First fit coloring of interval graphs
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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?.

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First Fit Coloring of Interval Graphs

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First fit coloring of interval graphs

First Fit Coloring of Interval Graphs

William T. Trotter

Georgia Institute of Technology

October 14, 2005


Interval graphs

Interval Graphs


First fit with left end point order provides optimal coloring

First Fit with Left End Point Order Provides Optimal Coloring


Interval graphs are perfect

Interval Graphs are Perfect

Χ = ω = 4


What happens with another order

What Happens with Another Order?


On line coloring of interval graphs

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

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

Dynamic Storage Allocation


How well does first fit do

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

Upper Bounds on FF(k)

Theorem: Kierstead (1988)

FF(k) ≤ 40k


Upper bounds on ff k1

Upper Bounds on FF(k)

Theorem: Kierstead and Qin (1996)

FF(k) ≤ 26.2k


Upper bounds on ff k2

Upper Bounds on FF(k)

Theorem: Pemmaraju, Raman and

Varadarajan(2003)

FF(k) ≤ 10k


Upper bounds on ff k3

Upper Bounds on FF(k)

Theorem: Brightwell, Kierstead

and Trotter (2003)

FF(k) ≤ 8k


Upper bounds on ff k4

Upper Bounds on FF(k)

Theorem: Narayansamy

and Babu (2004)

FF(k) ≤ 8k - 3


Analyzing first fit using grids

Analyzing First Fit Using Grids


The academic algorithm

The Academic Algorithm


Academic algorithm rules

Academic Algorithm - Rules

  • A Belongs to an interval

  • B Left neighbor is A

  • C Right neighbor is A

  • D Some terminal set of letters

    has more than 25% A’s

  • F All else fails.


A pierced interval

A Pierced Interval

A

B

C

C

D

B

A


The piercing lemma

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

Double Induction


Initial segment lemma

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

A Column Surviving at the End

  • b ≤ n/4

  • c ≤ n/4

  • n ≥ h+3

  • h ≤ 8a - 3


Lower bounds on ff k

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 k1

Lower Bounds on FF(k)

Theorem: Chrobak and Slusarek (1988)

FF(k) ≥ 4k - 9when k ≥ 4.


Lower bounds on ff k2

Lower Bounds on FF(k)

Theorem: Chrobak and Slusarek (1990)

FF(k) ≥ 4.4 k

when k is sufficiently large.


Lower bounds on ff k3

Lower Bounds on FF(k)

Theorem: Kierstead and Trotter (2004)

FF(k) ≥ 4.99 k

when k is sufficiently large.


A likely theorem

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

Tree-Like Walls


A negative result and a conjecture

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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