First Fit Coloring of Interval Graphs
William T. Trotter
Georgia Institute of Technology
October 14, 2005
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
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
Analyzing First Fit Using Grids
The Academic Algorithm
Academic Algorithm - Rules




A
B
C
D
F
Belongs to an interval
Left neighbor is A
Right neighbor is A
Some terminal set of letters
has more than 25% A’s
All else fails.
A Pierced Interval
A
B
C
C
D
B
A
The Piercing Lemma
Lemma: Every interval J is pierced by a
column of passing grades.
Proof: We use a double induction. Suppose
the interval J is at level j. We show that for
every i = 1, 2, …, j, there is a column of
grades passing at level i which is under
interval J
Double Induction
Initial Segment Lemma
Lemma: In any initial segment of n letters all
of which are passing,
a ≥ (n – b – c)/4
A Column Surviving at the End
1. b ≤ n/4
2. c ≤ n/4
3. n ≥ h+3
4. h ≤ 8a - 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)
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.