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The Online Track Assignment Problem

The Online Track Assignment Problem. Marc Demange, ESSEC Benjamin Leroy-Beaulieu, EPFL Gabriele di Stefano, L’Aquilla. Outline. The motivation The handled problems Online bounded coloring of permutation graphs Online coloring of overlap graphs. F. E. D. C. B. A. 18. 19. 20.

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The Online Track Assignment Problem

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  1. The Online Track Assignment Problem 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. F E D C B A 18 19 20 21 22 23 1 2 3 4 5 6 Motivation ? C D C B A Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring

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

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

  6. 6 5 2 1 4 3 A particular case 1 1 2 2 3 3 4 4 5 5 6 6 18 19 20 21 22 23 1 2 3 4 5 6 P = [5 2 1 4 3 6] P = [5 21436] Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring

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

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

  11. 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. In fact a reduction preserving competitive ratio: if the unbounded case is - competitive, then the bounded case is - competitive Permutation graphsbounded case + left to right b-First Fit: first fit with the bounded condition Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring

  12. 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’). Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring

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

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

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

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

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

  18. First Fit: 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 Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring

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

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

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

  22. 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 Overlap graphsunbounded case + left to right + bounded length Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring

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

  24. Conclusion Left-to-Right General FF Any FF Any ? Permutation graphs (bounded) ? ? Non constant Non constant Overlap Graphs (unbounded) ? ? Demange, di Stefano, Leroy-Beaulieu: Online Bounded Coloring

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