Online Colouring Problems in Overlap Graphs and their Complements e - - PowerPoint PPT Presentation

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Online Colouring Problems in Overlap Graphs and their Complements e - - PowerPoint PPT Presentation

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Online Colouring Problems in Overlap Graphs and their Complements e g n a , m e c e n D e c i c r S a M f o l o o h c S , y t i s r e v n i U T I M a l i R a r t s u A , e n r u o b u e l

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M a r c D e m a n g e R M I T U n i v e r s i t y , S c h

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S c i e n c e , M e l b

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r n e , A u s t r a l i a m a r c . d e m a n g e @ r m i t . e d u . a u M a r t i n O l s e n , B T E C H , A a r h u s U n i v e r s i t y , D e n m a r k m a r t i n

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b t e c h . a u . d k

Online Colouring Problems in Overlap Graphs and their Complements

  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018

P r e s e n t e d a t W A L C O M : A l g

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i t h m s a n d C

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p u t a t i

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, M a r c h 2 1 8 , D a k h a , B a n g l a d e s h

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M a r c D e m a n g e R M I T U n i v e r s i t y , S c h

  • l
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S c i e n c e , M e l b

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r n e , A u s t r a l i a m a r c . d e m a n g e @ r m i t . e d u . a u M a r t i n O l s e n , B T E C H , A a r h u s U n i v e r s i t y , D e n m a r k m a r t i n

  • @

b t e c h . a u . d k

Online Colouring Problems in Overlap Graphs and their Complements

(the art of BBQ down under)

  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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Motivation 1: stacking problem

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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Stacking problem

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 5

5/12 – 15/12

Stacking problem

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 6

5/12 – 15/12

18/09

18/09 – 26/09

Stacking problem

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 7

5/12 – 15/12

19/09

19/09–30/09

Stacking problem

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018

18/09 – 26/09

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SLIDE 8

5/12 – 15/12

26/09

19/09 – 30/09 26/09

Stacking problem

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018

18/09 – 26/09

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SLIDE 9

5/12 – 15/12

19/09

Stacking problem

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018

18/09 – 26/09 19/09–30/09

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SLIDE 10

5/12 – 15/12

20/09

20/09 – 28/09

Stacking problem

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018

18/09 – 26/09 19/09–30/09

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SLIDE 11

5/12 – 15/12

27/09

Stacking problem

6/12 – 21/12 27/09 – 5/10

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018

19/09–30/09 20/09 – 28/09

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SLIDE 12

Related graph problem

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018

6/12 – 21/12 19/09–30/09 20/09 – 28/09 18/09 – 26/09 27/09 – 5/10

18 20 19 5/10 26 28 30 27

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Related graph problem: incompatibility graph

19/09–30/09 20/09 – 28/09 18/09 – 26/09 27/09 – 5/10

Overlap graph

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018

18 20 19 5/10 26 28 30 27

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Overlap graphs as intersection graphs of chords (circle graphs)

A B C E D F

A B D C E F A B C D E F

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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Related graph problem: min colouring

19/09–30/09 20/09 – 28/09 18/09 – 26/09 27/09 – 5/10

Overlap graph

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018

18 20 19 5/10 26 28 30 27

Stack 1 Stack 2

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Motivation 2: track assignment

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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1 2 3 4 5 6 23 22 21 20 19 18 5 4 3 2 1 6

Midnight condition

5 2 1 4 3 6

Permutation graph

1 2 3 4 5 6

A particular case

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 18

1 2 3 4 5 6 23 22 21 20 19 18

Midnight condition

5 2 1 4 3 6

Permutation graph

1 2 3 4 5 6

A particular case

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 19

1 2 3 4 5 6 23 22 21 20 19 18

Midnight condition

5 2 1 4 3 6

Permutation graph

1 2 3 4 5 6

A particular case

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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Stacking problem / track assignment

  • Assign each item on a stack
  • Allays put / remove items on the top a stack (Last in first out)
  • The related incompatibility graph is an overlap / permutation graph
  • Minimising the number of stacks

Minimum colouring overlap / permutation graphs

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 21

On-line stacking / track assignment

  • Nothing is known about the future

(items are known when they arrive)

  • Departure time known at arrival
  • Assign stack at arrival, never on top of an item leaving earlier

On-line colouring overlap / permutation graphs

  • Eventually additional constraints: e.g. fixed capacity for each stack
  • Graph defines by intervals revealed from left to right

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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H-colouring in overlap graphs

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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H-colouring in overlap graphs

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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H-colouring in overlap graphs

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 25

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 26

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 27

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 28

A brochette

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Main ingredient: the BBQ strategy

  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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A BBQ arrangement

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Main ingredient: the BBQ strategy

  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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Main ingredient: the BBQ strategy

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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Main ingredient: the BBQ strategy

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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The BBQ strategy: main idea

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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The BBQ strategy: main idea

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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BBQ strategy: slice the steak into BBQ arrangements

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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The BBQ strategy:

  • Partition any instance into a minimum number of BBQ arrangements
  • Solve independently each arrangement using a specific colour set
  • Do it on-line

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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An ongoing project with Martin Olsen, Aarhus University, Denmark

Decomposition strategy: (assume first and are known)

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 37

An ongoing project with Martin Olsen, Aarhus University, Denmark

Decomposition strategy

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 38

An ongoing project with Martin Olsen, Aarhus University, Denmark

On-line algorithm: (unknown and )

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 39

An ongoing project with Martin Olsen, Aarhus University, Denmark

On-line reduction

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 40

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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Application to approximation

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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An ongoing project with Martin Olsen, Aarhus University, Denmark

Hardness result

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 43

An ongoing project with Martin Olsen, Aarhus University, Denmark

Hardness result

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 44

An ongoing project with Martin Olsen, Aarhus University, Denmark

Hardness result

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 45

An ongoing project with Martin Olsen, Aarhus University, Denmark

Hardness result

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 46

An ongoing project with Martin Olsen, Aarhus University, Denmark

Hardness result

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018

Bipartition

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SLIDE 47

An ongoing project with Martin Olsen, Aarhus University, Denmark

Hardness result

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018

Induction step

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SLIDE 48

An ongoing project with Martin Olsen, Aarhus University, Denmark

Hardness result

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 49

An ongoing project with Martin Olsen, Aarhus University, Denmark

Hardness result

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 50

An ongoing project with Martin Olsen, Aarhus University, Denmark

Hardness result

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 51

An ongoing project with Martin Olsen, Aarhus University, Denmark

Hardness result

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 52

An ongoing project with Martin Olsen, Aarhus University, Denmark

Hardness result

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 53

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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An ongoing project with Martin Olsen, Aarhus University, Denmark

Polynomial Approximation

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 55

An ongoing project with Martin Olsen, Aarhus University, Denmark

Conclusion

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  • Improves the competitive ratio for online colouring overlap graphs from

linear to a logarithmic factor

  • Competitive results for new colouring problems
  • Narrows the gap between competitive results and hardness result
  • New problem: partionning an overlap graph into permutation graphs
  • Competitive-preserving online reduction
  • Works on the graph and its complement
  • Future work: other generalised colouring problems (split & cocouloring)
  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 56

An ongoing project with Martin Olsen, Aarhus University, Denmark

On going project: excluding some interval configurations

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 57

An ongoing project with Martin Olsen, Aarhus University, Denmark

On going project: excluding some interval configurations

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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SLIDE 58

An ongoing project with Martin Olsen, Aarhus University, Denmark

Selected references

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018
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Special thanks: images.google.com – dreamstime.com

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  • M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018