Online Bin Packing with Advice Joan Boyar 1 , Shahin Kamali 2 , Kim - - PowerPoint PPT Presentation

Online Bin Packing with Advice

Joan Boyar1, Shahin Kamali2, Kim S. Larsen1, Alejandro L´

• pez-Ortiz2

March 8, 2014

1 University of Southern Denmark, Denmark 2 University of Waterloo, Canada 1 / 19 Online Bin Packing with Advice

Problem Statement

Bin Packing Problem

The input is a set of items of various sizes.

Item sizes are in range (0, 1] E.g., < 9, 3, 8, 5, 1, 1, 3, 2, 4, 2, 4, 5, 5, 8, 6, 4, 5 >.

The goal is to pack these items into a minimum number of bins of uniform capacity.

• 2 / 19

Online Bin Packing with Advice

Problem Statement

Bin Packing Problem

In the offline version of the problem, you have access to the whole set at the beginning.

E.g., you can sort items (E.g., First Fit Decreasing).

3 / 19 Online Bin Packing with Advice

Problem Statement

Bin Packing Problem

In the offline version of the problem, you have access to the whole set at the beginning.

E.g., you can sort items (E.g., First Fit Decreasing).

The problem is NP-hard [Garey and Johnson, 1979].

First Fit Decreasing has an approximation ratio of 11/9 ≈ 1.22 [Yue, 1991]. There is an asymptotic PTAS for the problem [Fernandez de la Vega and Lueker, 1981].

3 / 19 Online Bin Packing with Advice

Problem Statement

Online Bin Packing Problem

The input is a sequence of items of various sizes which are revealed in a sequential, online manner.

E.g., < 9, 3, 8, 5, 1, 1, 3, 2, 4, 2, 4, 5, 5, 8, 6, 4, . . . , 5 >.

Almost Any Fit algorithms

E.g., First Fit and Best Fit algorithms (widely applied).

Harmonic class of algorithms

Harmonic-based on placing items of almost equal sizes together.

4 / 19 Online Bin Packing with Advice

Algorithms

First Fit Algorithm

Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist.

5 / 19 Online Bin Packing with Advice

Algorithms

First Fit Algorithm

Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist.

< 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 >

9

5 / 19 Online Bin Packing with Advice

Algorithms

First Fit Algorithm

Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist.

< 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 >

3 9

5 / 19 Online Bin Packing with Advice

Algorithms

First Fit Algorithm

Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist.

< 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 >

3 8 3 8 9

5 / 19 Online Bin Packing with Advice

Algorithms

First Fit Algorithm

Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist.

< 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 >

3 8 5 3 8 9

5 / 19 Online Bin Packing with Advice

Algorithms

First Fit Algorithm

Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist.

< 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 >

3 8 5 1 5 3 8 9

5 / 19 Online Bin Packing with Advice

Algorithms

First Fit Algorithm

Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist.

< 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 >

3 8 5 1 1 1 1 5 3 8 9

5 / 19 Online Bin Packing with Advice

Algorithms

First Fit Algorithm

Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist.

< 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 >

3 8 5 1 1 3 1 1 3 5 3 8 9

5 / 19 Online Bin Packing with Advice

Algorithms

First Fit Algorithm

Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist.

< 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 >

3 8 5 1 1 3 2 1 1 3 5 3 8 9

5 / 19 Online Bin Packing with Advice

Algorithms

First Fit Algorithm

Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist.

< 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 >

3 8 5 1 1 3 2 4 1 1 2 3 4 5 3 8 4 9

5 / 19 Online Bin Packing with Advice

Algorithms

First Fit Algorithm

Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist.

< 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 >

3 8 5 1 1 3 2 4 2 1 1 2 2 3 4 5 3 8 4 9

5 / 19 Online Bin Packing with Advice

Algorithms

First Fit Algorithm

Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist.

< 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 >

3 8 5 1 1 3 2 4 2 4 1 1 2 2 3 4 5 3 8 4 9

5 / 19 Online Bin Packing with Advice

Algorithms

First Fit Algorithm

Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist.

< 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 >

3 8 5 1 1 3 2 4 2 4 5 1 1 2 2 3 4 5 5 3 8 4 9

5 / 19 Online Bin Packing with Advice

Algorithms

First Fit Algorithm

Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist.

