CS255 Spring 2019 Practice Final

To study for the final I would suggest you: (1) Know how to do (by heart) all the practice problems. (2) Go over your notes at least three times. Second and third time try to see how much you can remember from the first time. (3) Go over the homework problems. (4) Try to create your own problems similar to the ones I have given and solve them. (5) Skim the relevant sections from the book. (6) If you want to study in groups, at this point you are ready to quiz each other. The practice final is below. Here are some facts about the actual final: (a) It is comprehensive (b) It is closed book, closed notes. Nothing will be permitted on your desk except your pen (pencil) and test. (c) You should bring photo ID. (d) There will be more than one version of the test. Each version will be of comparable difficulty. (e) It is 10 problems (3pts each), 6 problems will be on materials since the midterm, 4 problems will be from the topics of the midterm. (f) Two problems will be exactly (less typos) off of the practice final, and one will be off of the practice midterm.

1. State the following number theory theorems we used when talking about RSA: (a) Lagrange's Theorem, (b) Chinese Remainder Theorem.
2. For each of the following equations, find all solutions or state no solutions exists: (a) `9x equiv 3 mod 60`. (b) `34x = 5 mod 38`. (c) `34x = 5 mod 57`
3. Let `p` be the smallest prime bigger than the day of the month you were born on. Let `q` be the first prime which is more than twice this. Let `e` be the smallest odd number bigger than 1 relatively prime to `phi(p cdot q)`. If these were used in RSA, what would be the public key, what would be the private key? Encode and decode the message 11.
4. Define the following concepts and give an example of each: (a) a formal language, (b) polynomial time reducible, (c) the class NP.
5. Sketch the proof of Cook's Theorem.
6. For each of the following problems, determine with proof or solid argument if it is in P, NPC, or likely NP - (NPC ∪ P): (a) log-VERTEX-COVER = `\{langle G,k rangle | G mbox( has a vertex cover of size ) log k \}`. (b) s-t-CONNECTIVITY = `\{langle G, s,t rangle | \mbox{Nodes } s,t \mbox{ are connected in graph } G\}`. (c) 3-COLORABILITY = `{langle G rangle | \mbox{There is an assignment of the colors red, green, blue}` `\mbox{ to vertices such that vertices connected by an edge always have different colors}}`.
7. Give a 2-aproximation algorithm for Vertex Cover and show why it is a 2-approximation.
8. Prove there is no p-time approximation algorithm for general TSP.
9. Give a 32/31-approximation algorithm for MAX-5SAT and prove it works.
10. Give the `TRIM(L, delta)` algorithm we used as part of our fully p-time approximation algorithm for SUBSET-SUM. Give an example of applying it to a list of integers.