Foundations of Data Science

Exercise 4.43 The threshold property seems to be related to uniform distributions. What
if we considered other distributions? Consider a model where i is selected from the set
{1, 2, . . . , n} with probability c(n) . Is there a threshold for perfect squares? Is there a
i
threshold for arithmetic progressions?
Exercise 4.44 Modify the proof that every increasing property of G(n, p) has a threshold
to apply to the 3-CNF satisfiability problem.
Exercise 4.45 Evaluate ( 1 − 1
2 k ) 2 k
for k=3, 5, and 7. How close is it to 1/e?
Exercise 4.46 Randomly generate clauses for a Boolean formula in 3-CNF. Compute
the number of solutions and the number of connected components of the solution set as a
function of the number of clauses generated. What happens?
Exercise 4.47 Consider a random process for generating a Boolean function f in conjunctive
normal form where each of c clauses is generated by placing each of n variables
in the clause with probability p and complementing the variable with probability 1 / 2 . What
is the distribution of clause sizes for various p such as p = 3 / n , 1 / 2 , other values? Experimentally
determine the threshold value of p for f to cease to be satisfied.
Exercise 4.48 For a random 3-CNF formula with n variables and cn clauses, what is
the expected number of satisfying assignments?
Exercise 4.49 Which of the following variants of the SC algorithm admit a theorem like
Theorem 4.21?
1. Among all clauses of least length, pick the first one in the order in which they appear
in the formula.
2. Set the literal appearing in most clauses independent of length to 1.
Exercise 4.50 Suppose we have a queue of jobs serviced by one server. There is a total
of n jobs in the system. At time t, each remaining job independently decides to join the
queue to be serviced with probability p = d/n, where d < 1 is a constant. Each job has a
processing time of 1 and at each time the server services one job, if the queue is nonempty.
Show that with high probability, no job waits more than Ω(ln n) time to be serviced once
it joins the queue.
Exercise 4.51 Consider G (n, p). Where is the phase transition for 2-colorability? Hint:
For p = d/n with d < 1, G(n, p) is acyclic, so it is bipartite and hence 2-colorable. When
pn → ∞, the expected number of triangles goes to infinity. Show that, almost surely, there
is a triangle. What does this do for 2-colorability?
Exercise 4.52 A vertex cover of size k for a graph is a set of k vertices such that one end
of each edge is in the set. Experimentally play with the following problem. For G(n, 1 2 ),
for what value of k is there a vertex cover of size k?
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