< 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 >

3 8 5 1 1 3 2 4 2 4 5 5 1 1 2 2 3 4 5 5 5 3 8 4 9

5 / 19 Online Bin Packing with Advice

Algorithms

First Fit Algorithm

Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist.

< 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 >

3 8 5 1 1 3 2 4 2 4 5 8 5 1 1 2 2 3 4 5 5 5 8 3 8 4 9

5 / 19 Online Bin Packing with Advice

Algorithms

First Fit Algorithm

Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist.

< 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 >

3 8 5 1 1 3 2 4 2 4 5 8 5 1 1 2 2 3 4 5 5 5 6 8 3 8 4 9

5 / 19 Online Bin Packing with Advice

Algorithms

First Fit Algorithm

Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist.

< 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 >

3 8 5 1 1 3 2 4 2 4 5 8 5 1 1 2 2 3 4 5 5 5 6 8 3 8 4 9 4

5 / 19 Online Bin Packing with Advice

Algorithms

First Fit Algorithm

Find the first bin which has enough space for the item, and place the item there Open a new bin if such bin does not exist.

< 9 3 8 5 1 1 3 2 4 2 4 5 5 8 6 4 5 >

3 8 5 1 1 3 2 4 2 4 5 8 5 5 4 1 1 2 2 3 4 5 5 5 5 6 8 3 8 4 9

5 / 19 Online Bin Packing with Advice

Worst Case Analysis

Competitive Analysis

Compare the performance of an online algorithm with an optimal

• ffline algorithm Opt:

Opt knows the whole sequence in the beginning.

Competitive ratio of an algorithm A is the maximum ratio between the cost of A and Opt for serving the same sequence.

6 / 19 Online Bin Packing with Advice

Worst Case Analysis

Competitive Analysis

Competitive ratio of Best Fit and First Fit are both 1.7 [Johnson et al., 1974]. The best existing online algorithm (Harmonic++) has a competitive ratio of 1.582 [Seiden, 2002]. No algorithm has a competitive ratio less than 1.54 [Vliet, 1992].

7 / 19 Online Bin Packing with Advice

Worst Case Analysis

Competitive Analysis

Competitive ratio of Best Fit and First Fit are both 1.7 [Johnson et al., 1974]. The best existing online algorithm (Harmonic++) has a competitive ratio of 1.582 [Seiden, 2002]. No algorithm has a competitive ratio less than 1.54 [Vliet, 1992]. Recall that offline First Fit Decreasing has an approximation ratio of 1.22.

A big gap between quality of online and offline solutions. What about an ‘almost online’ algorithm?

7 / 19 Online Bin Packing with Advice

Advice Complexity

Advice Model for Online Bin Packing Problem

Relax ‘absolutely no knowledge’ assumption behind

Look-ahead?

8 / 19 Online Bin Packing with Advice

Advice Complexity

Advice Model for Online Bin Packing Problem

Relax ‘absolutely no knowledge’ assumption behind

Look-ahead? Closed bin packing? (length of sequence is known to the algorithm)

8 / 19 Online Bin Packing with Advice

Advice Complexity

Advice Model for Online Bin Packing Problem

Relax ‘absolutely no knowledge’ assumption behind

Look-ahead? Closed bin packing? (length of sequence is known to the algorithm)

Under the advice model, the online algorithms are provided with b bits of advice:

The advice is generated by an offline oracle. The advice is written on a tape and can be accessed by the online algorithm at any time.

There are other advice models for bin packing [Renault et al., 2013].

8 / 19 Online Bin Packing with Advice

Advice Complexity

Relevant Questions

For a sequence of fixed length

How many bits of advice are required (sufficient) to achieve an

• ptimal solution?

How many bits of advice are sufficient to outperform all online algorithms? How good can the competitive ratio be with advice of linear/sublinear size?

9 / 19 Online Bin Packing with Advice

Advice Complexity

Optimal Solution with Advice

How many bits of advice are required (sufficient) to achieve an

• ptimal solution?

For each item, include an encoding of the index of the target bin for each item in Opt’s packing. n⌈log Opt(σ)⌉ bits of advice are sufficient to achieve an optimal solution.

