question Are there more (convincing) applications?

56 The end Open question Are there more (convincing) applications? Thanks for your attention