1. Chapter 10
NP-Complete Problems

2. Running times vs Input sizes

3. Decision and optimization
Decision problem: its solution has only two outcomes: yes or no
Optimization problem: concern with the minimization or maximization of a certain quantity
How to formulate a problem as a decision problem and an optimization problem?

4. Decision problem: ELEMENT UNIQUENESS
Input: A sequence S of integers
Question: Are there two elements in S that are equal?
Optimization problem: ELEMENT COUNT
Output: An element in S with highest frequency

5. Decision problem: COLORING
Input: An undirected graph G=(V,E) and a positive integer k1
Question: Is G k-colorable?, i.e., can G be colored using at most k colors in such a way that no two adjacent vertices have the same color?
Optimization problem: CHROMATIC NUMBER
Input: An undirected graph G=(V,E)
Output: The chromatic number (G) of G

6. Decision problem: CLIQUE
Input: An undirected graph graph G=(V,E) and a positive integer k
Question: Does G have a clique of size k?
Optimization problem: MAX-CLIQUE
Input: An undirected graph G=(V,E)
Output: A positive integer k, which is the maximum clique size in G

7. Decision and optimization
We usually can cast a given optimization problem as a related decision problem by imposing a bound on the value to be optimized.
If an optimization problem is easy, its related decision problem is easy as well.
If we can provide evidence that a decision problem is hard, we also provide evidence that its related optimization problem is hard.
In the study of NP-complete problems, and computational complexity or even computability in general, it is easier to restrict one’s attention to decision problems

8. The class P
Definition 10.1: Let A be an algorithm to solve a problem . We say that A is deterministic if, when presented with an instance of the problem , it has only one choice in each step throughout its execution. Thus, if A is run again and again on the same input instance, its output never changes.
Definition 10.2: The class of decision problems P consists of those decision problems whose yes/no solution can be obtained using a deterministic algorithm that runs in polynomial number of steps, i.e., in O(n^k) steps, for some nonnegative integer k, where n is the input size.

9. Some decision problems in the class P
SORTING
SET DISJOINTNESS
SHORTEST PATH
2-COLORING
2-SAT

10. Closed under complementation
A class of problems C is closed under complementation if for any problem C the complementation of  is also in C.
Example: 2-COLORING and NOT-2-COLORING
Theorem 10.1: The class P is closed under complementation

11. Nondeterministic algorithm
On input x, a nondeterministic algorithm consists of two phases:
a) The guessing phase: An arbitrary string of character y is generated. It may correspond to a solution to the input instance or not. It may not even be in the proper format of the desired solution. It may differ from one run to another of the nondeterministic algorithm. It is only required that this string be generated in a polynomial number of steps.
b) The verification phase: A deterministic algorithm verifies two things. First, it checks whether the generated solution string y is in the proper format. If it is not, the algorithm halts with the answer no. If y is on the proper format, the algorithm continues to check if it is a solution to the instance x of the problem, If it is indeed a solution to the instance x, it halts and answer yes; otherwise it halts and answer no. It is also required that this phase be completed in a polynomial number of steps.

12. The class NP
Definition 10.3: The class of decision problems NP consists of those decision problems for which there exists a nondeterminstic algorithm that runs in polynomial time
P is the class of decision problems that we can decide or solve using a deterministic algorithm that runs in polynomial time
NP is the class of decision problems that we can check or verify their solution using a deterministic algorithm that runs in polynomial time
Any problem in P is also in NP.

13. Reduce
Definition 10.4: Let  and ’ be two decision problems. We say that  reduces to ’ in polynomial time, symbolized as 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 ’ such that the answer to I is yes iff the answer to I’ is yes. Moreover, this transformation must be achieved in polynomial time.

14. NP-hard and NP-complete
Definition 10.5: A decision problem  is said to be NP-hard if for every problem ’ in NP, ’poly
Definition 10.6: A decision problem  is said to be NP-complete if 1)  is in NP, and 2) for every problem ’ in NP, ’poly
Theorem: If any NP-complete problem is polynomial-time solvable, then P=NP. Equivalently, if any problem in NP is not polynomial-time solvable, then no NP-complete problem is polynomial-time solvable.

15. The satisfiability problem
Given a boolean formula f, we say that it is in conjunctive normal form (CNF) if it is the conjunction of clauses. A clause is the disjunction of literals, where a literal is a boolean variable or its negation.
A formula is said to be satisfiable if there is a truth assignment to its variables that makes it true.
Decision problem: SATISFIABILITY
Input: A CNF boolean formula f
Question: Is f satisfiable?
Theorem 10.2: SATISFIABILITY is NP-complete

16. How to prove NP-completeness
Theorem 10.3: Let , ’ and ” be three decision problems such that poly’ and ’poly”, then poly”
Corollary 10.1: If  and ’ are two problems in NP such that ’poly and ’ is NP-complete, then  is NP-complete

17. How to prove NP-completeness
Theorem 10.3: Let , ’ and ” be three decision problems such that poly’ and ’poly”, then poly”
Corollary 10.1: If  and ’ are two problems in NP such that ’poly and ’ is NP-complete, then  is NP-complete
To prove a problem  is NP-complete, we need to show
1) NP
2) there is an NP-complete problem ’ such that ’poly

18. More NP-complete problems
CLIQUE
VERTEX COVER
INDEPENDENT SET
3-SAT
3-COLORING
3-dimensional matching
HAMILTONIAN PATH
PARTITION
KNAPSACK
BIN PACKING
SET COVER
MULTIPROCESSOR SCHEDULING
LONGEST PATH

19. The class co-NP
The class co-NP consists of those problems whose complements are in NP
Definition 10.7: A problem  is complete for the class co-NP if 1)  is in co-NP, and 2) for every problem ’ in co-NP, ’poly
Theorem 10.5: A problem  is NP-complete iff its complement is complete for the class co-NP
Theorem 10.6: The problem TAUTOLOGY is complete for the class co-NP
TAUTOLOGY is in P iff co-NP=P
TAUTOLOGY is in NP iff co-NP=NP

20. The class NPI
Theorem 10.7: If a problem  and its complement are NP-complete, then co-NP=NP
NPI: NP-Intermediate
Theorem (Ladner): If P≠NP then there is a set A in NP such that A is not in P and A is not NP-complete.

21. The relationships between 4 classes

22. Why we learn these
To become a good algorithm designer, you must understand the rudiments of the theory of NP-completeness. If you can establish a problem as NP-complete, you provide good evidence for its intractability. As an engineer, you would then do better spending your time developing an approximation algorithm or solving a tractable special case, rather than searching for a fast algorithm that solve the problem exactly.
Many natural and interesting problems that on the surface seem easy are in fact NP-complete. It is important to become familiar with this class of problems