[ 0 1 2 2 1 0 0 3 4 3 3 5 4 6 7 7 5 ]

0 1 2 3 4 5 6 7

< 9 3 8 2 5 1 1 3 4 2 4 5 5 8 6 4 5 > < 9 3 8 2 5 1 1 3 4 2 4 5 5 8 6 4 5 >

3 8 5 1 1 3 2 4 2 4 5 8 5 4 1 1 2 2 3 4 5 5 5 5 6 8 3 8 4 9 0 1 2 3 4 5 6 7

10 / 19 Online Bin Packing with Advice

Advice Complexity

Optimal Solution with Advice

How many bits of advice are required (sufficient) to achieve an

• ptimal solution?

For each item, include an encoding of the index of the target bin for each item in Opt’s packing. n⌈log Opt(σ)⌉ bits of advice are sufficient to achieve an optimal solution.

[ 0 1 2 2 1 0 0 3 4 3 3 5 4 6 7 7 5 ]

9 0 1 2 3 4 5 6 7

< 9 3 8 2 5 1 1 3 4 2 4 5 5 8 6 4 5 > < 9 3 8 2 5 1 1 3 4 2 4 5 5 8 6 4 5 >

3 8 5 1 1 3 2 4 2 4 5 8 5 4 1 1 2 2 3 4 5 5 5 5 6 8 3 8 4 9 0 1 2 3 4 5 6 7

10 / 19 Online Bin Packing with Advice

Advice Complexity

Optimal Solution with Advice

How many bits of advice are required (sufficient) to achieve an

• ptimal solution?

For each item, include an encoding of the index of the target bin for each item in Opt’s packing. n⌈log Opt(σ)⌉ bits of advice are sufficient to achieve an optimal solution.

[ 0 1 2 2 1 0 0 3 4 3 3 5 4 6 7 7 5 ]

9 0 1 2 3 4 5 6 7

< 9 3 8 2 5 1 1 3 4 2 4 5 5 8 6 4 5 >

3

< 9 3 8 2 5 1 1 3 4 2 4 5 5 8 6 4 5 >

3 8 5 1 1 3 2 4 2 4 5 8 5 4 1 1 2 2 3 4 5 5 5 5 6 8 3 8 4 9 0 1 2 3 4 5 6 7

10 / 19 Online Bin Packing with Advice

Advice Complexity

Optimal Solution with Advice

How many bits of advice are required (sufficient) to achieve an

• ptimal solution?

For each item, include an encoding of the index of the target bin for each item in Opt’s packing. n⌈log Opt(σ)⌉ bits of advice are sufficient to achieve an optimal solution.

[ 0 1 2 2 1 0 0 3 4 3 3 5 4 6 7 7 5 ]

9 0 1 2 3 4 5 6 7

< 9 3 8 2 5 1 1 3 4 2 4 5 5 8 6 4 5 >

3 8

< 9 3 8 2 5 1 1 3 4 2 4 5 5 8 6 4 5 >

3 8 5 1 1 3 2 4 2 4 5 8 5 4 1 1 2 2 3 4 5 5 5 5 6 8 3 8 4 9 0 1 2 3 4 5 6 7

10 / 19 Online Bin Packing with Advice

Advice Complexity

Optimal Solution with Advice

How many bits of advice are required (sufficient) to achieve an

• ptimal solution?

For each item, include an encoding of the index of the target bin for each item in Opt’s packing. n⌈log Opt(σ)⌉ bits of advice are sufficient to achieve an optimal solution.

[ 0 1 2 2 1 0 0 3 4 3 3 5 4 6 7 7 5 ]

9 0 1 2 3 4 5 6 7

< 9 3 8 2 5 1 1 3 4 2 4 5 5 8 6 4 5 >

3 8 2

< 9 3 8 2 5 1 1 3 4 2 4 5 5 8 6 4 5 >

3 8 5 1 1 3 2 4 2 4 5 8 5 4 1 1 2 2 3 4 5 5 5 5 6 8 3 8 4 9 0 1 2 3 4 5 6 7

10 / 19 Online Bin Packing with Advice

Advice Complexity

Optimal Solution with Advice

How many bits of advice are required (sufficient) to achieve an

• ptimal solution?

