CMPSCI 611: Advanced Algorithms
Micah Adler Problem Set 4 Out: November 16, 2004
Due: November 23, 2004
1. For this problem, you should use the Chernoff bounds Pr[B(n, p)≤(1 −δ)np]≤e−δ2np/2and
Pr[B(n, p)≥(1 + δ)np]≤e−δ2np/3.
(a) Say we throw a fair, six-sided die ntimes, and Xis the number of times that the result is a
1. We are interested in the probability that X≥n/4. Compare the best upper bounds on
this probability you can obtain using Markov’s inequality, Chebyshev’s inequality, and Chernoff
bounds.
(b) A manufacturer produces chips that are faulty independently with probability p. They can test
any chip to see if it is faulty, but would like to determine the value pwithout testing every chip. In
order to do so, they use the following procedure: choose nchips uniformly and independently at
random, and let Xbe the number of faulty chips encountered. Output X/n. We want to determine
how large nshould be so that we can be reasonably confident of the result. In particular, for a
given and δ, 0 < ,δ < 1, we want to find a value Nsuch that if n≥N, then
Pr[|X/n −p| ≤ p]≥1−δ.
Use the Chernoff bounds above to derive the smallest expression you can for a value of Nsuch
that this requirement is satisfied. Your expression for Nshould be in terms of ,δ, and p.
2. Let S={s1,...sk}be a set of finite binary strings. We say that a string xis a concatenation over Sif
it is equal to si1si2. . . sit, where for each ij, 1 ≤ij≤k. Note that the same string from Scan appear
more than once in the concatenation. Consider the Concatenation-Collision problem: given two sets
of finite binary strings Aand B, does there exist any xthat is a concatenation over both Aand B?
Show that the Concatenation-Collision problem is in NP.
(Hint: the witness you use should be a string xthat is a concatenation over both sets. Keep in mind,
however, that a witness must have size polynomial in the size of the input.)
3. An independent set in a graph is a subset Sof the nodes such that no pair of nodes in Shave an edge
between them. Finding the largest independent set in a graph is equivalent to finding the largest clique
in the compliment graph, and thus determining if a graph has an independent set larger than a given
kis NP-Complete.
Astrong independent set is a subset Sof the nodes such that no pair of nodes in Shave either an edge,
or a path of length two between them. In the decision version of the strongly independent set problem,
you are given a graph Gand an integer k, and you want to know if Ghas a strongly independent set
of size at least k.
Show that the strongly independent set problem is NP-Complete.
4. A Monotone CNF formula is a Boolean formula in Conjunctive Normal Form, where all literals are
non-negated. Thus (x1∨x2)∧(x2∨x3∨x4) is a Monotone CNF formula, but (x1∨x2)∧(x2∨x3∨x4)
is not. All Monotone CNF formulas are satisfiable, since we can simply set all the variables to True.
However, satisfying assignments where some of the variables are set to False can also exist.
In the Min-true Monotone CNF problem, we are given a Monotone CNF formula Φ and an integer k,
and we want to know if Φ has a satisfying assignment with at most kvariables set to True. Show that
the Min-true Monotone CNF problem is NP-Complete.
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