A Robust AFPTAS for Online Bin Packing with Polynomial Migration
A Robust AFPTAS for Online Bin Packing with Polynomial Migration Klaus Jansen
Kim-Manuel Klein
University of Kiel
July 2, 2014
Online Bin Packing
...
Online Bin Packing
...
Online Bin Packing
...
Online Bin Packing
Given an instance It = {i1 , . . . it } of items for each time t ∈ N and S a function s : I = t It → [0, 1]. Find for each t ∈ N an assignment Bt : {i1 , . . . , it } → N+ such P that i:Bt (i)=j s(i) ≤ 1 for all j. Goal: Minimize the number of bins maxi {Bt (i)} for each time t.
Competitive Ratio of Online Bin Packing
Best known algorithm: Ratio of 1.58889 (S.S. Seiden, 2002) Best known lower bound: Ratio of 1.54037 (J. Balogh, B. Jozsef, and G. Galambos, 2010)
Offline Bin Packing
APTAS: Approximation guarantee: (1 + )OPT + 1 Running time: poly(n) + f (1/) (W. Fernandez de la Vega and G.S. Lueker, 1981) AFPTAS: Approximation guarantee: (1 + )OPT + O(1/2 ) Running time: poly(n, 1/) (N. Karmarkar and R.M. Karp, 1982)
Repacking
Online Bin Packing with Repacking Ratio 1.33 repacking 7 items (G. Gambosi et al., 2000) Ratio 1.25 repacking O(log t) "shifting moves" (Z. Ivkovic and E.L. Lloyd, 1998) Ratio 1 + repacking amortized O(log t) "shifting moves" (Z. Ivkovic and E.L. Lloyd, 1997)
Repacking
Online Bin Packing with Repacking Ratio 1.33 repacking 7 items (G. Gambosi et al., 2000) Ratio 1.25 repacking O(log t) "shifting moves" (Z. Ivkovic and E.L. Lloyd, 1998) Ratio 1 + repacking amortized O(log t) "shifting moves" (Z. Ivkovic and E.L. Lloyd, 1997) Online Scheduling: Achieving ratio < 1.4659 requires repacking of Θ(t) jobs (S. Albers and M. Hellwig, 2012)
Migration
Migration Factor between Bt and Bt+1 1
X
s(it+1 ) j≤t:B (i )6=B t
j
s(ij )
t+1 (ij )
An algorithm is robust if the migration factor is bounded by a function f (1/). (P. Sanders, N. Sivadasan and M. Skutella, 2004).
Questions:
Can we achieve approximation guarantee: (1 + )OPT + g(1/) with migration f (1/)?
Questions:
Can we achieve approximation guarantee: (1 + )OPT + g(1/) with migration f (1/)? 1
YES Online Scheduling on identical machines (P. Sanders, N. Sivadasan, and M. Skutella, 2004)
Questions:
Can we achieve approximation guarantee: (1 + )OPT + g(1/) with migration f (1/)? 1
2
YES Online Scheduling on identical machines (P. Sanders, N. Sivadasan, and M. Skutella, 2004) YES Online Bin Packing (L. Epstein and A. Levin, 2006)
Questions:
Can we achieve approximation guarantee: (1 + )OPT + g(1/) with migration f (1/)? 1
2
3
YES Online Scheduling on identical machines (P. Sanders, N. Sivadasan, and M. Skutella, 2004) YES Online Bin Packing (L. Epstein and A. Levin, 2006) YES and NO Online Machine Covering (M. Skutella and J. Verschae 2010)
Questions:
Can we achieve approximation guarantee: (1 + )OPT + g(1/) with migration f (1/)? 1
2
3
YES Online Scheduling on identical machines (P. Sanders, N. 2 Sivadasan, and M. Skutella, 2004) Mf: 2O(1/ log (1/)) YES Online Bin Packing (L. Epstein and A. Levin, 2006) Mf: 2 2O(1/ log(1/)) YES and NO Online Machine Covering (M. Skutella and J. 2 Verschae 2010) amortized Mf: 2O(1/ log (1/)
Questions:
Can we achieve approximation guarantee: (1 + )OPT + g(1/) with migration f (1/)? 1
2
3
YES Online Scheduling on identical machines (P. Sanders, N. 2 Sivadasan, and M. Skutella, 2004) Mf: 2O(1/ log (1/)) YES Online Bin Packing (L. Epstein and A. Levin, 2006) Mf: 2 2O(1/ log(1/)) YES and NO Online Machine Covering (M. Skutella and J. 2 Verschae 2010) amortized Mf: 2O(1/ log (1/)
Can we achieve polynomial migration and polynomial running time?
