NP-COMPLETENESS

1
NP-COMPLETENESS

• PRESENTED BY
• TUSHAR KUMAR J.
• RITESH BAGGA

2
OVERVIEW

• Introduction.
• Examples of exponential-time algorithms.
• NPC Class of algorithms.
• Approaches to tackle NPC problems.
• Polynomial Reductions.
• Examples of NPC Problems.

3
Introduction

• Most of the algorithms, we came across so far
  takes either logarithmic time (O(logn)), linear
  time (O(n)) or polynomial time (O(nk), k2,3,4).
• But there are some problems, for which we do not
  have polynomial algorithm. Such problems falls
  under the Super polynomial category (taking
  exponential time).
• The following graph shows a clear distinction
  between the various categories under which the
  algorithm falls according to the running time
  they require -

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Introduction (1)
Exponential-time
2n
Polynomial-time
n10
F(n)
n3
n2
n
n
5
Examples of problem falling under the Exponential
time category

• Some of the simple problems, which are very
  similar to problems that requires polynomial time
  falling under the exponential time category are
• Longest simple path problem (similar to Shortest
  path problem, which requires polynomial time)
  suspected to require exponential time, since
  there is no known polynomial algorithm.
• Hamiltonian tour visiting all the vertices in a
  graph exactly once, by passing through edges more
  than once, requires exponential time (similar to
  Euler tour problem visiting all the vertices in
  a graph, by passing through each edge exactly
  once, requires polynomial time).

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Examples of problem falling under the Exponential
time category (1)

• A good example of a problem which cannot be
  solved in polynomial time is Satisfiability
  problem.
• Satisfiability problem (SAT problem)
• Basically a problem from Logic.
• Generally described using Boolean formula.
• A Boolean formula involves AND, OR, NOT operators
  and some variables.
• Ex (x or y) and (x or z), where x, y,
  z are boolean variables.
• Problem Definition Given a boolean formula
  containing n boolean variables, can you assign
  some values to these variables so that it can be
  true?????

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Examples of problem falling under the Exponential
time category (2)

• We generally deal with SAT problems having 2 or 3
  boolean variables in a clause. They are called as
  2SAT and 3SAT problems respectively.
• Ex 2SAT problem (x or y) and (y
  or z)
• 3SAT problem (x or y or
  z) and (x or y or z)
• where x, y, z, x (complement of x) and
  z (complement of z)
• (.) denotes a clause. In
  the above example for 2SAT problem, the clauses
  are (x or y) and (y or z).
  
• 2SAT problems can be solved in polynomial time.
• But 3SAT problems doesnt have any known
  polynomial time algorithm and it is historically
  found as the first problem which requires
  exponential time.

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Various classes of problems

• The problems generally falls into one of the
  three classes Polynomial (P), Non-deterministic
  Polynomial (NP) and NP Completeness (NPC).
• The figure following represents these classes

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Various classes of problems (1)

• P the class of problems that can be solved in
  polynomial time.
• NP the class of problems whose solutions can
  be verified in polynomial time.
• NPC the class of problems which belongs to NP
  and atleast as hard as all other problems in NP.

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Approaches to tackle NPC problems

• There are three approaches to tackle NPC
  problems. They are
• Decision problems
• Easy to solve when compare to other problems.
• Decision problems output the results as boolean
  value (Yes or No)
• Hence we convert all the problems into decision
  problems.

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Approaches to tackle NPC problems (1)

• Reduction Problems
• A means of comparing two problems.
• Suppose A and B are two problems, given that B
  is difficult.
• We suspect A is difficult.
• To confirm, we try to solve B using A as
  subroutine.
• For example
• We know that the Longest Path Problem (LPP)
  is hard and we can solve Hamiltonian path problem
  using LPP as subroutine. So we can conclude HPP
  is harder than LPP.

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Approaches to tackle NPC problems (2)

• Encodings
• Encoding is a process of converting a problem
  into a language is called as encoding.
• Strings are defined as sequence of alphabets (?).
• Languages are set of strings (i.e. we are
  interested in languages with unlimited set of
  strings.
• Some examples 3SAT equivalent to set of all
  formulas such that each formula is in correct
  syntax and it is satisfiable.
• Suppose we have 20 variables, so we can use 0,1
  to represent these 20 variables, to represent
  OR and to represent AND. Thus we can encode
  these formulas using set of the above characters
  all strings over 0,1,,
• Now the problem of solving 3SAT (any decision
  problem can be reduced to language recognition
  problem).

?
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Polynomial Reductions

• Given two problems P1 and P2, if all strings in
  P1 can be mapped to strings in P2 using a mapping
  algorithm in polynomial time. we can say that P2
  is polynomially harder than P1.
• P1 ltp P2
• Formal definition of NP- complete
• A problem P belongs to NP- complete if
  it belongs to the class NP and P1 belongs
  to NP, P1 ltp P2

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Polynomial Reductions (1)
S
S
P2
P1
Polynomial-time Matching-algorithm
lt
P1
P2
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Cooks Theorem

• Stephen Cook founded the first NP- complete
  problem (3SAT) .
• 3SAT problem was proved to be the most difficult
  problem in the class of NP- complete problems.
• To prove a new problem p is NP- complete, we
  need to
• prove p is in NP
• Any known NP- complete problem p

p
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Examples of NPC problems

• 3SAT We will prove that it is a NPC problem.
  Suppose we have m clauses with n variables (3
  in each clause), so we need to try all possible
  sets of solutions. Thus we have 2n sets of
  possible solutions. So it is a NPC problem.
• Clique It is a subgraph of a graph which
  contains all possible edges between each pair of
  vertices in the subgraph. To prove clique is a
  NPC problem, we compare with 3SAT problem. It can
  be reduced to decision problem, where input G,K
  
• output Yes, if clique with atleast
  K nodes exists,
• No, otherwise

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Examples of NPC problems (1)

• For example (x1 OR x2 OR x3) AND (x1 OR x2
  OR X3) AND (x1 OR x2 OR x3). This is a 3SAT
  problem and we will create a graph from it. We
  will put all possible edges, except edges in the
  same clause and between a variable and its
  negation.
• Graph showing
  Clique

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Examples of NPC problems (2)

• Note Red nodes denotes negation.

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Examples of NPC problems (3)

• For 3SAT problem to be satisfiable, one variable
  from each clause should be true. Suppose there
  are m clauses, then it is satisfiable if the
  equivalent graph has a clique of size atleast
  m.
• If it is given that 3SAT problem is satisfiable,
  then we can select one variable from each clause
  to form a clique of size atleast m, as there
  will be atleast m inter-connected nodes (one
  true node from each clause).

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Examples of NPC problems (4)

• Maximum Independent Set Problem It is defined
  as a set of vertices with no edges in between.
  Its equivalent decision problem is, given a graph
  G and some number t. Thus there exists an
  independent set of size gt K?
• A clique problem can be reduced to maximum
  independent set problem. Given a problem to find
  a clique of size K in a graph G is equivalent
  to finding a maximum independent set
• in G (complement of G).

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Examples of NPC problems (5)
1
2
2
3
3
4
6
5
6
4
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Graph G has a clique of size 4 (2,3,4,5)
Graph G has a maximum independent set of size 4
(2,3,4,5)