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CSC 250

Theory of Computation

Smith Computer Science



Lecture Notes 34: UnRecognizable Ls and Poly-Time Reductions in NP-C


Outline

This class we'll discuss:



Grading for the last part of the course

The following are the ways in which we will grade the final part of course:

  1. We will work on Problem Set 8 during next week of classes... IN-Class. The idea is that you use the class time as a sort of "Discuss and Solve" session, and I get to give you instant feedback.
    • I will grade each of you during class when you are done with it
    • If you finish it and have nothing else to show me (see next point), you are free to leave!

  2. For your recovery points, you have:
    1. One resubmit assignment (5 questions ...that you got wrong in A1-A6)
    2. A points-back resubmission of your midterm
    The BEST way for me to grde you on these two is IN-CLASS while we work on assignment 8.
    If you have already made a video, don't worry, you can also show it to me in-class and I'll give you instant feedback and let you tweak the submissions.





Recap: P vs NP vs NP-Hard vs NP-Complete




































Activity [2 minutes] Find, at least 3 more!:
(Wait; then Click)


















More Polynomial-Time Reductions in NP-Complete





















































































Activity [2 minutes]
Complete the following:
What would a Certificate look like for each of the following Problems?
  • Deciding if \( \Phi \) belongs in SAT
  • Deciding if \( \Phi \) belongs in 3SAT
  • Deciding if \( < G, k> \) belongs in CLIQUE
  • Deciding if \( < G, k> \) belongs in Independent-Set














Activity [2 minutes] How would you Prove this?:
(Wait; then Click)




























Upshot:

If you find k vertices connected in the graph, then they MUST be between vertices in different clauses, which means there is a combination that could be simultaneously TRUE in each clause... making \(\Phi\) == True!































TIP:

What is the most obvious vertex-cover of a graph \(G\>)?

What is the most obvious independent-set of a graph \(G\>)?



As you make the I-Set greater... what happens to the vertex-cover?

Is there a maximum I-Set in a graph \(G\>)?

What is the relation with the vertex-cover of a graph \(G\>)?


Activity [2 minutes] How would you Prove this?:
(Wait; then Click)










Upshot:

The largest I-Set must have, as its complement, the smallest vertex-cover!










Before next class (Wednesday 12/08)


[Due for everyone]
TODO
Review Decidable langs, Undecidable Langs, Un-recognizable Langs, Reductions (T, M), P, NP, NP-Hard, and NP-C, and P-Reductions.

[Optional]
TODO