For each item, include an encoding of the index of the target bin for each item in Opt’s packing. n⌈log Opt(σ)⌉ bits of advice are sufficient to achieve an optimal solution.

[ 0 1 2 2 1 0 0 3 4 3 3 5 4 6 7 7 5 ]

9 0 1 2 3 4 5 6 7

< 9 3 8 2 5 1 1 3 4 2 4 5 5 8 6 4 5 >

3 8 2 5

< 9 3 8 2 5 1 1 3 4 2 4 5 5 8 6 4 5 >

3 8 5 1 1 3 2 4 2 4 5 8 5 4 1 1 2 2 3 4 5 5 5 5 6 8 3 8 4 9 0 1 2 3 4 5 6 7

10 / 19 Online Bin Packing with Advice

Advice Complexity

Optimal Solution with Advice

In fact (n − 2 Opt(σ)) log Opt(σ) bits of advice are required to achieve an optimal packing.

11 / 19 Online Bin Packing with Advice

Advice Complexity

Optimal Solution with Advice

In fact (n − 2 Opt(σ)) log Opt(σ) bits of advice are required to achieve an optimal packing.

Theorem To achieve an optimal packing for a sequence of size n and

• ptimal cost Opt(σ), it is sufficient to receive n⌈log Opt(σ)⌉

bits of advice. Moreover, any online algorithm requires at least (n − 2 Opt(σ)) log Opt(σ) bits of advice to achieve an optimal packing.

11 / 19 Online Bin Packing with Advice

Advice Complexity

Optimal Solution with Advice

Assume the sequence is formed by m = o(n) distinct items which have size larger than a fixed value ǫ.

12 / 19 Online Bin Packing with Advice

Advice Complexity

Optimal Solution with Advice

Assume the sequence is formed by m = o(n) distinct items which have size larger than a fixed value ǫ. Idea: encode the whole input set in roughly m log n bits.

Create an (almost) optimal packing and maintain it as items appear!

12 / 19 Online Bin Packing with Advice

Advice Complexity

Optimal Solution with Advice

Assume the sequence is formed by m = o(n) distinct items which have size larger than a fixed value ǫ. Idea: encode the whole input set in roughly m log n bits.

Create an (almost) optimal packing and maintain it as items appear!

It is sufficient to read m⌈log(n + 1)⌉ + o(log n) bits of advice to achieve an (almost optimal) packing.

12 / 19 Online Bin Packing with Advice

Advice Complexity

Optimal Solution with Advice

Indeed, this upper bound is almost tight.

13 / 19 Online Bin Packing with Advice

Advice Complexity

Optimal Solution with Advice

Indeed, this upper bound is almost tight. Consider a family of sequences which start by n/2 items of size ǫ.

Let X denote the number of ways that these n/2 items can be packed.

1

Example: n=30, m=6 (bin capacities scaled up by 12) sequence: <1(15) ... sequence: <1(15) ... 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

13 / 19 Online Bin Packing with Advice

Advice Complexity

Optimal Solution with Advice

Indeed, this upper bound is almost tight. Consider a family of sequences which start by n/2 items of size ǫ.

Let X denote the number of ways that these n/2 items can be packed. For each partial packing, complete the sequence with items which fill the empty spaces.

1

12 11 Example: n=30, m=6 (bin capacities scaled up by 12) sequence: <1(15) 11 11 11 11 10 10 9 8 12(7)> sequence: <1(15) 11 11 11 11 11 11 11 8 8 12(6) >… 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 11 11 11 11 11 11 11 11 11 10 10 9 8 8 8 12 12 12 12 12 12 12 12 12 12 12 12

13 / 19 Online Bin Packing with Advice

Advice Complexity

Optimal Solution with Advice

Indeed, this upper bound is almost tight. Consider a family of sequences which start by n/2 items of size ǫ.

Let X denote the number of ways that these n/2 items can be packed. For each partial packing, complete the sequence with items which fill the empty spaces. Each sequence requires a distinct advice, and consequently advice of size lg X is required.