Our Result for Online Bin Packing: Approximation scheme with running time poly(t, 1/) and migration factor poly(1/).
Overview Robust Algorithms
optimum ILP solution y 0
rounding
Packing Bt for instance It
Overview Robust Algorithms
optimum ILP solution y 0
rounding
Packing Bt for instance It
ky 0 − y 00 k∞ bounded
optimum ILP solution y 00
rounding
Packing Bt+1 for instance It+1
Sensitivity Analysis
Goal Let y 0 be an optimum ILP solution of min {kx k1 |Ax ≥ b 0 , x ≥ 0}. Find an optimum ILP solution y 00 of min {kx k1 |Ax ≥ b 00 , x ≥ 0} such that ky 00 − y 0 k∞ is small.
Sensitivity Analysis
Goal Let y 0 be an optimum ILP solution of min {kx k1 |Ax ≥ b 0 , x ≥ 0}. Find an optimum ILP solution y 00 of min {kx k1 |Ax ≥ b 00 , x ≥ 0} such that ky 00 − y 0 k∞ is small. Theorem (Cook et al., 1986): There exists an optimum ILP solution y 00 with ky 00 − y 0 k∞ ≤ n∆(kb 00 − b 0 k∞ + 2).
Sensitivity Analysis
Goal Let y 0 be an optimum ILP solution of min {kx k1 |Ax ≥ b 0 , x ≥ 0}. Find an optimum ILP solution y 00 of min {kx k1 |Ax ≥ b 00 , x ≥ 0} such that ky 00 − y 0 k∞ is small. Theorem (Cook et al., 1986): There exists an optimum ILP solution y 00 with ky 00 − y 0 k∞ ≤ n∆(kb 00 − b 0 k∞ + 2). Problem: The number of variables n and the largest subdeterminant ∆ can only bounded by an exponential term in 1/.
Theorem Consider the LP min {kx k1 |Ax ≥ b, x ≥ 0} with A ∈ Rm×n ≥0 and let x 0 be an approximate fractional solution with kx 0 k1 ≤ (1 + δ)OPT for δ > 0 and kx 0 k1 ≥ α( 1δ + 1).
Theorem Consider the LP min {kx k1 |Ax ≥ b, x ≥ 0} with A ∈ Rm×n ≥0 and let x 0 be an approximate fractional solution with kx 0 k1 ≤ (1 + δ)OPT for δ > 0 and kx 0 k1 ≥ α( 1δ + 1). Then there exists a solution x 00 with kx 00 k1 ≤ (1 + δ)OPT − α and kx 0 − x 00 k1 ≤ 2α (1/δ + 1).
Prove the feasibility of the following linear program: Ax ≥ b x ≥0 X
xi ≤ (1 + δ)OPT − α x ≥ x 0 − α(1/δ + 1)
x0 kx 0 k1
x ≤ x 0 + α(1/δ + 1)
x OPT kx 0 k1
Prove the feasibility of the following linear program: Ax ≥ b x ≥0 X
xi ≤ (1 + δ)OPT − α x ≥ x 0 − α(1/δ + 1)
x0 kx 0 k1
x ≤ x 0 + α(1/δ + 1)
x OPT kx 0 k1
A feasible solution is x 00 = (1 −
α(1/δ+1) 0 kx 0 k1 )x
+
α(1/δ+1) OPT . kx 0 k1 x
Algorithm Let x 0 be a LP solution with kx 0 k ≤ (1 + δ)OPT and kx 0 k ≥ α(1/δ + 1). Set x fix := x 0 −
α(1/δ+1) kx 0 k1
x 0 and b var := b − A(x fix )
Solve the LP xˆ = min {kx k1 |Ax ≥ b var , x ≥ 0} Generate a new solution x 00 = x fix + xˆ
Algorithm Let x 0 be a LP solution with kx 0 k ≤ (1 + δ)OPT and kx 0 k ≥ α(1/δ + 1). Set x fix := x 0 −
α(1/δ+1) kx 0 k1
x 0 and b var := b − A(x fix )
Solve the LP xˆ = min {kx k1 |Ax ≥ b var , x ≥ 0} Generate a new solution x 00 = x fix + xˆ The algorithm returns a feasible LP solution x 00 with kx 00 k1 ≤ (1 + δ)OPT − α and distance kx 00 − x 0 k1 ≤ 2α(1/δ + 1).
