1. The Online Track Assignment Problem Marc Demange, ESSEC Benjamin Leroy-Beaulieu, EPFL Gabriele di Stefano, L’Aquilla Marc Demange, ESSEC Benjamin Leroy-Beaulieu, EPFL Gabriele di Stefano, L’Aquilla

2. Outline The motivation The handled problems Online bounded coloring of permutation graphs Online coloring of overlap graphs

3. Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring 3 ARRIVALDEPARTURE A6 p.m6 a.m B7 p.m1 a.m C8 p.m4 a.m D9 p.m5 a.m E10 p.m2 a.m F11 p.m3 a.m ABC C D ? 123456232221201918 B C D E F A Motivation

4. Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring 4 Motivation (II) Time Overlap Graph A C B E D ABCDE

5. Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring 5 A particular case 123456232221201918 5 4 3 2 1 6 Midnight condition 52143 6 Permutation graph 1 2 3 4 5 6

6. Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring 6 P = [5 2 1 4 3 6 ] 123456232221201918 5 4 3 2 1 6 P = [5 2 1 4 3 6 ] 5 4 3 2 1 6 52143 6 A particular case

7. Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring 7 Motivation (III) b Resources are scarce  Bounded Coloring Number of docks is limited  Upper Bound on : k Bounded case

8. Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring 8 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)  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)

9. Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring 9 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.  Vertices are delivered one by one.  Left to right model  general model  At each delivery, decide for a color.  Performance measure: competitive ratio c.

10. Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring 10 Permutation graphs: unbounded case (Online Coloring) – general model  First-Fit (Permutation bipartite):  Upper Bound (Comparability):  First-Fit (Permutation bipartite):  Upper Bound (Comparability): [ Leroy-Beaulieu - MD, 2006]

11. Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring 11 Permutation graphs bounded case + left to right b-First Fit: first fit with the bounded condition In fact a reduction preserving competitive ratio: if the unbounded case is - competitive, then the bounded case is - competitive b-First Fit is an optimal online algorithm guaranteeing a competitive ratio If and b are bounded by a fixed constant, then it gives an asymptotic competitive schema.

12. Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring 12 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’). Permutation graphs bounded case + left to right (II)

13. Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring 13 Permutation graphs bounded case + left to right (III)

14. Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring 14 2D-representation Arrival Departure

15. Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring 15 Permutation graphs bounded case + left to right (IV)

16. Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring 16 2 1 Permutation graphs bounded case + left to right Lower bound of every algorithm

17. Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring 17 [Bouille, Plumettaz, 2006] Permutation graphs bounded case + general model

18. Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring 18 Overlap graphs unbounded case + left to right First Fit: 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

19. Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring 19 All other intervals are included in the grey area k Overlap graphs unbounded case + left to right (II) Schech of proof: It is possible to force k colors on a stable set like this:

20. Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring 20 Overlap graphs unbounded case + left to right (III) How to force 2 colors:

21. Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring 21 Overlap graphs unbounded case + left to right (IV) k colors k+1 colors are forced

22. Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring 22 Overlap graphs unbounded case + left to right + bounded length There is an online algorithm guaranteeing colors, where L (l) is the maximum (minimum) length l(t) new set of colors After using different sets of colors, we can use the first one again

23. Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring 23 Overlap graphs unbounded case + left to right + bounded length This algorithm can be improved in order to guarantee a competitive ratio of:

24. Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring 24 Model Algo Lower Bound Upper Bound Lower Bound Upper Bound Permutation graphs (bounded) Overlap Graphs (unbounded) ? ? Non constant Left-to-RightGeneral FFAnyFFAny Conclusion ? ? ? Non constant