1

12 11 Example: n=30, m=6 (bin capacities scaled up by 12) sequence: <1(15) 11 11 11 11 10 10 9 8 12(7)> sequence: <1(15) 11 11 11 11 11 11 11 8 8 12(6) >… 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 11 11 11 11 11 11 11 11 11 10 10 9 8 8 8 12 12 12 12 12 12 12 12 12 12 12 12

13 / 19 Online Bin Packing with Advice

Advice Complexity

Optimal Solution with Advice

Theorem It is sufficient to read m⌈log(n + 1)⌉ + o(log n) bits of advice to achieve an (almost optimal) packing. At least (m − 3) log n − 2m log m bits of advice are required to achieve an optimal solution.

14 / 19 Online Bin Packing with Advice

Advice Complexity

Breaking the Lower Bound

Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5.

All online algorithms have a competitive ratio of at least 1.54.

15 / 19 Online Bin Packing with Advice

Advice Complexity

Breaking the Lower Bound

Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5.

All online algorithms have a competitive ratio of at least 1.54.

Receive the number of items in range (1/2, 2/3]. Reserve a space of size 2/3 for each of them, apply FF for other items.

15 / 19 Online Bin Packing with Advice

Advice Complexity

Breaking the Lower Bound

Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5.

All online algorithms have a competitive ratio of at least 1.54.

Receive the number of items in range (1/2, 2/3]. Reserve a space of size 2/3 for each of them, apply FF for other items.

< 9 3 6 5 1 1 3 2 4 2 4 5 6 8 6 4 5 >

15 / 19 Online Bin Packing with Advice

Advice Complexity

Breaking the Lower Bound

Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5.

All online algorithms have a competitive ratio of at least 1.54.

Receive the number of items in range (1/2, 2/3]. Reserve a space of size 2/3 for each of them, apply FF for other items.

< 9 3 6 5 1 1 3 2 4 2 4 5 6 8 6 4 5 >

2

15 / 19 Online Bin Packing with Advice

Advice Complexity

Breaking the Lower Bound

Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5.

All online algorithms have a competitive ratio of at least 1.54.

Receive the number of items in range (1/2, 2/3]. Reserve a space of size 2/3 for each of them, apply FF for other items.

< 9 3 6 5 1 1 3 2 4 2 4 5 6 8 6 4 5 >

2 9

15 / 19 Online Bin Packing with Advice

Advice Complexity

Breaking the Lower Bound

Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5.

All online algorithms have a competitive ratio of at least 1.54.

Receive the number of items in range (1/2, 2/3]. Reserve a space of size 2/3 for each of them, apply FF for other items.

< 9 3 6 5 1 1 3 2 4 2 4 5 6 8 6 4 5 >

2 9 3

15 / 19 Online Bin Packing with Advice

Advice Complexity

Breaking the Lower Bound

Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5.

All online algorithms have a competitive ratio of at least 1.54.

Receive the number of items in range (1/2, 2/3]. Reserve a space of size 2/3 for each of them, apply FF for other items.

< 9 3 6 5 1 1 3 2 4 2 4 5 6 8 6 4 5 >

2 9 3 6

15 / 19 Online Bin Packing with Advice

Advice Complexity

Breaking the Lower Bound

Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5.

All online algorithms have a competitive ratio of at least 1.54.

Receive the number of items in range (1/2, 2/3]. Reserve a space of size 2/3 for each of them, apply FF for other items.

< 9 3 6 5 1 1 3 2 4 2 4 5 6 8 6 4 5 >

2 9 3 6 5

15 / 19 Online Bin Packing with Advice

Advice Complexity

Breaking the Lower Bound

Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5.

All online algorithms have a competitive ratio of at least 1.54.

Receive the number of items in range (1/2, 2/3]. Reserve a space of size 2/3 for each of them, apply FF for other items.

< 9 3 6 5 1 1 3 2 4 2 4 5 6 8 6 4 5 >

2 9 3 6 5 1

15 / 19 Online Bin Packing with Advice

Advice Complexity

Breaking the Lower Bound

Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5.

All online algorithms have a competitive ratio of at least 1.54.

Receive the number of items in range (1/2, 2/3]. Reserve a space of size 2/3 for each of them, apply FF for other items.

< 9 3 6 5 1 1 3 2 4 2 4 5 6 8 6 4 5 >

2 9 3 6 5 1 1

15 / 19 Online Bin Packing with Advice

Advice Complexity

Breaking the Lower Bound

Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5.

All online algorithms have a competitive ratio of at least 1.54.

