[Solved] 685-621 Homework2

1. Problem 1
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CLRS 34.3-2: Show that the _P relation is a transitive relation on languages. That is, show that if L_{1 P} L₂ and L_{2 P} L₃, then L_{1 P} L3.

2. Problem 2
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Recall the definition of a complete graph K_n is a graph with n vertices such that every vertex is connected to every other vertex. Recall also that a clique is a complete subset of some graph. The graph coloring problem consists of assigning a color to each of the vertices of a graph such that adjacent vertices have different colors and the total number of colors used is minimized. We define the chromatic number of a graph G to be this minimum number of colors required to color graph G. Prove that the chromatic number of a graph G is no less than the size of the maximal clique of G.

3. Problem 3 Note this is a Collaborative Problem

30 Points Total

Suppose youre helping to organize a summer sports camp, and the following problem comes up. The camp is supposed to have at least one counselor whos skilled at each of the n sports covered by the camp (baseball, volleyball, and so on). They have received job applications from m potential counselors. For each of the n sports, there is some subset of the m applicants qualified in that sport. The question is For a given number k < m, is is possible to hire at most k of the counselors and have at least one counselor qualified in each of the n-sports? Well call this the Efficient Recruiting Problem. Prove that Efficient Recruiting is NP-complete.

4. Problem 4 Note this is a Collaborative Problem

30 Points Total

We start by defining the Independent Set Problem (IS). Given a graph G = (V,E), we say a set of nodes S V is independent if no two nodes in S are joined by an edge. The Independent Set Problem, which we denote IS, is the following. Given G, find an independent set that is as large as possible. Stated as a decision problem, IS answers the question: Does there exist a set S V such that |S| k? Then set k as large as possible. For this problem, you may take as given that IS is NP-complete.

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Table 1: Customer Tracking Table

A store trying to analyze the behavior of its customers will often maintain a table A where the rows of the table correspond to the customers and the columns (or fields) correspond to products the store sells. The entry A[i,j] specifies the quantity of product j that has been purchased by customer i. For example, Table 1 shows one such table.

One thing that a store might want to do with this data is the following. Lets say that a subset S of the customers is diverse if no two of the customers in S have ever bought the same product (i.e., for each product, at most one of the customers in S has ever bought it). A diverse set of customers can be useful, for example, as a target pool for market research.

We can now define the Diverse Subset Problem (DS) as follows: Given an m n array A as defined above and a number k m, is there a subset of at least k customers that is diverse?

Prove that DS is NP-complete.