Algorithm Design

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Chapter 8 NP and Computational Intractability
designated node r ~ V and a designated set of "target nodes" T __ V - {r}.
Each node v has a switching time sv, which is a positive integer.
At time 0, the node r generates a message that it would like ev~ node
in T to receive. To accomplish this, we want to find a scheme whereby r
tells some of its neighbors (in sequence), who in turn tell some of their
neighbors, and so on, until every node in T has received the message. More
formally, a broadcast scheme is defined as follows. Node r may send a
copy of the message to one of its neighbors at time 0; this neighbor will
receive the message at time 1. In genera!, at time t >_ 0, any node u that
has already received the message may send a copy of the message to
one of its neighbors, provided it has not sent a copy of the message in
any of the time steps t - su + 1, t - su + 2 ..... t - 1. (This reflects the role
of the switching time; v needs a pause of su - 1 steps between successive
sendings of the message. Note that if sv = 1, then no restriction is imposed
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by this.)
The completion time of the broadcast scheme is the ~u_m time t
by which all nodes in T have received the message. The Broadcast Time
Problem is the following: Given the input described above, and a bound
b, is there a broadcast scheme with completion time at most b?
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Prove that Broadcast Time is NP-complete.
Example. Suppose we have a directed graph G = (V, E), with V =
{r, a, b, c}; edges (r, a), (a, b), (r, c); the set T = {b, c}; and switching time
sv = 2 for each v ~ V. Then a broadcast scheme with minimum completion
time would be as follows: r sends the message to a at time 0; a sends
the message to b at time 1; r sends the message to c at time 2; and the
scheme completes at time 3 when c receives the message. (Note that a can
send the message as soon as it receives it at time 1, since this is its first
sending of the message; but r cannot send the message at time 1 since
sr = 2 and it sent the message at time 0.)
Given a directed graph G, a cycle cover is a set of node-disjoint cycles
so that each node of G belongs to a cycle. The Cycle Cover Problem asks
whether a given directed graph has a cycle cover.
(a) Show that the Cycle Cover Problem can be solved in pol~ rnomial time.
(b)
(Hint: Use Bipartite Matching.)
Suppose we require each cycle to have at most three edges. Show that
determining whether a graph G has such a cycle cover is NP-complete.
Suppose you’re consulting for a company in northern New Jersey that
designs communication networks, and they come to you with the follow-
Notes and Further Reading
ing problem. They’re studying a specific n-node communication network,
modeled as a directed graph G = (V, E). For reasons of fault tolerance, they
want to divide up G into as many virtual "domains" as possible: A domain
in G is a set X of nodes, of size at least 2, so that for each pair of nodes
u, v ~ X there are directed paths from u to v and v to a that are contained
entirely in X.
Show that the following Domain Decomposition Problem is NP-complete.
Given a directed graph G = (V, E) and a number k, can V be partitioned
into at least k sets, each of which is a domain?
Notes and Further Reading
In the notes to Chapter 2, we described some of the early work on formalizing
computational efficiency using polynomial time; NP-completeness evolved
out of this work and grew into its central role in computer science following
the papers of Cook (1971), Levin (1973), and Karp (1972). Edmonds (1965)
is credited with drawing particular attention to the class of problems in
?,f~P ~ co-~f~P--those with "good characterizations." His paper also contains
the explicit conjecture that the Traveling Salesman Problem cannot be solved
in polynomial time, thereby prefiguring the P ~ NP question. Sipser (1992) is
a useful guide to all of this historical context.
The book by Garey and Johnson (1979) provides extensive material on NPcompleteness
and concludes with a very useful catalog of known NP-complete
problems. While this catalog, necessarily, only covers what was known at the
time of the book’s publication, it is still a very useful reference when one
encounters a new problem that looks like it might be NP-complete. In the
meantime, the space of known NP-complete problems has continued to expand
dramatically; as Christos Papadimitriou said in a lecture, "Roughly 6,000
papers every year contain an NP-completeness result. That means another
NP-complete problem has been discovered since lunch." (His lecture was at
2:00 in the afternoon.)
One can interpret NP-completeness as saying that each individual NPcomplete
problem contains the entire complexity of NP hidden inside it. A
concrete reflection of this is the fact that several of the Ni~-complete problems
we discuss here are the subject of entire books: the Traveling Salesman is the
subject of Lawler et al. (1985); Graph Coloring is the subject of Jensen and Toft
(1995); and the Knapsack Problem is the subject of Marte!lo and Toth (1990).
NP-completeness results for scheduling problems are discussed in the survey
by Lawler et al. (1993).
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