1. NP-Complete Problems Problems in Computer Science are classified into
Tractable: There exists a polynomial time algorithm that solves the problem
O(nk)
Intractable: Unlikely for a polynomial time algorithm solution to exist
NP-Complete Problems

2. Decision Problems vs. Optimization Problems
Decision Problem: Yes/no answer
Optimization Problem: Maximization or minimization of a certain quantity
When studying NP-Completeness, it is easier to deal with decision problems than optimization problems

3. Element Uniqueness Problem
Decision Problem: Element Uniqueness
Input: A sequence of integers S
Question: Are there two elements in S that are equal?
Optimization Problem: Element Count
Output: An element in S of highest frequency
What is the algorithm to solve this problem? How much does it cost?

4. Coloring a Graph Decision Problem: Coloring
Input: G=(V,E) undirected graph and k, k > 0.
Question: Is G k-colorable?
Optimization Problem: Chromatic Number
Input: G=(V,E) undirected graph
Output: The chromatic number of G,(G)
i.e. the minimum number (G) of colors needed to color a graph in such a way that no two adjacent vertices have the same color.

5. Cliques
Definition: A clique of size k in G, for some +ve integer k, is a complete subgraph of G with k vertices.
Decision Problem
Input:
Question:
Optimization Problem
Output:

6. From Decision To Optimization
For a given problem, assume we were able to find a solution to the decision problem in polynomial time. Can we find a solution to the optimization problem in polynomial time also?

7. Deterministic Algorithms
Definition: Let A be an algorithm to solve problem. A is called deterministic if, when presented with an instance of the problem , it has only one choice in each step throughout its execution.
If we run A again and again, is there a possibility that the output may change?
What type of algorithms did we have so far?

8. The Class P
Definition: The class of decision problems P consists of those whose yes/no solution can be obtained using a deterministic algorithm that runs in polynomial time of steps, i.e. O(nk), where k is a non-negative integer and n is the input size.

9. Examples
Sorting: Given n integers, are they sorted in non-decreasing order?
Set Disjointness: Given two sets of integers, are they disjoint?
Shortest path:
2-coloring:
Theorem: A graph G is 2-colorable if and only if G is bipartite

10. Closure Under Complementation
A class C of problems is closed under complementation if for any problem   C the complement of  is also in C.
Theorem: The class P is closed under complementation

11. Non-Deterministic Algorithms
A non-deterministic algorithm A on input x consists of two phases:
Guessing: An arbitrary “string of characters y” is generated in polynomial time. It may
Correspond to a solution
Not correspond to a solution
Not be in proper format of a solution
Differ from one run to another
Verification: A deterministic algorithm verifies
The generated “string of characters y” is in proper format
Whether y is a solution
in polynomial time

12. Non-Deterministic Algorithms (Cont.)
Definition: Let A be a nondeterministic algorithm for a problem . We say that A accepts an instance I of  if and only if on input I, there exists a guess that leads to a yes answer.
Does it mean that if an algorithm A on a given input I leads to an answer of no for a certain guess, that it does not accept it?
What is the running time of a non-deterministic algorithm?

13. The Class NP
Definition: The class of decision problems NP consists of those decision problems for which there exists a nondeterministic algorithm that runs in polynomial time

14. Example
Show that the coloring problem belongs to the class of NP problems

15. P and NP Problems
What is the difference between P problems and NP Problems?
We can decide/solve problems in P using deterministic algorithms that run in polynomial time
We can check or verify the solution of NP problems in polynomial time using a deterministic algorithm
What is the set relationship between the classes P and NP?

16. NP-Complete Problems
Definition: Let  and ’ be two decision problems. We say that ’ reduces to  in polynomial time, denoted by ’poly , if there exists a deterministic algorithm A that behaves as follows: When A is presented with an instance I’ of problem ’, it transforms it into an instance I of problem  in polynomial time such that the answer to I’ is yes if and only if the answer to I is yes.

17. NP-Hard and NP-Complete
Definition: A decision problem  is said to be NP-hard if ’ NP, ’poly .
Definition: A decision problem  is said to be NP-complete if
 NP
’ NP, ’poly .
What is the difference between an NP-complete problem and an NP-hard problem?

18. Conjunctive Normal Forms
Definition: A clause is the disjunction of literals, where a literal is a boolean variable or its negation
E.g., x1  x2  x3  x4
Definition: A boolean formula f is said to be in conjunctive normal form (CNF) if it is the conjunction of clauses.
E.g., (x1  x2)  (x1  x5)  (x2  x3  x4  x6)
Definition: A boolean formula f is said to be satisfiable if there is a truth assignment to its variables that makes it true.

19. The Satisfiability Problem
Input: A CNF boolean formula f.
Question: Is f satisfiable?
Theorem: Satisfiability is NP-Complete
Satisfiability is the first problem to be proven as NP-Complete
The proof includes reducing every problem in NP to Satisfiability in polynomial time.

20. Transitivity of poly
Theorem: Let , ’, and ’’ be three decision problems such that  poly ’ and ’poly ’’. Then poly ’’.
Proof:
Corollary: If , ’ NP such that ’ poly  and ’  NP-complete, then   NP-complete
How can we prove that   NP-hard?
How can we prove that   NP-complete?

21. Proving NP-Completeness
SAT
3-CNF-SAT
Clique
Hamiltonian Cycle
Vertex-Cover
Traveling Salesman
Subset-Sum

22. Example
Show that the traveling salesman problem is NP-complete, assuming that the Hamiltonian cycle problem is NP-complete.