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 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.

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.

Dynamic Storage Allocation

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).

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

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

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

Analyzing First Fit Using Grids

The Academic Algorithm

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

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

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.

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

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

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

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.

Tree-Like Walls

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.