# Lab 9¶

The exercises for this week are available below for download as a PDF file.

Green Exercises

- Exercise 1 aims to familiarise you with the satisfiability problem (SAT).
- Exercises 2 and 3 walk you through mostly completed proofs of NP-completeness of two simple problems.
- Exercises 4--6 introduce you to three popular NP-complete problems: CLIQUE, VERTEX-COVER, and HAMILTONIAN CYCLE.

Orange Exercises

- Exercise 7 asks you to show that the Subset-Sum Problem (SSP) is NP-complete.
- Exercise 8 asks you to show that CLIQUE is NP-complete, using reduction from VERTEX-COVER.
- Exercise 9 asks you to show that a modification of CLIQUE, called HALF-CLIQUE, is NP-complete.
- Exercise 10 asks you to show that the class
**P**is closed under union, concatenation and complementation operations (like the regular languages).

There are no red exercises this week.

Todo

In advance of your lab session, you need to:

- Refresh your knowledge of the key concepts covered in the lecture videos.
- Attempt to solve the green exercises.
- Then attempt the orange exercises.
- Highlight any challenging exercises to ask about them during your lab sessions.
- If you want to attempt the advanced/optional (red/blue) exercises then do so.