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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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