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First Fit Coloring of Interval Graphs

First Fit Coloring of Interval Graphs. William T. Trotter Georgia Institute of Technology. Interval Graphs. First Fit with Left End Point Order Provides Optimal Coloring. Interval Graphs are Perfect. Χ = ω = 4. What Happens with Another Order?. On-Line Coloring of Interval Graphs.

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First Fit Coloring of Interval Graphs

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  1. First Fit Coloring of Interval Graphs William T. Trotter Georgia Institute of Technology

  2. Interval Graphs

  3. First Fit with Left End Point Order Provides Optimal Coloring

  4. Interval Graphs are Perfect Χ = ω = 4

  5. What Happens with Another Order?

  6. On-Line Coloring of Interval Graphs Suppose the vertices of an interval graph are presented one at a time by a Graph Constructor. In turn, Graph Colorer must assign a legitimate color to the new vertex. Moves made by either player are irrevocable.

  7. Optimal On-Line Coloring • Theorem (Kierstead and Trotter, 1982) • There is an on-line algorithm that will use at most 3k-2 colors on an interval graph G for which the maximum clique size is at most k. • This result is best possible. • The algorithm does not need to know the value of k in advance. • The algorithm is not First Fit. • First Fit does worse when k is large.

  8. Dynamic Storage Allocation

  9. How Well Does First Fit Do? • For each positive integer k, let FF(k) denote the largest integer t for which First Fit can be forced to use t colors on an interval graph G for which the maximum clique size is at most k. • Woodall (1976) FF(k) = O(k log k).

  10. Upper Bounds on FF(k) Theorem: Kierstead (1988) FF(k) ≤ 40k

  11. Upper Bounds on FF(k) Theorem: Kierstead and Qin (1996) FF(k) ≤ 26.2k

  12. Upper Bounds on FF(k) Theorem: Pemmaraju, Raman and Varadarajan(2003) FF(k) ≤ 10k

  13. Analyzing First Fit Using Grids

  14. The Academic Algorithm

  15. Upper Bounds on FF(k) Theorem: Brightwell, Kierstead and Trotter (2003) FF(k) ≤ 8k

  16. Upper Bounds on FF(k) Theorem: Narayansamy and Babu(2004) FF(k) ≤ 8k - 3

  17. Lower Bounds on FF(k) Theorem: Kierstead and Trotter (1982) There exists ε > 0 so that FF(k) ≥ (3 + ε)k when k is sufficiently large.

  18. Lower Bounds on FF(k) Theorem: Chrobak and Slusarek (1988) There exists ε > 0 so that FF(k) ≥ 4k - 9 when k ≥ 4.

  19. Lower Bounds on FF(k) Theorem: Chrobak and Slusarek (1990) FF(k) ≥ 4.4 k when k is sufficiently large.

  20. Lower Bounds on FF(k) Theorem: Kierstead and Trotter (2004) FF(k) ≥ 4.99 k when k is sufficiently large.

  21. A Likely Theorem Our proof that FF(k) ≥ 4.99 k is computer assisted. However, there is good reason to believe that we can actually write out a proof to show: For every ε > 0, FF(k) ≥ (5 – ε) k when k is sufficiently large.

  22. Tree-Like Walls

  23. A Negative Result and a Conjecture However, we have been able to show that the Tree-Like walls used by all authors to date in proving lower bounds will not give a performance ratio larger than 5. As a result it is natural to conjecture that As k tends to infinity, the ratio FF(k)/k tends to 5.

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