1. ICS 353: Design and Analysis of Algorithms
King Fahd University of Petroleum & Minerals
Information & Computer Science Department
ICS 353: Design and Analysis of Algorithms
NP-Complete Problems

2. Reading Assignment
M. Alsuwaiyel, Introduction to Algorithms: Design Techniques and Analysis, World Scientific Publishing Co., Inc
Chapter 10 Sections 1 – 4

3. Problems in Computer Science are classified into
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

4. 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

5. 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?

6. Decision Problem: Coloring
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.

7. Clique
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:

8. Vertex Cover
Definition: A vertex cover of an undirected graph G=(V,E) is a subset C V such that each edge e=(x,y)  E is incident to at least one vertex in C, i.e. x  C or y  C.
Decision Problem
Input:
Question:
Optimization Problem
Output:

9. Independent Set
Definition: Given an undirected graph G=(V,E) a subset S V is called an independent set if for each pair of vertices x, y  S, (x,y)  E
Decision Problem
Input:
Question:
Optimization Problem
Output:

10. 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?

11. 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?

12. 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.

13. Sorting: Given n integers, are they sorted in non-decreasing order?
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

14. 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

15. 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

16. 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?

17. 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

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

19. 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?

20. 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.

21. 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?

22. 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.

23. 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.

24. 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?

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

26. Example NP-Complete Problems
3-CNF-SAT
Input: Boolean formula f in CNF, such that each clause consists of exactly three literals.
Question: Is f satisfiable.
Hamiltonian Cycle
Input: G = (V,E), undirected graph.
Does G have a cycle that visits each vertex exactly once (Hamiltonian Cycle)?
Traveling Salesman
Input: A set of n cities with their intercity distances and an integer k.
Question: Does there exist a tour of length less than or equal to k? A tour is a cycle that visits each vertex exactly once.

27. Example 1
Show that the traveling salesman problem is NP-complete, assuming that the Hamiltonian cycle problem is NP-complete.

28. Example 2 Prove that the Problem Clique is NP-Complete. Proof:
Clique  NP
Clique  NP-Hard
SAT poly Clique
Given an instance of satisfiability f = C1 C2  …  Cm with m clauses and n boolean variables x1, x2, …, xn we construct a graph G = (V, E), where V is the set of all occurrences of the 2n literals x1, x2, …, xn, and E = {(xi, xj) | xi  x’j and xi and xj belong to two different clauses}.
Lemma 10.1 f is satisfiable if and only if G has a clique of size m.

29. Example 3 Prove that the problem Vertex Cover is NP-Complete Proof:
Given an instance of satisfiability f = C1 C2  …  Cm with m clauses and n boolean variables x1, x2, …, xn we construct a graph G = (V, E) as follows:
xi  f, two vertices, xi and x’i are added to V with the edge (xi, x’i) added to E.
 Cj containing nj literals, add the nj vertices to V and add to E all the edges that make of the nj vertices a clique.
w  Cj, connect (w, xw) where xw has been constructed in 1. These are called connection edges.
Let k = n + Sigma_(j=1)^(m) (nj – 1)

30. Example 4 Prove that the problem Independent Set is NP-Complete Proof:
Vertex Cover poly Independent Set
S is an independent set iff V – S is a vertex cover

31. Example NP-Complete Problems (Cont.)
Subset Sum
3-Coloring
3D-Matching
Hamiltonian Path
Partition
Knapsack
Bin Packing
Set Cover
Multiprocessor Scheduling
Longest Path …