Improve Packing:
Let Bt be a packing of instance It with maxi Bt (i) ≤ (1 + δ)OPT . Find a packing Bt0 with maxi Bt0 (i) ≤ (1 + δ)OPT − 1 such that migration factor between Bt and Bt0 is small.
ILP solution y 0
rounding
Packing Bt for instance It
LP solution x 0
ILP solution y 0
rounding
Packing Bt for instance It
LP solution
x0
ILP solution y 0
rounding
Packing Bt for instance It
kx 0 − x 00 k1 = O(1/)
LP solution x 00
LP solution
x0
ILP solution y 0
rounding
Packing Bt for instance It
kx 0 − x 00 k1 = O(1/)
LP solution x 00
ILP solution y 00
LP solution
x0
ILP solution
y0
rounding
Packing Bt for instance It
kx 0 − x 00 k1 = O(1/)
ky 0 − y 00 k1 = O(m/)
LP solution x 00
ILP solution y 00
LP solution
x0
ILP solution
y0
rounding
Packing Bt for instance It
kx 0 − x 00 k1 = O(1/)
ky 0 − y 00 k1 = O(m/)
LP solution x 00
ILP solution y 00
rounding
Improved packing Bt0 for instance It
What are the remaining problems:
Keep the number of non-zero components small
What are the remaining problems:
Keep the number of non-zero components small Avoid calculation of optimum LP solutions
What are the remaining problems:
Keep the number of non-zero components small Avoid calculation of optimum LP solutions Dynamic rounding technique
Main Result We obtain a fully robust AFPTAS for the online bin packing problem with migration factor O(1/4 ) and running time O(M(1/2 )1/4 + t + 1/2 log(2 t)).
Open Questions:
Smaller migration factor and running time Lower bounds for migration factor Dynamic bin packing (allow departing of items) Use LP-techniques for other online problems
Open Questions:
Smaller migration factor and running time Lower bounds for migration factor Dynamic bin packing (allow departing of items) Use LP-techniques for other online problems Thank you!
Kim-Manuel Klein
University of Kiel
July 2, 2014
Online Bin Packing
...
Online Bin Packing
...
Online Bin Packing
...
Online Bin Packing
Given an instance It = {i1 , . . . it } of items for each time t ∈ N and S a function s : I = t It → [0, 1]. Find for each t ∈ N an assignment Bt : {i1 , . . . , it } → N+ such P that i:Bt (i)=j s(i) ≤ 1 for all j. Goal: Minimize the number of bins maxi {Bt (i)} for each time t.
Competitive Ratio of Online Bin Packing
Best known algorithm: Ratio of 1.58889 (S.S. Seiden, 2002) Best known lower bound: Ratio of 1.54037 (J. Balogh, B. Jozsef, and G. Galambos, 2010)
Offline Bin Packing
APTAS: Approximation guarantee: (1 + )OPT + 1 Running time: poly(n) + f (1/) (W. Fernandez de la Vega and G.S. Lueker, 1981) AFPTAS: Approximation guarantee: (1 + )OPT + O(1/2 ) Running time: poly(n, 1/) (N. Karmarkar and R.M. Karp, 1982)
Repacking
Online Bin Packing with Repacking Ratio 1.33 repacking 7 items (G. Gambosi et al., 2000) Ratio 1.25 repacking O(log t) "shifting moves" (Z. Ivkovic and E.L. Lloyd, 1998) Ratio 1 + repacking amortized O(log t) "shifting moves" (Z. Ivkovic and E.L. Lloyd, 1997)
Repacking
Online Bin Packing with Repacking Ratio 1.33 repacking 7 items (G. Gambosi et al., 2000) Ratio 1.25 repacking O(log t) "shifting moves" (Z. Ivkovic and E.L. Lloyd, 1998) Ratio 1 + repacking amortized O(log t) "shifting moves" (Z. Ivkovic and E.L. Lloyd, 1997) Online Scheduling: Achieving ratio < 1.4659 requires repacking of Θ(t) jobs (S. Albers and M. Hellwig, 2012)
Migration
Migration Factor between Bt and Bt+1 1
X
s(it+1 ) j≤t:B (i )6=B t
j
s(ij )
t+1 (ij )
An algorithm is robust if the migration factor is bounded by a function f (1/). (P. Sanders, N. Sivadasan and M. Skutella, 2004).