Receive the number of items in range (1/2, 2/3]. Reserve a space of size 2/3 for each of them, apply FF for other items.

< 9 3 6 5 1 1 3 2 4 2 4 5 6 8 6 4 5 >

2 9 3 6 5 1 1 3

15 / 19 Online Bin Packing with Advice

Advice Complexity

Breaking the Lower Bound

Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5.

All online algorithms have a competitive ratio of at least 1.54.

Receive the number of items in range (1/2, 2/3]. Reserve a space of size 2/3 for each of them, apply FF for other items.

< 9 3 6 5 1 1 3 2 4 2 4 5 6 8 6 4 5 >

2 9 3 6 5 1 1 3 2

15 / 19 Online Bin Packing with Advice

Advice Complexity

Breaking the Lower Bound

Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5.

All online algorithms have a competitive ratio of at least 1.54.

Receive the number of items in range (1/2, 2/3]. Reserve a space of size 2/3 for each of them, apply FF for other items.

< 9 3 6 5 1 1 3 2 4 2 4 5 6 8 6 4 5 >

2 9 3 6 5 1 1 3 2 4

15 / 19 Online Bin Packing with Advice

Advice Complexity

Breaking the Lower Bound

Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5.

All online algorithms have a competitive ratio of at least 1.54.

Receive the number of items in range (1/2, 2/3]. Reserve a space of size 2/3 for each of them, apply FF for other items.

< 9 3 6 5 1 1 3 2 4 2 4 5 6 8 6 4 5 >

2 9 3 6 5 1 1 3 2 4 2

15 / 19 Online Bin Packing with Advice

Advice Complexity

Breaking the Lower Bound

Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5.

All online algorithms have a competitive ratio of at least 1.54.

Receive the number of items in range (1/2, 2/3]. Reserve a space of size 2/3 for each of them, apply FF for other items.

< 9 3 6 5 1 1 3 2 4 2 4 5 6 8 6 4 5 >

2 9 3 6 5 1 1 3 2 4 2 4

15 / 19 Online Bin Packing with Advice

Advice Complexity

Breaking the Lower Bound

Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5.

All online algorithms have a competitive ratio of at least 1.54.

Receive the number of items in range (1/2, 2/3]. Reserve a space of size 2/3 for each of them, apply FF for other items.

< 9 3 6 5 1 1 3 2 4 2 4 5 6 8 6 4 5 >

2 9 3 6 5 1 1 3 2 4 2 4 5

15 / 19 Online Bin Packing with Advice

Advice Complexity

Breaking the Lower Bound

Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5.

All online algorithms have a competitive ratio of at least 1.54.

Receive the number of items in range (1/2, 2/3]. Reserve a space of size 2/3 for each of them, apply FF for other items.

< 9 3 6 5 1 1 3 2 4 2 4 5 6 8 6 4 5 >

2 9 3 6 5 1 1 3 2 4 2 4 5 6

15 / 19 Online Bin Packing with Advice

Advice Complexity

Breaking the Lower Bound

Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5.

All online algorithms have a competitive ratio of at least 1.54.

Receive the number of items in range (1/2, 2/3]. Reserve a space of size 2/3 for each of them, apply FF for other items.

< 9 3 6 5 1 1 3 2 4 2 4 5 6 8 6 4 5 >

2 9 3 6 5 1 1 3 2 4 2 4 5 6 8

15 / 19 Online Bin Packing with Advice

Advice Complexity

Breaking the Lower Bound

Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5.

All online algorithms have a competitive ratio of at least 1.54.

Receive the number of items in range (1/2, 2/3]. Reserve a space of size 2/3 for each of them, apply FF for other items.

< 9 3 6 5 1 1 3 2 4 2 4 5 6 8 6 4 5 >

2 9 3 6 5 1 1 3 2 4 2 4 5 6 8 6

15 / 19 Online Bin Packing with Advice

Advice Complexity

Breaking the Lower Bound

Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5.

All online algorithms have a competitive ratio of at least 1.54.

Receive the number of items in range (1/2, 2/3]. Reserve a space of size 2/3 for each of them, apply FF for other items.

