The Online Track Assignment Problem

1
The Online Track Assignment Problem

• Marc Demange, ESSEC
• Benjamin Leroy-Beaulieu, EPFL
• Gabriele di Stefano, LAquilla

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Outline

• The motivation
• The handled problems
• Online bounded coloring of permutation graphs
• Online coloring of overlap graphs

3
Motivation
  ARRIVAL DEPARTURE
A 6 p.m 6 a.m
B 7 p.m 1 a.m
C 8 p.m 4 a.m
D 9 p.m 5 a.m
E 10 p.m 2 a.m
F 11 p.m 3 a.m
?
C
A
B
C
D
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Motivation (II)
Time
Overlap Graph
5
A particular case
Permutation graph
1
1
2
2
3
3
4
4
5
5
6
6
1
2
3
4
5
6
23
22
21
20
19
18
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A particular case
1
1
2
2
3
3
4
4
5
5
6
6
1
2
3
4
5
6
23
22
21
20
19
18
P 5 2 1 4 3 6
P 5 2 1 4 3 6
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Motivation (III)
Bounded case
b
Resources are scarce ? Bounded Coloring
Number of docks is limited ? Upper Bound on
k
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The related coloring problems

• On permutation graphs (midnight condition)
• Unbounded polynomial
• Bounded NP-hard (Jansen 98)
• For fixed b and k, polynomial in k-colorable
  permutation graphs Leroy-Beaulieu - MD, 2007,
  ongoing work
• On overlap graphs
• Unbounded case NP-hard for (Unger)

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Online Coloring

• Vertices are delivered one by one.
• Left to right model
• general model
• At each delivery, decide for a color.
• Performance measure competitive ratio c.

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Permutation graphs unbounded case (Online
Coloring) general model

• First-Fit (Permutation bipartite)
• Upper Bound (Comparability)

• Leroy-Beaulieu - MD, 2006

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Permutation graphsbounded case left to right
b-First Fit first fit with the bounded condition
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Permutation graphsbounded case left to right
(II)
Lemma Consider G(V,E). Let V be the vertices
colored with unsaturated colors, and G be the
subgraph induced by V. Then
Proof Two vertices of V have the same color in
bFF(G) iff they have the same color in FF(G).
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Permutation graphsbounded case left to right
(III)
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2D-representation
Arrival
Departure
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Permutation graphsbounded case left to right
(IV)
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Permutation graphsbounded case left to
rightLower bound of every algorithm
1
2
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Permutation graphsbounded case general model
Bouille, Plumettaz, 2006
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Overlap graphsunbounded case left to right
For any online algorithm and any K, it is
possible to force K colors on a bipartite
overlap graph revealed from left to right, so it
is not possible to guarantee a constant
competitive ratio
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Overlap graphsunbounded case left to right (II)
Schech of proof
It is possible to force k colors on a stable set
like this
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Overlap graphsunbounded case left to right
(III)
How to force 2 colors
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Overlap graphsunbounded case left to right (IV)
k1 colors are forced
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Overlap graphsunbounded case left to right
bounded length
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Overlap graphsunbounded case left to right
bounded length
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Conclusion
Model
Algo
Lower Bound
Upper Bound
Lower Bound
Upper Bound
Left-to-Right
General
FF
Any
FF
Any
?
Permutation graphs (bounded)
?
?
Non constant
Non constant
Overlap Graphs (unbounded)
?
?