Homework 4

Due Date: May 13, 2016

Don't forget to follow the Homework Guidelines.

1. Do Problem 3.15(b)

2. Do Problem 3.16(d).

3. Do Problem 4.21.

4. Do Exercise 5.1

5. Do Exercise 5.2

6. Do Exercise 5.4. Here is a paraphrase of the problem statement.

Conjecture: If A  ≤ _m B and B is a regular language, then A is a regular language.

Either give a proof showing the conjecture is true, or give a counterexample showing it is false, i.e., give a non-regular language A, a regular language B, and a mapping reduction from A to B.

7. Do Problem 5.9

8. Do Problem 7.35. Use these definitions of DOMINATING-SET and VERTEX-COVER.

DOMINATING-SET = { ⟨G, k⟩ : undirected graph G has a dominating set of size  ≤ k }

VERTEX-COVER = { ⟨G, k⟩ : undirected graph G has a vertex cover of size  ≤ k }

9. Choose an NP-complete language you know and show that it is mapping reducible to SET-PACKING in polynomial time. SET-PACKING is the following language:

SET-PACKING = { ⟨C, k⟩ : C is a collection of finite sets and C contains  ≥ k disjoint sets }