Polynomial Church-Turing thesis

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Polynomial Church-Turing thesis

• A decision problem can be solved in polynomial
  time by using a reasonable sequential model of
  computation if and only if it can be solved in
  polynomial time by a Turing machine.

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The complexity class P

• P the class of decision problems (languages)
  decided by a Turing machine so that for some
  polynomial p and all x, the machine terminates
  after at most p(x) steps on input x.
• By the Polynomial Church-Turing Thesis, P is
  robust with respect to changes of the machine
  model.
• Is P also robust with respect to changes of the
  representation of decision problems as languages?

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How to encode max flow instance?
java MaxFlow 60161300000101200
04001400090020
000704000000
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java MaxFlow 111111 111111111111111111111111111
11 1111111111111111111111 1111111111
11111111 11111111111111111111111111111
11111111111
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Ford-Fulkerson

• Ford-Fulkerson algorithm is not a polynomial time
  algorithm if input is encoded in binary.
• Ford-Fulkerson is a polynomial time algorithm if
  input is ecoded in unary.

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Polynomial time computable maps

• f 0,1 ! 0,1 is called polynomial time
  computable if for some polynomial p,
• - For all x, f(x) p(x).
• - Lf 2 P.

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Polynomial time computable maps

• A map is polynomial time computable if and only
  if there is a Turing machine that on every input
  x accepts after at most a polynomial number of
  steps and leaves f(x) on its tape when
  terminating.

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Good and polynomially equivalent representations

• A representation is good if the language of valid
  representations is in P.
• Two different representations of objects (say
  graphs, numbers) are called polynomially
  equivalent if we may translate between them using
  polynomial time computable maps.
• Ex Adjacency matrices vs. Edge lists
• Ex Binary vs. Decimal
• Counterexample Binary vs. Unary

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Robustness of Representation

• Given two good, polynomially equivalent
  representations of the instances of a decision
  problem, resulting in languages L1 and L2 we have
• L1 2 P iff L2 2 P.

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Terminology

• When we say, Problem X can be solved in
  polynomial time, we mean Lbinary X 2 P, i.e., we
  assume binary representation of integers of
  input.
• If we want to say LunaryX 2 P, i.e., assume unary
  representation of integers, we say Problem X can
  be solved in pseudopolynomial time,

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Rigorous Formalization
Problems Languages
Efficient Algorithms Turing Machines, P
Search Problems NP
Reductions Polynomial Reductions
Universal Search Problems NPC
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Search Problems NP

• L is in NP iff there is a language L in
  P and a polynomial p so that

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Intuition

• The y-strings are the possible solutions to the
  instance x.
• We require that solutions are not too long and
  that it can be checked efficiently if a given y
  is indeed a solution (we have a simple search
  problem)

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P vs. NP

• P is a subset of NP
• Is PNP? Then any simple search problem has a
  polynomial time algorithm.
• This is the most famous open problem of
  mathematical computer science!

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P vs. NP and mathematics

• If PNP, mathematicians may be replaced by (much
  more reliable) computers
• PNP ) There is an algorithmic procedure that
  takes as input any formal math statement and
  always outputs its shortest formal proof in time
  polynomial in the length of the proof.
• This is usually regarded as evidence that P and
  NP are different.

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Rigorous Formalization
Problems Languages
Efficient Algorithms Turing Machines, P
Search Problems NP
Reductions Polynomial Reductions
Universal Search Problems NPC
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Reductions

• A reduction r of L1 to L2 is a polynomial time
  computable map so that
• 8 x x 2 L1 iff r(x) 2 L2
• We write L1 L2 if there is a reduction of
  L1 to L2.
• Intuition Efficient software for L2 can also be
  used to efficiently solve L1.

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Example

• LTSP LILP

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TSP as ILP, compact formulation
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Properties of reductions

• Transitivity
• L1 L2 Æ L2 L3 ) L1 L3
• Follows from Polynomial Church-Turing thesis.

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Properties of reductions

• Downward closure of P
• L1 L2 Æ L2 2 P ) L1 2 P.
• Follows from Polynomial Church-Turing thesis.

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NP-hardness

• A language L is called NP-hard iff
• 8 L 2 NP L L
• Intuition Software for L is strong enough to be
  used to solve any simple search problem.
• Proposition If some NP-hard language is in P,
  then PNP.

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NPC

• A language L 2 NP that is NP-hard is called
  NP-complete.
• NPC the class of NP-complete problems.
• Proposition
• L 2 NPC ) L 2 P iff PNP.

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Usefulness of NPC

• Languages in NPC are the least likely problems in
  NP to be in P.
• Suppose we would like to find a worst case
  efficient algorithm for L 2 NPC.
• If we believe that P is not NP, we know that no
  worst case efficient algorithm exists.
• If we have no opinion about P vs. NP, we know
  that if we find a worst case efficient algorithm
  for L, well earn 1,000,000.

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How to establish NP-hardness

• Thousands of natural problems are NP-complete
• Empiric fact Most natural problems in NP are
  either in P or NP-hard.
• Lemma If L1 is NP-hard and L1 L2 then L2 is
  NP-hard.
• We need to establish one problem to be NP-hard,
  the rest follows using chains of reductions. Cook
  (1972) established SAT to be NP-hard.

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TSP
HAMILTONIAN CYCLE
MIN VERTEX COLORING
SAT
MAX INDEPENDENT SET
SET COVER
ILP
MILP
KNAPSACK
TRIPARTITE MATCHING
BINPACKING