< 9 3 6 5 1 1 3 2 4 2 4 5 6 8 6 4 5 >

2 9 3 6 5 1 1 3 2 4 2 4 5 6 8 6 4

15 / 19 Online Bin Packing with Advice

Advice Complexity

Breaking the Lower Bound

Advice of size ⌈log n⌉ is sufficient to achieve a competitive ratio of 1.5.

All online algorithms have a competitive ratio of at least 1.54.

Receive the number of items in range (1/2, 2/3]. Reserve a space of size 2/3 for each of them, apply FF for other items.

< 9 3 6 5 1 1 3 2 4 2 4 5 6 8 6 4 5 >

2 9 3 6 5 1 1 3 2 4 2 4 5 6 8 6 4 5

15 / 19 Online Bin Packing with Advice

Advice Complexity

Advice of Linear Size

An online algorithm which receives two bits of advice per request (plus an additive lower order term).

Achieves a competitive ratio of 4/3 + ε, for any positive value of ε. A variety of bin packing techniques are used in the proof.

16 / 19 Online Bin Packing with Advice

Advice Complexity

Advice of Linear Size

An online algorithm which receives two bits of advice per request (plus an additive lower order term).

Achieves a competitive ratio of 4/3 + ε, for any positive value of ε. A variety of bin packing techniques are used in the proof.

< 9 3 1 8 2 5 3 4 3 4 5 5 8 6 4 5 >

3 8 5 3 2 4 4 5 8 5 4 3 4 5 5 5 5 6 8 3 8 4 9 3 1

16 / 19 Online Bin Packing with Advice

Advice Complexity

Advice of Linear Size

An online algorithm which receives two bits of advice per request (plus an additive lower order term).

Achieves a competitive ratio of 4/3 + ε, for any positive value of ε. A variety of bin packing techniques are used in the proof.

< 9 3 1 8 2 5 3 4 3 4 5 5 8 6 4 5 >

3 8 5 3 2 4 4 5 8 5 4 3 4 5 5 5 5 6 8 3 8 4 9 3 Good bins 1

16 / 19 Online Bin Packing with Advice

Advice Complexity

Advice of Linear Size

An online algorithm which receives two bits of advice per request (plus an additive lower order term).

Achieves a competitive ratio of 4/3 + ε, for any positive value of ε. A variety of bin packing techniques are used in the proof.

< 9 3 1 8 2 5 3 4 3 4 5 5 8 6 4 5 >

3 8 5 3 2 4 4 5 8 5 4 3 4 5 5 5 5 6 8 3 8 4 9 3 Good bins Bad bins (type 1) Bad bins (type 2) Bad bins (type 3) 1

16 / 19 Online Bin Packing with Advice

Advice Complexity

Advice of Linear Size

An online algorithm which receives two bits of advice per request (plus an additive lower order term).

Achieves a competitive ratio of 4/3 + ε, for any positive value of ε. A variety of bin packing techniques are used in the proof.

< 9 3 1 8 2 5 3 4 3 4 5 5 8 6 4 5 >

3 8 5 3 2 4 4 5 8 5 4 3 4 5 5 5 5 6 8 3 8 4 9 3 Good bins Bad bins (type 1) Bad bins (type 2) Bad bins (type 3) 1 Advice: < G B2 G G G B2 B3 B3 B3 B2 B2 B2 B1 B2 B2 B2 >

16 / 19 Online Bin Packing with Advice

Advice Complexity

Advice of Linear Size

An online algorithm which receives two bits of advice per request (plus an additive lower order term).

Achieves a competitive ratio of 4/3 + ε, for any positive value of ε. A variety of bin packing techniques are used in the proof.

Place items of each group separately:

G items: 9 1 8 2. B1 items: 8. B2 items: 3 5 4 5 5 6 4 5. B3 items: 3 4 3.

