On the limitations of efficient computation

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On the limitations of efficient computation

• Oded Goldreich
• Weizmann Institute of Science

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Like Mathematics, typically, TOC does not engage
in solving specific problems but rather studies
the solvability of classes of problems.
Specifically, it studies the efficient
solvability of computational problems.

• Theory of Computation a meta-discipline

Individual problems consists of infinitely many
instances, and the problem specifics a set of
valid solutions for each instance.
Classes of problems Problems set of
instance-solution pairs Instances finite
object (repred by a finite
sequence of bits)
Categories
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Problems of full information.So what is the
problem here? The problem is to transform one
representation of the information to another
(more explicit representation).

• Problems (i.e., computational problems)

answer implicitin (input) data
explicit answer in output
transformation data manipulation
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Search problems binary relations specifying
valid instance-solution pairs.

• Type of questions The P-vs-NP Question

• Finding solutions given an instance, find a
  valid solution.
• Checking validity of solutions given an
  instance-solution pair, determine whether the
  solution is valid.

The P-vs-NP QuestionIs solving harder than
checking?

• P class of search problems for which
  solutions can be found efficiently.
• NP class of search problems for which the
  validity of solutions can be checked efficiently.

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Solving seems harder than checking
Example 1 Sudoku
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Solving seems harder than checking, cont.
Example 2 Factoring integers
6917 ? 8893
61512881
Example 3 Solving a system of linear (or
quadratic) equations
2x 3y ? z 9 x ? y 3z
?4 ?3x 2y z 0
x 1 y 2 z ?1
2x2 3xy ? z 0 x ? y2 3yz
?5 ?3x 2xy 2xz 3
x ?1 y 1 z ?1
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Solving seems harder than checking, cont.
Example 4 Coloring a Map (with 3 colors)
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Not all problems in NP are in PThere exist
search problems for which solutions can be
efficiently checked (for correctness), and yet
can not be found efficiently (when given an
instance).For these problems, solving is harder
than checking.
I mean significantly harder

• The P-vs-NP Conjecture

• Finding solutions given an instance, find a
  valid solution.
• Checking validity of solutions given an
  instance-solution pair, determine whether the
  solution is valid.

• P class of search problems for which
  solutions can be found efficiently.
• NP class of search problems for which the
  validity of solutions can be checked efficiently.

Exhaustive search
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Assuming that not all problems in NP are in P,
can we pinpoint such a problem (i.e., a problem
for which finding solutions is harder than
verifying their correctness)?Yes we can
furthermore, a host of natural problem are such!
E.g., solving systems of quadratic equations
(SQE).

• NP-Completeness (universal problems in NP)

If you can efficiently solve (i.e., find
solutions to)any such (NP-complete) problem,
then you can efficiently solve all problems in
NP.
How is this done? By encoding instances of any
problem in NP (e.g., factoring, 3-color) as
instances of the universal problem (e.g.,
SQE). Indeed amazing we can factor a composite
number (or 3-color a geometric map) by solving a
related system of quadratic equations, where
the system can be efficiently obtained from the
number (or map).
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If NP is not contained in P, in a strong sense,
then we get Cryptography and pseudorandomness.
I.e., hard to solve on typical cases and not only
on worst cases.

• The Bright Side of Hardness

The hardness conjecture we need (one-way
functions) There are efficient processes that
cannot be efficiently reversed/inverted (not even
on typical cases).
The inverting task is always in NP.
Cryptography systems that are easy to use but
hard to abuse.

• P class of search problems for which
  solutions can be found efficiently.
• NP class of search problems for which the
  validity of solutions can be checked efficiently.

pseudorandomness object that look random
although they are not truly random.
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• Pseudorandomness (exists if OWFs exist)

A (huge) world determined using an
unlimited amount of randomness
We cannot distinguish these two worlds (i.e., a
truly random world vs. a pseudorandom one)
An alternative world determined using a very
small amount of randomness
An alternative world determined using a very
small amount of randomness
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• Pseudorandomness (exists if OWFs exist),
  revised

A truly random function from n-digit numbers to
n-digit numbers, where (say) n100.
We cannot distinguish these two cases (i.e., a
truly random function vs. a pseudorandom one).
A function from n-digit numbers to n-digit
numbers, determined by a random n-digit number.
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The End

• The slides of this talk are available at
  http//www.wisdom.weizmann.ac.il/oded/T/eff-comp
  .ppt
• A related textbook is available at
• http//www.wisdom.weizmann.ac.il/oded/bc-book
  .html