Questions:
Can we achieve approximation guarantee: (1 + )OPT + g(1/) with migration f (1/)?
Questions:
Can we achieve approximation guarantee: (1 + )OPT + g(1/) with migration f (1/)? 1
YES Online Scheduling on identical machines (P. Sanders, N. Sivadasan, and M. Skutella, 2004)
Questions:
Can we achieve approximation guarantee: (1 + )OPT + g(1/) with migration f (1/)? 1
2
YES Online Scheduling on identical machines (P. Sanders, N. Sivadasan, and M. Skutella, 2004) YES Online Bin Packing (L. Epstein and A. Levin, 2006)
Questions:
Can we achieve approximation guarantee: (1 + )OPT + g(1/) with migration f (1/)? 1
2
3
YES Online Scheduling on identical machines (P. Sanders, N. Sivadasan, and M. Skutella, 2004) YES Online Bin Packing (L. Epstein and A. Levin, 2006) YES and NO Online Machine Covering (M. Skutella and J. Verschae 2010)
Questions:
Can we achieve approximation guarantee: (1 + )OPT + g(1/) with migration f (1/)? 1
2
3
YES Online Scheduling on identical machines (P. Sanders, N. 2 Sivadasan, and M. Skutella, 2004) Mf: 2O(1/ log (1/)) YES Online Bin Packing (L. Epstein and A. Levin, 2006) Mf: 2 2O(1/ log(1/)) YES and NO Online Machine Covering (M. Skutella and J. 2 Verschae 2010) amortized Mf: 2O(1/ log (1/)
Questions:
Can we achieve approximation guarantee: (1 + )OPT + g(1/) with migration f (1/)? 1
2
3
YES Online Scheduling on identical machines (P. Sanders, N. 2 Sivadasan, and M. Skutella, 2004) Mf: 2O(1/ log (1/)) YES Online Bin Packing (L. Epstein and A. Levin, 2006) Mf: 2 2O(1/ log(1/)) YES and NO Online Machine Covering (M. Skutella and J. 2 Verschae 2010) amortized Mf: 2O(1/ log (1/)
Can we achieve polynomial migration and polynomial running time?
Our Result for Online Bin Packing: Approximation scheme with running time poly(t, 1/) and migration factor poly(1/).
Overview Robust Algorithms
optimum ILP solution y 0
rounding
Packing Bt for instance It
Overview Robust Algorithms
optimum ILP solution y 0
rounding
Packing Bt for instance It
ky 0 − y 00 k∞ bounded
optimum ILP solution y 00
rounding
Packing Bt+1 for instance It+1
Sensitivity Analysis
Goal Let y 0 be an optimum ILP solution of min {kx k1 |Ax ≥ b 0 , x ≥ 0}. Find an optimum ILP solution y 00 of min {kx k1 |Ax ≥ b 00 , x ≥ 0} such that ky 00 − y 0 k∞ is small.
Sensitivity Analysis
Goal Let y 0 be an optimum ILP solution of min {kx k1 |Ax ≥ b 0 , x ≥ 0}. Find an optimum ILP solution y 00 of min {kx k1 |Ax ≥ b 00 , x ≥ 0} such that ky 00 − y 0 k∞ is small. Theorem (Cook et al., 1986): There exists an optimum ILP solution y 00 with ky 00 − y 0 k∞ ≤ n∆(kb 00 − b 0 k∞ + 2).
Sensitivity Analysis
Goal Let y 0 be an optimum ILP solution of min {kx k1 |Ax ≥ b 0 , x ≥ 0}. Find an optimum ILP solution y 00 of min {kx k1 |Ax ≥ b 00 , x ≥ 0} such that ky 00 − y 0 k∞ is small. Theorem (Cook et al., 1986): There exists an optimum ILP solution y 00 with ky 00 − y 0 k∞ ≤ n∆(kb 00 − b 0 k∞ + 2). Problem: The number of variables n and the largest subdeterminant ∆ can only bounded by an exponential term in 1/.
Theorem Consider the LP min {kx k1 |Ax ≥ b, x ≥ 0} with A ∈ Rm×n ≥0 and let x 0 be an approximate fractional solution with kx 0 k1 ≤ (1 + δ)OPT for δ > 0 and kx 0 k1 ≥ α( 1δ + 1).
