Algorithm Design

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Chapter 8 NP and Computational Intractability
local farmers’ markets, gathering fresh fruits and vegetab!es for the new
health-food restaurant they’ve opened, Chez Alanisse.
In the course of trying to save money on ingredients, they’ve come
across the following thorny problem. There is a large set of n possible raw
ingredients they could buy, I1, I2 ..... In (e.g., bundles of dandelion greens,
jugs of rice vinegar, and so forth), ingredient Ij must be purchased in units
of size s(j) grams (any purchase must be for a whole number of units),
and it costs c(j) dollars per unit. Also, it remains safe to use for t(j) days
from the date of purchase.
Now, over the next k days, they want to make a set of k different daffy
specials, one each day. (The order in which they schedule the specials
is up to them.) The i t~ daffy special uses a subset Si _ {I1, I2 ..... Ia} of
the raw ingredients. Specifically, it reqmres a(i, j) grams of ingredient Ij.
And there’s a final constraint: The restaurant’s rabidly loyal customer
base only remains rabidiy loyal if they’re being served the freshest meals
available; so for each daffy special, the ingredients Si are partitioned into
two subsets: those that must be purchased on the very day when i_he daffy
special is being offered, and those that can be used any day while they’re
still safe. (For example, the mesclun-basfl salad special needs to be macie
with basil that has been purchased that day; but the arugula-basfl pesto
with Cornel! dairy goat cheese special can use basil that is several days
37.
old, as long as it is still safe.)
This is where the opportunity to save money on ingredients comes
up. Often, when they buy a unit of a certain ingredient I~, they don’t need
the whole thing for the special they’re making that day. Thus, if they can
follow up quickly with another special that uses I~ but doesn’t require it to
be fresh that day, then they can save money by not having to purchase I)
again. Of course, scheduling the basil recipes close together may make it
harder to schedule the goat cheese recipes close together, and so forth-this
is where the complexity comes in.
So we define the Daily Special Scheduling Problem as follows: Given
data on ingredients and recipes as above, and a budget x, is there a way to
schedule the k daffy specials so that the total money spent on ingredients
over the course of all k days is at most x?
Prove that Daily Special Scheduling is NP-complete.
There are those who insist that the initial working title for Episode XXVII
of the Star Wars series was "IP = ~f~P"--but this is surely apocryphal. In anY
case, ff you’re so inclined, it’s easy to find NP-complete problems lurking
just below the surface of the original Star Wars movies.
Exercises
Consider the problem faced by Luke, Leia, and friends as they tried to
make their way from the Death Star back to the hidden Rebel base. We can
view the galaxy as an undirected graph G = (V, E), where each node is a
star system and an edge (u, v) indicates that one can travel directly from u
to v. The Death Star is represented by a node s, the hidden Rebel base by a
node t. Certain edges in t~s graph represent !onger distances than others;
thus each edge e has an integer length ee >- O. Also, certain edges represent
routes that are more heavily patrolled by evil Imperial spacecraft; so each
edge e also has an Integer risk r e >_ O, indicating the expected amount
of damage incurred from special-effects-intensive space battles if one
traverses this edge.
It would be safest to travel through the outer rim of the galaxy, from
one quiet upstate star system to another; but then one’s ship would run
out of fuel long before getting to its destination. Alternately, it would be
quickest to plunge through the cosmopolitan core of the galaxy; but then
there would be far too many Imperial spacecraft to deal with. In general,
for any path P from s to t, we can define its total length to be the sum of
the lengths of all its edges; and we can define its total risk to be the sum
of the risks of all its edges.
So Luke, Leia, and company are looking at a complex type of shortestpath
problem in this graph: they need to get from s to t along a path whose
total length and total risk are both reasonably small. In concrete terms, we
can phrase the Galactic Shortest-Path Problem as follows: Given a setup
as above, and integer bounds L and R, is there a path from s to t whose
total length is at most L, and whose total risk is at most R?
Prove that Galactic Shortest Path is NP-complete.
38. Consider the following version of the Steiner Tree Problem, which we’ll
refer to as Graphical Steiner Tree. You are given an undirected graph
G = (V, E), a set X _ V of vertices, and a number k. You want to decide
whether there is a set F ~ E of at most k edges so that in the graph (V, F),
X belongs to a single connected component.
Show that Graphical Steiner Tree is NP-complete.
39. The Directed Disjoint Paths Problem is defined as follows. We are given
a directed graph G and/c pairs of nodes (Sl, tl), (s~, t 2) ..... (sk, tk). The
problem is to decide whether there exist node-disjoint paths P1, P2 ..... Pk
so that Pi goes from & to t~.
Show that Directed Disjoint Paths is NP-complete.
40. Consider the following problem that arises in the design of broadcasting
schemes for networks. We are given a directed graph G = (V, E), with a
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