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auxiliary slides follow

• taken from related talks

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A common feature Easy to check correctness
A search problem consists of an infinite (or
huge) set of instances and a concise/simple
specification of valid solutions.
NP All search problems for which valid
solutions can be efficiently recognized that is,
there exists an efficient procedure that, when
given an instance-solution pair, determines
whether or not the solution is valid.
The fact that it is easy to check correctness of
candidate solutions yields an obvious (but slow)
way of finding solutions to all problems, called
exhaustive search Just try all potential
solutions and check the correctness of each of
them, halting with a correct solution once it is
found.
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Exhaustive search an obvious (but slow) way of
finding solutions to all NP problems
Trying all potential solutions, we find a correct
one once it is tried.
Example 1 Sudoku (with 50/81 entries missing).
950 possibilities.
Example 2 Factoring 8-digit integers. 10000
possibilities.(When factoring 200-digit
integers, there are 10100 possibilities!)
Example 3 3-coloring a map (with 20 countries).
320 possibilities.
Are there better ways to find solutions?
Example 1a Solving systems of linear equations.
YES!
Example 1b Solving systems of quadratic
equations. (prob.) NO!
Example 2a 2-coloring a (2-colorable) map. YES!
Example 2b 3-coloring a (3-colorable) map.
(prob.) NO!
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The P-vs-NP Question (version 1 search problems)
P All search problems that can be solved
efficiently that is, there exists an efficient
procedure that, when given an instance of the
problem, finds a valid solution to that instance
(or indicates that none exists).
NP All search problems for which valid
solutions can be efficiently recognized that is,
there exists an efficient procedure that, when
given an instance-solution pair, determines
whether or not the solution is valid.
It is widely believed that there are problems in
NP ? P. For example 3-Coloring, TSP, Knapsack,
Solve-Quad-Equations, and also Factoring.
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Decision problems (a formulation)
Again, a problem is not a specific instance
(e.g.,a specific Sudoko puzzle), but rather the
general form/class/type (e.g., Sudoku). A
decision problem consists of an infinite (or
huge) set of instances and a concise/simple
specification of YES-instnaces (a set of
instances having a desired property).
A generic example (having solutions w.r.t a
search problem) An instance is an instance of a
search problem, and the question is whether this
instance has a solution.
Example 1 (Sudoku) An instance is any 9-by-9
rectangle partially filled with single digits,
and the question is whether it can be augmented
such that
Example 2 (coloring maps) An instance is a map,
and the question is whether it is 3-colrable
(i.e., whether there exists a 3-coloring of the
areas such that no two adjacent areas are
assigned the same color).
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Focus on the problem of 3-colorability
Coloring a Map with 3 colors such that no two
adjacent areas are assigned the same color.
THM Every map can be colored with four
colors. Some maps can be colored with three
colors, this one not (e.g., Belarus is surrounded
by five neighbors). The decision problem Given
a map, determine whether it is 3-colorable.
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The P-vs-NP Question (version 2 decision
problems)
P All decision problems that can be solved
efficiently that is, there exists an efficient
procedure that, when given an instance of the
problem, determines whether it is a YES-instance.
NP All decision problems for which each
YES-instance has an efficiently verifiable
certificate that is, there exists an efficient
procedure that, when given an instance-certificate
pair, determines whether or not the certificate
is valid.
It is widely believed that there are problems in
NP ? P. For example 3-Coloring, TSP, Knapsack,
Solve-Quad-Equations.
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Universal problems (NP-completeness)
NP All decision problems for which each
YES-instance has an efficiently verifiable
certificate that is, there exists an efficient
procedure that, when given an instance-certificate
pair, determines whether or not the certificate
is valid.
P All decision problems that can be solved
efficiently that is, there exists an efficient
procedure that, when given an instance of the
problem, determines whether it is a YES-instance.
A problem in NP is called NP-complete if the
ability to efficiently solve it implies the
ability to efficiently solve any problem in
NP. How can such problems possibly exist? (Let
alone how can we prove that they exist?) P?NP ?
NP-complete problems are hard to solve. For
example 3-Coloring, TSP, Knapsack,
Solve-Quad-Equations.
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Universal problems (NP-completeness) continued
A (decision) problem C in NP is called
NP-complete if the ability to efficiently solve
it implies the ability to efficiently solve any
(decision) problem D in NP. How can NP-complete
problems possibly exist? (Let alone how can we
prove that they exist?) Idea An efficient
transformation of instances of (any NP) problem
D into instances of problem C such that
YES-instances are mapped to YES-instances (and
NO-instances are mapped to NO-instances). For
example 3-Coloring, TSP, Knapsack,
Solve-Quad-Equations are all NP-complete.
Instances, say, of FACTORING (in NP) can be
mapped to any of them.
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Universal problems (NP-completeness) 2nd
cont.
A (decision) problem C in NP is called
NP-complete if the ability to efficiently solve
it implies the ability to efficiently solve any
(decision) problem D in NP. Shown via an
efficient transformation of instances of (any
NP) problem D into instances of problem C such
that YES-instances are mapped to YES-instances
(and NO-instances are mapped to NO-instances).
YES
YES
D
C
NO
NO
(e.g.,Factor)
(e.g.,3Color)
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The End

• The slides of this talk are available at
  http//www.wisdom.weizmann.ac.il/oded/T/p-vs-np.
  ppt
• A related textbook is available at
• http//www.wisdom.weizmann.ac.il/oded/bc-book
  .html