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

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

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– Input: A sequence of integers S– Output: An element in S of highest frequency

• What is the algorithm to solve this problem? How much does it cost?

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.

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– Input:– Output:

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?

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?

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.

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

Closure Under Complementation

• A class CC of problems is closed under complementation if for any problem CC the complement of is also in CC.

• Theorem: The class P is closed under complementation

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

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?

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

Example

• Show that the coloring problem belongs to the class of NP problems

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?

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.

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?

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.

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.

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

Proof:– How can we prove that NP-hard?– How can we prove that NP-complete?

Proving NP-Completeness

SAT

3-CNF-SAT

Subset-Sum

Clique Hamiltonian Cycle

Vertex-Cover Traveling Salesman

Example

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