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

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Interval Graphs
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First Fit with Left End Point Order
Provides Optimal Coloring
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Interval Graphs are Perfect
? ? 4
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What Happens with Another Order?
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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.
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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.

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Dynamic Storage Allocation
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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).

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Upper Bounds on FF(k)
Theorem Kierstead (1988)
FF(k) 40k
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Upper Bounds on FF(k)
Theorem Kierstead and Qin (1996)
FF(k) 26.2k
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Upper Bounds on FF(k)
Theorem Pemmaraju, Raman and
Varadarajan(2003)
FF(k) 10k
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Upper Bounds on FF(k)
Theorem Brightwell, Kierstead and Trotter
(2003) FF(k) 8k
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Upper Bounds on FF(k)
Theorem Narayansamy and Babu (2004)
FF(k) 8k - 3
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Analyzing First Fit Using Grids
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The Academic Algorithm
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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 As
• F All else fails.

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A Pierced Interval
A B C C D B A
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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
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Double Induction
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Initial Segment Lemma
Lemma In any initial segment of n letters
all of which are passing, a
(n b c)/4
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A Column Surviving at the End

• b n/4
• c n/4
• n h3
• h 8a - 3

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Lower Bounds on FF(k)
Theorem Kierstead and Trotter (1982) There
exists e gt 0 so that FF(k) (3 e)k
when k is sufficiently large.
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Lower Bounds on FF(k)
Theorem Chrobak and Slusarek (1988)
FF(k) 4k - 9 when k 4.
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Lower Bounds on FF(k)
Theorem Chrobak and Slusarek (1990)
FF(k) 4.4 k when k is sufficiently
large.
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Lower Bounds on FF(k)
Theorem Kierstead and Trotter (2004)
FF(k) 4.99 k when k is sufficiently
large.
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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 e gt 0, FF(k) (5 e) k when
k is sufficiently large.
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Tree-Like Walls
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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.