Algorithm Design

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Chapter 10 Extending the Limits of Tractability
~ colonngs of (a, b, }’~
and {a", b", c"} must be /
~.nsistent..)
Figure 10.3 (a) Cutting through the cycle in an instance of Circular-Arc Coloring, and
then u~rolling it so it becomes, in (b), a collection of intervals on a line.
pieces of the same path Pi on G, it’s not clear how to take the differing colors
of P~ and P~’ and infer from this how to color Pi on G. For example; having
~ a get the set of intervals pictured in
sliced open the cycle in Figure 10.a(), we
Figure 10.3 (b). Suppose we compute a coloring so that the intervals in the first
row get the color 1, those in the second row get the color 2, and those in the
~d row get the color 3. Then we don’t have an obvious way to figure out a
color for a and c.
This suggests a way to formalize the relationship between the instance
Circular-Arc Coloring in G and the instance of Interval Colorifig in G’.
10.3 Coloring a Set of Circular Arcs
(10.9) The paths in G can be k-colored if and only if the paths in G’ can be
k-colored subject to the additional restriction that P~ and P[’ receive the same
color, for each i = 1, 2 .....
Proof. If the paths in G can be k-colored, then we simply use these as the colors
in G’, assigning each of P[ and P[’ the color of Pi- In the resulting coloring, no
two paths with the same color have an edge in common.
Conversely, suppose the paths in G’ can be k-colored subject to the
additional restriction that P[ and P[’ receive the same color, for-each i =
1, 2 ..... k. Then we assign path Pi (for i < k) the common color of P[ and
P[’; and we assign path Pj (for j > k) the color that Pj gets in G’. Again, under
this coloring, no two paths with the same color have an edge in common. []
We’ve now transformed our problem into a search for a coloring of the
paths in G’ subject to the condition in (10.9): The paths P[ and P[’ (for 1