Algorithm Design

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Chapter 8 NP and Computational Intractability
the total weight of the objects selected. This, for example, is what happened in
the proof of (8.24), showing that Scheduling with Release Times and Deadlines
is NP-complete.
At the same time, one must heed the warning that Subset Sum only
becomes hard with truly large integers; when the magnitudes of the input
numbers are bounded by a polynomial function of n, the problem is solvable
in polynomial time by dynamic programming.
Constraint Satisfaction Problems
Finally, we considered basic constraint satisfaction problems, including Circuit
Safisfiability, SAT, and B-SAT. Among these, the most useful for the purpose of
designing reductions is B-SAT.
o 3-SAT: Given a set of clauses C1 ..... Ck, each of length B, over a set of
variables X = {xl ..... xn}, does there exist a satisfying truth assignment?
Because of its expressive flexibility, B-SAT is often a useful starting point for
reductions where none of the previous five categories seem to fit naturally onto
the problem being considered. In designing B-SAT reductions, it helps to recall
the advice given in the proof of (8.8), that there are two distinct ways to view
an instance of B-SAT: (a) as a search over assignments to the variables, subject
to the constraint that al! clauses must be satisfied, and (b) as a search over
ways to choose a single term (to be satisfied) from each clause, subject to the
constraint that one mustn’t choose conflicting terms from different clauses.
Each of these perspectives on B-SAT is useful, and each forms the key idea
behind a large number of reductions.
Solved Exercises
Solved Exercise 1
You’re consulting for a small high-tech company that maintains a high-security
computer system for some sensitive work that it’s doing. To make sure this
system is not being used for any i~icit purposes, they’ve set up some logging
software that records the IP addresses that all their users are accessing over
time. We’ll assume that each user accesses at most one IP address in any given
minute; the software writes a log file that records, for each user ~ and each
minute m, a value I(~, m) that is equal to the IP address (if any) accessed by
user u during minute m. It sets I(~, m) to the null symbol 3_ if u did not access
any IP address during minute m.
The company management iust learned that yesterday the system was
used to lannch a complex attack on some remote sites. The attack was carried
out by accessing t distinct IP addresses over t consecutive minutes: In minute
Solved Exercises
1, the attack accessed address is; in minute 2, it accessed address i2; and so
on, up to address i t in minute t.
Who could have been responsible for carrying out this attack? The company
checks the logs and finds to its surprise that there’s no single user u who
accessed each of the IP addresses involved at the appropriate time; in other
words, there’s no u so that I(u, ra) = i m for each minute ra from I to t.
So the question becomes: What if there were a small coalition of k users
that collectively might have carried out the attack? We will say .a subset S of
users is a suspicious coalition if, for each minute m from 1 to t, tiiere is at least
one user u ~ S for which I(u, m) = im. (In other words, each IP address Was
accessed at the appropriate time by at least one user in the coalition. )
The Suspicious Coalition Problem asks: Given the collection of all values
I(u, ra), and a number k, is there a suspicious coalition of size at most
Solution First of all, Suspicious Coalition is clearly in ~:P: If we were to be
shown a set S of users, we could check that S has size at most k, and that
for each minute ra from 1 to t, at least one of the users in S accessed the IP
address im.
Now we want to find a known NP-complete problem and reduce it to
Suspicious Coalition. Although Suspicious Coalition has lots of features (users,
minutes, IP addresses), it’s very clearly a covering problem (following the
taxonomy described in the chapter): We need to explain all t suspicious
accesses, and we’re allowed a limited number of users (k) with which to do
this. Once we’ve decided it’s a covering problem, it’s natural to try reducing
Vertex Cover or Set Cover to it. And in order to do this, it’s useful to push most
of its complicated features into the background, leaving just the bare-bones
features that will be used to encode Vertex Cover or Set Cover.
Let’s focus on reducing Vertex Cover to it. In Vertex Cover, we need to cover
every edge, and we’re only al!owed k nodes. In Suspicious Coalition, we need
to "cover" all the accesses, and we’re only allowed k users. This parallelism
strongly suggests that, given an instance of Vertex Cover consisting of a graph
G = (V, E) and a bound k, we should construct an instance of Suspicious
Coalition in which the users represent the nodes of G and the suspicious
accesses represent the edges.
So suppose the graph G for the Vertex Cover instance has ra edges
el .....%, and ej = (vj, wj). We construct an instance of Suspici6us Coalition
as follows. For each node of G we construct a user, and for each edge
et = (vt, wt) we construct a minute t. (So there will be m minutes total.) In
minute t, the users associated with the two ends of et access an IP address it,
and all other users access nothing. Finally, the attack consists of accesses to
addresses il, i2 ..... irn in minutes 1, 2 ..... m, respectively.
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