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

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 a . M u d e . t i m r @ e g n a m e d . c r a m , n e s l O n i r t a M , y t s i r e v i n U s u h r a A , H C E T B k r a m n e D k d . u a . h c e t b @ s h o e d n a g l i n t a r B a a , m h k a D , 8 0 1 2 h r c a M n , o t i a u t p m o C d n a s m h i t o r g l A M : O C L A W t a d e n t e s e P r M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018

Online Colouring Problems in Overlap Graphs and their Complements (the art of BBQ down under) 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 a . M u d e . t i m r @ e g n a m e d . c r a m , n e s l O n i r t a M , y t s i r e v i n U s u h r a A , H C E T B k r a m n e D k d . u a . h c e t b @ o n i t r a m M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018

Motivation 1: stacking problem M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 3

Stacking problem M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 4

Stacking problem 5/12 – 15/12 M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 5

Stacking problem 18/09 5/12 – 15/12 18/09 – 26/09 M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 6

Stacking problem 19/09 19/09–30/09 5/12 – 15/12 18/09 – 26/09 M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 7

Stacking problem 26/09 19/09 – 30/09 26/09 5/12 – 15/12 18/09 – 26/09 M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 8

Stacking problem 19/09 5/12 – 15/12 18/09 – 26/09 19/09–30/09 M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 9

Stacking problem 20/09 20/09 – 28/09 5/12 – 15/12 18/09 – 26/09 19/09–30/09 M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 10

Stacking problem 27/09 20/09 – 28/09 5/12 – 15/12 19/09–30/09 6/12 – 21/12 27/09 – 5/10 M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 11

Related graph problem 18 19 20 26 27 28 30 5/10 20/09 – 28/09 27/09 – 5/10 19/09–30/09 18/09 – 26/09 6/12 – 21/12 M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 12

Related graph problem: incompatibility graph Overlap graph 18 19 20 26 27 28 30 5/10 27/09 – 5/10 19/09–30/09 18/09 – 26/09 20/09 – 28/09 M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 13

Overlap graphs as intersection graphs of chords (circle graphs) A D B D C F E C E F A B A B D C F E M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 14

Related graph problem: min colouring Overlap graph 18 19 20 26 27 28 30 5/10 Stack 2 Stack 1 27/09 – 5/10 19/09–30/09 18/09 – 26/09 20/09 – 28/09 M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 15

Motivation 2: track assignment M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 16

A particular case 5 2 1 4 3 6 Midnight condition Permutation graph 1 1 2 2 3 3 4 4 5 5 6 6 18 19 20 21 22 23 1 2 3 4 5 6 M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 17

A particular case 5 2 1 4 3 6 Midnight condition Permutation graph 1 2 3 4 5 6 18 19 20 21 22 23 1 2 3 4 5 6 M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 18

A particular case 5 2 1 4 3 6 Midnight condition Permutation graph 18 19 20 21 22 23 1 2 3 4 5 6 5 2 1 4 3 6 M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 19

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 M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 20

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

H-colouring in overlap graphs M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 22

H-colouring in overlap graphs M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 23

H-colouring in overlap graphs M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 24

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

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

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

Main ingredient: the BBQ strategy A brochette M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 28

Main ingredient: the BBQ strategy A BBQ arrangement M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 29

Main ingredient: the BBQ strategy M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 30

Main ingredient: the BBQ strategy M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 31

The BBQ strategy: main idea M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 32

The BBQ strategy: main idea M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 33

BBQ strategy : slice the steak into BBQ arrangements M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 34

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 M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 35

Decomposition strategy: (assume first and are known) An ongoing project with Martin Olsen, Aarhus University, Denmark M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 36

Decomposition strategy An ongoing project with Martin Olsen, Aarhus University, Denmark M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 37

On-line algorithm: (unknown and ) An ongoing project with Martin Olsen, Aarhus University, Denmark M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 38

On-line reduction An ongoing project with Martin Olsen, Aarhus University, Denmark M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 39

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

Application to approximation M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 41

Hardness result An ongoing project with Martin Olsen, Aarhus University, Denmark M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 42

Hardness result An ongoing project with Martin Olsen, Aarhus University, Denmark M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 43

Hardness result An ongoing project with Martin Olsen, Aarhus University, Denmark M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 44

Hardness result An ongoing project with Martin Olsen, Aarhus University, Denmark M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 45

Hardness result Bipartition An ongoing project with Martin Olsen, Aarhus University, Denmark M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 46

Induction step Hardness result An ongoing project with Martin Olsen, Aarhus University, Denmark M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 47

Hardness result An ongoing project with Martin Olsen, Aarhus University, Denmark M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 48

Hardness result An ongoing project with Martin Olsen, Aarhus University, Denmark M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 49

Hardness result An ongoing project with Martin Olsen, Aarhus University, Denmark M. Demange – Monash Discrete Math Research Group meeting – 18/09/2018 50