1

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.

2

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?

3

java MaxFlow 6#0|16|13|0|0|0#0|0|10|12|0|0 #0|4|0|0|14|0#0|0|9|0|0|20 #0|0|0|7|0|4|#0|0|0|0|0|0

How to encode max flow instance?

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java MaxFlow 111111#|1111111111111111|1111111111111|||#||1111111111|111111111111||#|1111|||11111111111111|#||111111111|||11111111111111111111#|||1111111||1111 #|||||

5

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

9

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 P=NP? 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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.Clay Mathematics Institute

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P versus NP

The Hodge Conjecture

The Poincaré Conjecture

The Riemann Hypothesis

Yang-Mills Existence and Mass Gap

Navier-Stokes Existence and Smoothness

The Birch and Swinnerton-Dyer Conjecture

Announced 16:00, on Wednesday, May 24, 2000 Collège de France

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• If P=NP, mathematicians may be replaced by (much more reliable) computers:

P=NP ) 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.

P vs. NP and mathematics

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Rigorous Formalization

“Problems” Languages

“Efficient Algorithms” Turing Machines, P

“Search Problems” NP

“Reductions” Polynomial Reductions

“Universal Search Problems” NPC

20

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 P=NP.

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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 P=NP].

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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, we’ll 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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SAT

ILPMILP

MAX INDEPENDENT SET

MIN VERTEX COLORING

HAMILTONIAN CYCLE TSP

TRIPARTITE MATCHING

SET COVER

KNAPSACK

BINPACKING