Question:

Algorithm : Reduction (Polynomial-time reduction, Packings and coverings)

: More reduction (Satisfiabiliy, Hamilton Cycle)

(Other references: CLRS ch 34 (& 34.5.3), KT 8.1, 8.2, 8.5)

Relationships on social networks have a very obvious representation as a graph. Many social networks use a symmetric friend relation. We can say that if there is an edge between accounts u and v that u and v are friends.

There is an interesting principle from social network theory called triadic closure. This is the observation that if two people a and b in a social network have a friend c in common, then there is an increased likelihood that a and b will become friends. What we would like to do is to identify clusters of friends, where a group of users are all friends with each other.

For a given social network graph G, consider the following problems:

(a) Does there exist a cluster of friends that involves a third of the accounts represented in the graph? That is, if G has n accounts, determine whether there is such a cluster of size exactly floor(n/3) n .

(b) Does there exist a cluster of friends that involves at least five of the accounts represented in the graph? That is, if G has n accounts, determine whether there 1 is such a cluster of size at least 5.

One of these problems is known to be solvable in polynomial time and the other is not. Determine which is which.

For the problem that is known to be solvable in polynomial time, give a brief justification.

For the problem that is not known to be solvable in polynomial time, give a reduction to one of the problems listed at the top (Packings and Coverings Satisfiability, Hamiltonian cycle)

Question 46 1 pts Which of the following is not a demand function O P = 30 . Q 0 Q : 30 .P OP- 30-Q 30 - P+Q D Question 47 1 pts Left shoes and right shoes are substitutes Q True False D Question 48 1 pts The law of demand is a fairly weak law, In fact this is one of the weakest laws in economics. True O False D Question 49 1 pts Luxury goods are more clastic then necessities. FaheExercise 7. [10 points] Estimators' Properties You know a random variable X has IE[X] = ,u. The value for its variance V[X] = 02 is nite but unknown and you want to estimate it. a) Consider the following estimator: 1 n 62 := 2X? n i=1 a) Use the weak law of large numbers to construct the probability limit of 62. That is 62 ? Exercise 5.7 The weak law of large numbers states that, if X1, X2, . . . are independent and identically > distributed random variables with mean ,1; and standard deviation 0 0 we have Iim P ( > 5) z 0. IIIx (i) Calculate EiSn] and Var [3"] for S9 = i 2;! X\". iz-Xn i=1 (ii) Use Chebychev's inequality to prove the weak law of large numbers. Example C.2Weak law of large numbers (convergence in pr. of the sample mean). Let {X} i = l, 2.. \"k be a sequence of uncorrelated scalar random variables with common mean u and uniformly bounded, not necessarily identical variances E[(X.- [02] s c 0 i=1 as k > no, where the second equality follows by the uncorrelatedness. So, X} is n1.s. convergent to 11.. By the implications in Figure Cl, is is also pr. convergent, which is the weak law of large numbers (eg, Laha and Rohatgi, 1979, pp. 6769). A special case of this law is that the sample mean of a sequence of independent, identically distributed (i.i.d.) random variables with nite variance converges in pr. to the true mean of the sequence. In fact, the sample mean for this special case will converge in the stronger as. sense (the strong law of large numbers), but the proof techniques for the strong law are more difcult than those for the weak law above- El