< 9 3 1 8 2 5 3 4 3 4 5 5 8 6 4 5 >

3 8 5 3 2 4 4 5 8 5 4 3 4 5 5 5 5 6 8 3 8 4 9 3 Good bins Bad bins (type 1) Bad bins (type 2) Bad bins (type 3) 1 Advice: < G B2 G G G B2 B3 B3 B3 B2 B2 B2 B1 B2 B2 B2 >

16 / 19 Online Bin Packing with Advice

Advice Complexity

Another Lower Bound

An advice of linear size is required to achieve a competitive ratio better than 9/8. Reduction order: Binary guessing problem [Emek et al., 2011, B¨

• ckenhauer et al., 2013] −

→ separation problem − → bin packing problem

17 / 19 Online Bin Packing with Advice

Advice Complexity

Another Lower Bound

An advice of linear size is required to achieve a competitive ratio better than 9/8. Reduction order: Binary guessing problem [Emek et al., 2011, B¨

• ckenhauer et al., 2013] −

→ separation problem − → bin packing problem Binary guessing problem (with known history)

Guess the next bit in a bitstring revealed in an

• nline manner

0 1 0 ? An advice of linear size is required to correctly guess more than half bits.

17 / 19 Online Bin Packing with Advice

Advice Complexity

Another Lower Bound

An advice of linear size is required to achieve a competitive ratio better than 9/8. Reduction order: Binary guessing problem [Emek et al., 2011, B¨

• ckenhauer et al., 2013] −

→ separation problem − → bin packing problem Binary guessing problem (with known history)

Guess the next bit in a bitstring revealed in an

• nline manner

0 1 0 ? An advice of linear size is required to correctly guess more than half bits.

Binary separation problem

For a sequence of n1 + n2 items decide whether an item belongs to the n1 smaller items or n2 larger items. .5(s) .75(l) .625(s) .7104875(?)

17 / 19 Online Bin Packing with Advice

Advice Complexity

Discussion

The bounds for the size of advice are tight for getting the optimal solution.

18 / 19 Online Bin Packing with Advice

Advice Complexity

Discussion

The bounds for the size of advice are tight for getting the optimal solution. It is not clear at least how many bits are required to break the lower bound for online algorithms.

We know ⌈log n⌉ bits are sufficient.

18 / 19 Online Bin Packing with Advice

Advice Complexity

Discussion

The bounds for the size of advice are tight for getting the optimal solution. It is not clear at least how many bits are required to break the lower bound for online algorithms.

We know ⌈log n⌉ bits are sufficient.

Advice of linear size is required to achieve a competitive ratio better than 9/8.

In fact, with linear advice, one can achieve a competitive ratio of 1 + ǫ for any constant value of ǫ [Renault et al., 2013].

18 / 19 Online Bin Packing with Advice

Advice Complexity

Discussion

The bounds for the size of advice are tight for getting the optimal solution. It is not clear at least how many bits are required to break the lower bound for online algorithms.

We know ⌈log n⌉ bits are sufficient.

Advice of linear size is required to achieve a competitive ratio better than 9/8.

In fact, with linear advice, one can achieve a competitive ratio of 1 + ǫ for any constant value of ǫ [Renault et al., 2013].

What if advice appears in an online manner? [Renault et al., 2013]

18 / 19 Online Bin Packing with Advice

Advice Complexity

Bibliography

B¨

• ckenhauer, H.-J.; Hromkovic, J.; Komm, D.; Krug, S.; Smula, J.; and

Sprock, A. (2013).

”The String Guessing Problem as a Method to Prove Lower Bounds on the Advice Complexity”. In COCOON ’13, volume 7936 of LNCS, pages 493–505.

Emek, Y.; Fraigniaud, P.; Korman, A.; and Ros´ en, A. (2011).

”Online computation with advice”.

• Theoret. Comput. Sci., 412(24), pp. 2642 – 2656.

Fernandez de la Vega, W. and Lueker, G. (1981).

”Bin packing can be solved within 1 + ǫ in linear time”. Combinatorica, 1, pp. 349–355.

Garey, M. R. and Johnson, D. S. (1979).

Computers and Intractability: A Guide to the theory of of NP-Completeness. Freeman and Company, San Francisco.

Johnson, D. S.; Demers, A.; Ullman, J. D.; Garey, M. R.; and Graham, R. L. (1974).

”Worst-case performance bounds for simple one-dimensional packing algorithms”. SIAM J. Comput., 3, pp. 256–278.

Renault, M. P.; Ros´ en, A.; and van Stee, R. (2013).

”Online Algorithms with Advice for Bin Packing and Scheduling Problems”. CoRR, abs/1311.7589.

Seiden, S. S. (2002).

”On the online bin packing problem”.

• J. ACM, 49, pp. 640–671.

19 / 19 Online Bin Packing with Advice