The document discusses three classes of decision problems: 1) P problems that can be solved quickly in polynomial time. 2) NP problems where a "YES" answer has a proof checkable in polynomial time. 3) co-NP problems where a "NO" answer has a proof checkable in polynomial time. It then defines NP-Complete problems as the hardest problems in NP, and explains that 3SAT is a famous NP-Complete problem involving finding a variable assignment that satisfies a Boolean formula of clauses with 3 variables each. The document provides methods for proving other problems like Clique and Independent Set are also NP-Complete by reducing 3SAT to them in polynomial time.

1. Module 3

7. Three classes of decision problems
• P is the set of decision problems that can be
solved in polynomial time.3 Intuitively, P is the
set of problems that can be solved quickly.
• NP is the set of decision problems with the
following property: If the answer is YES, then
there is a proof of this fact that can be checked in
polynomial time.
• co-NP is the opposite of NP. If the answer to a
problem in co-NP is NO, then there is a proof of
this fact that can be checked in polynomial time.

8. Three classes of decision problems

14. NP-Completeness
• How would you define NP-Complete?
• They are the “hardest” problems in NP
P
NP
NP-Complete

15. Definition of NP-Complete
• Q is an NP-Complete problem if:
• 1) Q is in NP
• 2) every other NP problem polynomial time
reducible to Q

21. SAT
• 3SAT:
– n variables (taking values 0 and 1)
– m clauses, each being OR of 3 variables or their
negations
• E.g. (¬x1) x2 (¬x3)
– 3SAT formula: AND of these m clauses
• E.g. ((¬x1) x2 x3) ((¬x2) x4 (¬x5)) (x1 x3 x5)
• 3SAT Problem: Is there an assignment of
variables s.t. the formula evaluates to 1?
– i.e. satisfying all clauses.

22. Hard
• 3SAT is known as an NP-complete problem.
– Very hard: no polynomial algorithm is known.
– Conjecture: no polynomial algorithm exists.
– If a polynomial algorithm exists for 3SAT, then
polynomial algorithms exist for all NP problems.
• Later in this course

26. NP-Complete
• To prove a problem is NP-Complete show a
polynomial time reduction from 3-SAT
• Other NP-Complete Problems:
– PARTITION
– SUBSET-SUM
– CLIQUE
– HAMILTONIAN PATH (TSP)
– GRAPH COLORING
– MINESWEEPER (and many more)

27. NP-Completeness Proof Method
• To show that Q is NP-Complete:
• 1) Show that Q is in NP
• 2) Pick an instance, R, of your favorite NP-
Complete problem (ex: Φ in 3-SAT)
• 3) Show a polynomial algorithm to transform
R into an instance of Q

28. Example: Clique
• CLIQUE = { <G,k> | G is a graph with a clique
of size k }
• A clique is a subset of vertices that are all
connected
• Why is CLIQUE in NP?

29. Reduce 3-SAT to Clique
• Pick an instance of 3-SAT, Φ, with k clauses
• Make a vertex for each literal
• Connect each vertex to the literals in other
clauses that are not the negation
• Any k-clique in this graph corresponds to a
satisfying assignment

31. Example: Independent Set
• INDEPENDENT SET = { <G,k> | where G has an
independent set of size k }
• An independent set is a set of vertices that
have no edges
• How can we reduce this to clique?

32. Independent Set to CLIQUE
• This is the dual problem!