Algorithm Design

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Chapter 10 Extending the Limits of TractabiliW
Figure 10.1 Aninstance of the Circular-Arc Coloring Problem with six arcs (a, b, c, d, e, f)
on a four-node cycle.
P~ so that overlapping paths receive different wavelengths. We wil! refer to
this as a valid assignment of wavelengths to the paths. Figure 10.1 shows a
sample instance of this problem. In this instance, there is a valid assignment
using k = 3 wavelengths, by assigning wavelength 1 to the pat!gs a and e,
wavelength 2 to the paths b and f, and wavelength 3 to the paths c. and d.
From the figure, we see that the underlying cycle network can be viewed as a
circle, and the paths as arcs on this circle; hence we will refer to this special
case of Path Coloring as the Circular-Arc Coloring Problem.
The Complexity of Circular-Arc Coloring It’s not hard to see that Circular-
Arc Coloring can be directly reduced to Graph Coloring. Given an instance of
Circular-Arc Coloring, we define a graph H that has a node zi for each path
pi, and we connect nodes zi and zj in H if the paths Pi and Pj share an edge
in G. Now, routing all streams using k wavelengths is simply the problem
of coloring H using at most k colors. (In fact, this problem is yet
application of graph coloring in which the abstract ’cmo , since they
different wavelengths of light, are actually colors.)
10.3 Coloring a Set of Circular Arcs
Note that this doesn’t imply that Circular-Arc Coloring is NP-complete-all
we’ve done is to reduce it to a known NP-complete problem, which doesn’t
tell us anything about its difficulty. For Path Co!oring on general graphs, in fact,
it is easy to reduce from Graph Coloring to Path Coloring, thereby establishing
that Path Coloring is NP-complete. However, this straightforward reduction
does not work when the underlying graph is as simple as a cycle. So what is
the complexity of Circular-Arc Coloring?
It turns out that Circular-Arc Coloring can be shown to be NP-complete
using a very complicated reduction. This is bad news for people working
with optical networks, since it means that optimal wavelength assignment
is unlikely to be efficiently solvable. But, in fact, the known reductions that
show Circular-Arc Coloring is NP-complete all have the following interesting
property: The hard instances of Circular-Arc Coloring that they produce all
involve a set of available wavelengths that is quite large. So, in particular,
these reductions don’t show that the Circular-Arc Coloring is hard in the case
when the number of wavelengths is small; they leave open the possibility that
for every fixed, constant number of wavelengths k, it is possible to solve the
wavelength assignment problem in time polynomial in n (the size of the cycle)
and m (the number of paths). In other words, we could hope for a running
time of the form we saw for Vertex Cover in Section 10.1: O(f(k).p(n, m)),
where f(.) may be a rapidly growing function but p(., .) is a polynomial.
Such a running time would be appealing (assuming f(.) does not grow too
outrageously), since it would make wavelength assignment potentially feasible
when the number of wavelengths is small. One way to appreciate the challenge
in obtaining such a running time is to note the following analogy: The general
Graph Coloring Problem is already hard for three colors. So if Circular-Arc
Coloring were tractable for each fixed number of wavelengths (i.e., colors) k,
it would show that it’s a special case of Graph Coloring with a qualitatively
different complexity.
The goal of this section is to design an algorithm with this type of running
time, O(f(k). p(n, m)). As suggested at the beginning of the section, the
algorithm itself builds on the intuition we developed in Section 10.2 when
solving Maximum-Weight Independent Set on trees. There the difficult search
inherent in finding a maximum-weight independent set was made tractable
by the fact that for each node u in a tree T, the problems in the components
of T-{u} became completely decoupled once we decided whether or not to
include u in the independent set. This is a specific example of the general
principle of fixing a small set of decisions, and thereby separating the problem
into smaller subproblems that can be dealt with independently.
The analogous idea here will be to choose a particular point on the cycle
and decide how to color the arcs that cross over this point; fixing these degrees
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