COSC 336: P, NP, NPcomplete problems

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COSC 336 P, NP, NP-complete problems
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Polynomial versus Exponential Time

• Most of our algorithms so far have been O(log N),
  O(N), O(N log N) or O(N2) running time for inputs
  of size N
• These are all polynomial time algorithms
• Their running time is O(Nk) for some k gt 0
• Exponential time BN is asymptotically worse than
  any polynomial function Nk for any k
• For any k, Nk is ?(BN) for any constant B gt 1

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How bad is exponential time?
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Feasible/Unfeasible Problems

• Question When we should declare a problem to be
  practically solvable?
• The answer is YES, it it can be solved by an alg.
  running in O(n), O(n log n), O(n2).
• The answer is NO, if it can only be solved in
  exponential time.
• Where do we draw the line?
• Answer at polynomial time. O(nk), for a
  constant k.
• DEF A problem is feasible if it can be solved by
  an algorithm running in time O(nk) for constant
  k.
• Otherwise, it is unfeasible.

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Polynomial-time reduction

• DEF Problem A is polynomial-time reducible to
  problem B, if there exists a polynomial-time
  algorithm that solves A using an oracle for
  problem B.
• Think of oracle as a subroutine that solves B. A
  call to this subroutine counts as one operation.
• Notation A ?p B
• Allows us to compare problems.
• We can say If we could solve B in poly. time,
  we could solve A in poly. Time.
• Or Problem B is not easier than problem A.

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Example of reduction

• Problem A (FACTORING). Given n (where n p
  q , and p and q are prime numbers), find p and
  q.
• Problem B (MOD SQRT) Given n (where n p q,
  and p and q are prime numbers), find x ? 1,
  such that x2 1 (mod n).
• FACTORING ?p MOD SQRT. (shown in class)
• FACTORING is believed to be hard (no poly.-time
  algorithm is known for it)
• This means that if we have access to an algorithm
  that solves MOD SQRT, then we could solve
  FACTORING fast.

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Key Theorem on poly-time reductions

• TH If A ?p B, and B is feasible, then A is
  feasible.
• Also if A is not feasible, then B is not
  feasible.

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Decision Problems

• A decision problem is of the form
• Input something (e.g., a graph, a list of
  numbers, )
• Question Does the input have a certain property?
  Answer YES or NO.
• Example 1
• Input n , a positive integer.
• Question Is n prime?
• Example 2
• Input G, a graph.
• Question Is G connected?
• We restrict our discussion to decision problems
  only.
• Why The theory is more clear in this case, and
  typically most problems reduce do decision
  problems.

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The Complexity Class P

• The set P is defined as the set of all decision
  problems that can be solved in polynomial worst
  case time
• Also known as the polynomial time complexity
  class
• All decision problems that have some algorithm
  whose running time is O(Nk) for some k
• Examples of problems in P Is a graph connected,
  is alist sorted, is there a path of cost k
  between s and t, etc.

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The Complexity Class NP

• Definition NP is the set of all problems for
  which a given candidate solution can be checked
  in polynomial time
• Example of a problem in NP
• Hamiltonian circuit problem Why is it in NP?
• Given a candidate path, can test in linear time
  if it is a Hamiltonian circuit just check if
  all vertices are visited exactly once in the
  candidate path (except start/finish vertex)

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NP- formal definition

• A decision problem B is in NP if there exists a
  polynomial-time algorithm VER with the following
  properties
• (1) for every YES instance x of B, there exists
  w of length polyn. in the length of x, such that
  VER(x,w) YES
• (2) for every NO instance of B, for all w of
  length polyn, in the length of x,
• VER(x,w) NO.
• So, for YES instance, there exists a witness w,
  that the Verifier algorithm can check is a
  solution for x
• And, for NO instances, there is no false witness
  that can fool the Verifier algorithm.

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Why NP?

• NP stands for Nondeterministic Polynomial time
• Why nondeterministic? Corresponds to
  algorithms that can guess a solution (if it
  exists) ? the solution is then verified to be
  correct in polynomial time
• Nondeterministic algorithms are not useful in
  practice purely theoretical idea invented to
  understand how hard a problem could be
• Examples of problems in NP
• Hamiltonian circuit Given a candidate path, can
  test in linear time if it is a Hamiltonian
  circuit
• Sorting Can test in linear time if a candidate
  ordering is sorted
• Are any other problems in P also in NP?

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More Revelations About NP

• Are any other problems in P also in NP?
• YES! All problems in P are also in NP
• Notation P ? NP
• If you can solve a problem in polynomial time,
  can definitely verify a solution in polynomial
  time
• Question Are all problems in NP also in P?
• Is NP ? P?
• I wished I could tell you that NP ? P, but I
  cant
• Instead, I will tell you about NP-Complete
  problems

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Another NP Problem

• SAT Given a formula in Boolean logic, e.g.
• determine if there is an assignment of values to
  the variables that makes the formula true (1).
• 3-SAT Same but for boolean formulas having the
  following special format formula is an AND of
  clauses and each clause is an OR of exactly 3
  literals
• Why is it in NP?

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NP-Complete problems

• DEF A problem B is NP complete if
• (1) B is in NP
• (2) Any problem A in NP is poly-time reducible to
  B
• Thus In some sense B is the hardest problem in
  NP, because if we could solve it, then we could
  solve all problems in NP efficiently.

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3-SAT is NP-Complete

• Thus, if we knew how to solve SAT in poly-time,
  then there is a way to solve ANY NP problem in
  polynomial time !!!
• This is strong evidence that SAT is not feasible.
• After the Cook-Levin theorem has been proved,
  thousands of other problems have been shown to be
  NP-complete.

Theorem (Cook-Levin, 1971) 3-SAT is NP-complete.
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How to show that a given problem is NP-complete

• TH If A is NP-complete, B is in NP, and A ?p B,
  then B is NP-complete.
• Why (sketch of proof) Any problem C can be
  solved in poly time with oracle access to A
  (because A is NP complete). We can replace
  queries to the A-oracle, by a poly-time algorithm
  that has oracle access to B. In this way we can
  solve C in poly-time with oracle access to B.
• CLIQUE problem
• Input a graph G (V,E) and a natural number k.
• Question Does G have a clique of size k?
• TH 3-SAT ?p CLIQUE. (shown in class). So,
  CLIQUE is NP-complete.
• VERTEX COVER problem
• Input a graph G (V,E) and a natural number k.
• Question Are there k vertices so that it every
  edge of G is adjacent to at least one of them?
• TH CLIQUE ?p VERTEX COVER. (shown in class).
  So, VERTEX COVER is NP-complete.

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• SUBSET SUM problem
• Input A set of numbersA k_1, k_2, , k_n
  and additional number t
• Question Is there a subset of A, whose sum is
  equal to t ?
• Th 3-SAT ?p SUBSET SUM. So SUBSET SUM is
  NP-complete.

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A Great Book with a list of NP-complete problems

• Computers and Intractability A Guide to the
  Theory of NP-Completeness, by Michael S. Garey
  and David S. Johnson