COS 423 Theory of Algorithms Spring 2005
Assignment 6
Answer each problem below. This assignment is due Wednesday, April 6 at the beginning of lecture. Col-
laboration is allowed (according to the rules specified in the handout). If you work with a group, be sure to
clearly acknowledge the other members of your study group on the first page.
When proving NP-completeness, it helps a great deal to start the reduction from the “right” base problem.
Your base problem must appear in the course materials - come to office hours if you want a hint.
Read Chapter 8 in Kleinberg-Tardos.
1. Prove that 3-Sat polynomial reduces to LowDiameterClustering. See Problem 8.20 for a descrip-
tion of LowDiameterClustering.
2. Prove that 3-Sat polynomial reduces to DominatingSet. See Problem 8.29 for a description of
DominatingSet.Hint: show that VertexCover polynomial reduces to DominatingSet and use
the fact that 3-Sat polynomial reduces to VertexCover.
3. Since the time you solved Problem 4.7, El Goog has purchased 16 additional supercomputers. Now,
up to 17 jobs can be pre-processed at the same time. However, once a job begins its pre-processing on
a supercomputer, its pre-processing must be completed on that supercomputer without interruption.
Your goal is to sequence the jobs (and assign them to available supercomputers) so as to minimize the
completion time of the last job to finish. Prove that 3-Sat polynomial reduces to ElGoog.
4. Finding the prime factorization of an integer xis a fundamental problem in cryptography and math-
ematics. Show that finding the factorization problem is polynomial-time equivalent to the following
decision version of the problem.
Factor: Given an integer xand an integer y≤x,doesxhave a prime factor less than y?
The integers xand yare represented in binary, so your reduction must be polynomial in the number of
bits used to encode xand y. That is, the reduction should use O(nk) time for some constant k,where
nis the number of bits in the binary representation of x.
5. Which of the following are possible? Answer each question independently. No explanation is necessary.
(a) P6=NP.
(b) P6=NP, but there exists a polynomial-time algorithm for factoring integers.
(c) P=NP, but there does not exist a polynomial-time algorithm for factoring integers.
(d) P=NP, but the fastest polynomial-time algorithm (from an asymptotic viewpoint) to factor an
n-bit integer takes Θ(n1717)time.
(e) Next week Alice proves P=NP, but her proof is so highly non-constructive that finding a
polynomial-time algorithm for 3-Sat remains an open reserach question.
(f) Next week Bob proves that the P=NP question cannot be resolved using the axioms of mathe-
matics (e.g., Zermelo-Fraenkel set theory).
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