Theorem Consider the LP min {kx k1 |Ax ≥ b, x ≥ 0} with A ∈ Rm×n ≥0 and let x 0 be an approximate fractional solution with kx 0 k1 ≤ (1 + δ)OPT for δ > 0 and kx 0 k1 ≥ α( 1δ + 1). Then there exists a solution x 00 with kx 00 k1 ≤ (1 + δ)OPT − α and kx 0 − x 00 k1 ≤ 2α (1/δ + 1).
Prove the feasibility of the following linear program: Ax ≥ b x ≥0 X
xi ≤ (1 + δ)OPT − α x ≥ x 0 − α(1/δ + 1)
x0 kx 0 k1
x ≤ x 0 + α(1/δ + 1)
x OPT kx 0 k1
Prove the feasibility of the following linear program: Ax ≥ b x ≥0 X
xi ≤ (1 + δ)OPT − α x ≥ x 0 − α(1/δ + 1)
x0 kx 0 k1
x ≤ x 0 + α(1/δ + 1)
x OPT kx 0 k1
A feasible solution is x 00 = (1 −
α(1/δ+1) 0 kx 0 k1 )x
+
α(1/δ+1) OPT . kx 0 k1 x
Algorithm Let x 0 be a LP solution with kx 0 k ≤ (1 + δ)OPT and kx 0 k ≥ α(1/δ + 1). Set x fix := x 0 −
α(1/δ+1) kx 0 k1
x 0 and b var := b − A(x fix )
Solve the LP xˆ = min {kx k1 |Ax ≥ b var , x ≥ 0} Generate a new solution x 00 = x fix + xˆ
Algorithm Let x 0 be a LP solution with kx 0 k ≤ (1 + δ)OPT and kx 0 k ≥ α(1/δ + 1). Set x fix := x 0 −
α(1/δ+1) kx 0 k1
x 0 and b var := b − A(x fix )
Solve the LP xˆ = min {kx k1 |Ax ≥ b var , x ≥ 0} Generate a new solution x 00 = x fix + xˆ The algorithm returns a feasible LP solution x 00 with kx 00 k1 ≤ (1 + δ)OPT − α and distance kx 00 − x 0 k1 ≤ 2α(1/δ + 1).
Improve Packing:
Let Bt be a packing of instance It with maxi Bt (i) ≤ (1 + δ)OPT . Find a packing Bt0 with maxi Bt0 (i) ≤ (1 + δ)OPT − 1 such that migration factor between Bt and Bt0 is small.
ILP solution y 0
rounding
Packing Bt for instance It
LP solution x 0
ILP solution y 0
rounding
Packing Bt for instance It
LP solution
x0
ILP solution y 0
rounding
Packing Bt for instance It
kx 0 − x 00 k1 = O(1/)
LP solution x 00
LP solution
x0
ILP solution y 0
rounding
Packing Bt for instance It
kx 0 − x 00 k1 = O(1/)
LP solution x 00
ILP solution y 00
LP solution
x0
ILP solution
y0
rounding
Packing Bt for instance It
kx 0 − x 00 k1 = O(1/)
ky 0 − y 00 k1 = O(m/)
LP solution x 00
ILP solution y 00
LP solution
x0
ILP solution
y0
rounding
Packing Bt for instance It
kx 0 − x 00 k1 = O(1/)
ky 0 − y 00 k1 = O(m/)
LP solution x 00
ILP solution y 00
rounding
Improved packing Bt0 for instance It
What are the remaining problems:
Keep the number of non-zero components small
What are the remaining problems:
Keep the number of non-zero components small Avoid calculation of optimum LP solutions
What are the remaining problems:
Keep the number of non-zero components small Avoid calculation of optimum LP solutions Dynamic rounding technique
Main Result We obtain a fully robust AFPTAS for the online bin packing problem with migration factor O(1/4 ) and running time O(M(1/2 )1/4 + t + 1/2 log(2 t)).
Open Questions:
Smaller migration factor and running time Lower bounds for migration factor Dynamic bin packing (allow departing of items) Use LP-techniques for other online problems
Open Questions:
Smaller migration factor and running time Lower bounds for migration factor Dynamic bin packing (allow departing of items) Use LP-techniques for other online problems Thank you!
