Algorithm Design

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Chapter 8 NP and Computational Intractability
Prove that the problem of Winner Determination in Combinatorial
Auctions is NP-complete.
iw We’ve seen the Interval Scheduling Problem in C.hapteors " 1. a~.d 4~er~ 1
e consider a computationally much harder version oI it mat we l~ can
Multiple Interval Scheduling. As before, you have a processor that is
available to run jobs over some period of time (e.g., 9 A.M. to 5 P.M).
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People submit jobs to run on the processor; the processor can only
work on one job at any single point in time. Jobs in this model, however,
are more comphcated than we’ve seen in the past: each job requires a
set of intervals of time during which it needs to use the processor. Thus,
for example, a single job could require the processor from ~0 A.M. tO
11 A.M., and again from 2 P.M. tO 3 P.M.. If you accept this job, it ties up
your processor during those two hours, but you could sti~ accept jobs
that need any other time periods (including the hours from 11 A.M. tO
2 A.M.).
NOW you’re given a set of n jobs, each specified by a set of time
intervals, and you want to answer the following question: For a given
number k, is it possible to accept at least k of the jobs so that no two of
the accepted jobs have any overlap in time?
Show that Ivlultiple Interval Scheduling is NP-complete.
You’re sitting at your desk one day when a FedF~x package arrives for -
you. Inside is a cell phone that begins to ring, and you’re not entixely
surprised to discover that it’s your friend Neo, whom you haven’t heard
from in quite a while. Conversations with Neo al! seem to go the same
way: He starts out with some big melodramatic justification for why he’s
calling, but in the end it always comes down to him trying to get you to
vohinteer your time to help ~th some problem he needs to solve.
This time, for reasons he can’t go into (something having .to do
with protecting an u~derground city from killer robot probes), he and
a few associates need to monitor radio signals at various points on the
electromagnetic spectrum. Specffica~y, there are n different frequencies
that need monitoring, and to do this they have available a collection of
sensors.
There are two components to the monitoring pr oblem.
¯ A set L of m geographic locations at which sensors canbe placed; and
¯ A set $ of b interference sources, each of which blocks certain frequencies
at certain locations. Specifically, each interference source i
is specified by a pair (F~, L~), where F~ is a subset of the frequencies
and L~ is a subset of the locations; it signifies that (due to radio inter-
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Exercises
ference) a sensor placed at any location in the set L~ will not be able
to receive signals on any frequency in the set F v
We say that a subset L’ __c L of locations is sufficient if, for each of the n
frequencies j, there is some location in L’ where frequencyj is not blocked
by any interference source. Thus, by placing a sensor at each location in
a sufficient set, you can successfully monitor each of the n fre~luencies.
They have k sensors, and hence they want to know whether ~]~ere is
a sufficient set of locations of size at most k. We’ll call this an instance
of the Nearby Electromagnetic Observation Problem: Given frequencies,
locations, interference sources, and a parameter k, is there a sufficient
set of size at most k?
Example. Suppose we have four frequencies {f~, f2, f~, f4} and four locations
{el, Q, e~, Q}. There are three interference sources, with
(F1, L,) ---- ({fl, f2}, {el, e2, ~3})
(F~, L~) = ({£, f~},
Then there is a sufficient set of size 2: We can choose locations Q and Q
(since f~ and fz are not blocked at Q, and f~ and f~ are not blocked at Q).
Prove that Nearby Electromagnetic Observation is NP-complete.
Consider the problem of reasoning about the identity of a set from the
size of its intersections with other sets. You are given a finite set U of size
n, and a collection A1 ..... Am of subsets of U. You are also given numbers
q ..... cm. The question is: Does there exist a set X ~ U so that for each
i = 1, 2 ..... m, the cardinali W of X ¢~ A~ is equal to Q? We will call this an
instance of the Intersection Inference Problem, with input U, {Ai}, and {q}.
Prove that Intersection Inference is NP-complete.
You are given a directed graph G = (V, E) with weights we on its edges e ~ E.
The weights can be negative or positive. The Zero-Weight-Cycle Problem
is to decide ff there is a simple cycle in G so that the sum of the edge
weights on this cycle is exactly 0. Prove that this problem is NP-complete.
You’ve been asked to help some organizational theorists analyze data on
group decision-making, in particular, they’ve been lboking at a dataset
that consists of decisions made by a particular governmental policy
committee, and they’re trying to decide whether it’s possible to identify
a small set of influential members of the committee.
Here’s how the committee works. It has a set M = {rn~ ..... rn~} of n
members, and over the past year it’s voted on t different issues. On each
issue, each member can vote either "Yes," "No," or "Abstain"; the overall
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