Computability and Complexity

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Computability and Complexity
12-1
The Class P
Computability and Complexity Andrei Bulatov
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Computability and Complexity
12-2
The Class of Easy Problems
Requirements to such a class

• Practical
• Must capture the idea of tractable
  problems
• Theoretical
• Must be robust, that is independent of
  the model of computation

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Computability and Complexity
12-3
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Computability and Complexity
12-4
Different Models of Computation
Lemma 1. We can simulate t steps of a
k-tape TM with an equivalent one-tape TM
in steps.
Lemma 2. We can simulate t steps of a RAM
computation with a 3-tape TM in
steps.
It seems that improving our machines we can
obtain any polynomial speed-up!
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Computability and Complexity
12-5
Different Encodings
Lemma 3. For any number n, the length of
the encoding of n in base and the
length of the encoding of n in base are
related by a constant factor (provided
)
Lemma 4. For any graph G, the length of
the encoding of G as an adjacency matrix,
and the length of the encoding of G as a list
of edges, are both related by a polynomial
factor to the number of vertices
It seems that improving encodings we can shorten
the input by a polynomial factor!
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Computability and Complexity
12-6
The Class P
Definition
Note

• The class P contains all classes
  TIMEt(n), where t(n) is a
• polynomial of finite degree

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Computability and Complexity
12-7
Robustness of P
The class P is the same for all existing
models of computation (except for quantum
computation)
The class P is the same for all existing
methods of encoding
The class P is said to be robust it is a
mathematical object that does not depend on the
exact details of the computational model or
encoding
In fact, P can be define in ways that do not
refer to machines at all (but we will not pursue
that idea here)
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Computability and Complexity
12-8
P ? doable?
The class P is a reasonable mathematical model
of the class of problems which are tractable or
solvable in practice
However, the correspondence is not exact

• When the degree of the polynomial is high then
  the time grows
• so fast that in practice the problem is not
  solvable
• The constants may also be very large

(These problems usually seem to arise only in
artificially created problems for naturally
defined problems the polynomials and
constants are not too large)

• Some algorithms, exponential in principal,
  works satisfactory
• in practice (e.g. Simplex method for
  Linear Programming)

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Computability and Complexity
12-9
Proving a Problem is in P

• The most direct way to show that a problem is
  in P is to give a
• a polynomial time algorithm that solves it
• Even a naive polynomial time algorithm often
  provides a good
• insight into how the problem can be solved
  efficiently
• To find such an algorithm we generally need
  to identify an approach
• to the problem that is considerably better
  than brute-force search
• Because of robustness, we do not generally
  need to specify all the
• details of the machine model or even the
  encoding (e.g. pseudo-code
• is sufficient)

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Computability and Complexity
12-10
A Problem in P
(When g1 this is equivalent to the problem
Relatively Prime)
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Computability and Complexity
12-11
Proof GCD(a,b) can be computed by
Euclids algorithm
Function GCD(a,b) integer Begin if b0
then GCD(a,b) a else GCD
GCD(b,a mod b) end
This algorithm polynomial in the length of the
input numbers (logarithmic in their value)
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Computability and Complexity
12-12
The Satisfiability Problem
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Computability and Complexity
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Resolution
Axioms a set of clauses
Proof rules

• resolution
• C (X ? C?), D ? ??X ? D??
• C, D ?? ?C? ? D??
• (?C? ? D?? is called the resolvent of C
  and D)

R
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Computability and Complexity
12-14
Example
Proof
This algorithm works in , because
the number of all possible 2-clauses is related
by a quadratic factor to the number of variables
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Computability and Complexity
12-15
A Better Algorithm for 2-Satisfiability

• Declare all clauses unsatisfied and literals
  unassigned
• Select an arbitrary unassigned variable X and
  assign X true, ?X false
• Select an unsatisfied clause
• Repeat step 3, until either

• if neither literal is assigned, ignore
• if either literal is assigned true, declare
  clause satisfied
• if both are false, restart the algorithm setting
  X false, and ?X true. If a conflict occurs
  again, declare unsatisfiable
• if one literal false, other unsatisfied, set the
  other to true, its negation to false, and
  declare the clause satisfied

• all clauses are satisfied, return satisfiable
• All clauses remaining unsatisfied have no
  variables assigned. In this case return to step 2

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Computability and Complexity
12-16
Polynomial-time reducibility
Another way to prove that a problem is in P is
to use a reduction
Definition A language A is polynomially
reducible to a language B , denoted A ? B, if
a polynomial-time computable function f exists
such that for all
This fundamental tool of complexity theory was
introduced by Karp in 1972
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Computability and Complexity
12-17
Key Property of Polynomial-time Reducibility
Lemma A ? B and B ? P implies A ? P
Proof The composition of polynomials is a
polynomial
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Computability and Complexity
12-18
Reductions in P
Almost all members of P can be reduced to each
other
Theorem If B is any language in
P, B ? ?, B ? ?, then for any A ?
P, A ? B
Proof Choose w ? B and some w? ? B.
Define the function f by setting
Since A ? P, f is computable in
polynomial-time, and is a reduction from A to B
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Computability and Complexity
12-19
Colourability

• Let G ? (V,E) be a graph, with nodes V
  and edges E
• A function f V ? 1,,n is a (proper)
  colouring if connected
• nodes are assigned different values (?
  colours)

(When k 2 this is equivalent to Bipartite)
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Computability and Complexity
12-20
2-Colourability ? 2-Satisfiability
We can reduce 2-COLOURABILITY to
2-SATISFIABILITY

• For each vertex the graph we create a
  variable
• For each edge we add two clauses

This translation of a 2-COLOURABILITY problem
to a 2-SATISFIABILITY problem is computable in
polynomial time. Now we check it satisfies the
reducibility condition

• If graph is 2-colourable, use 2-colouring to
  assign truth values to
• variables (one colour is true, the other
  false)
• If formula satisfiable, define 2-colouring by
  setting true variables
• to colour 1 and false to colour 2, if
  adjacent vertices get same
• colour one of the associated clauses is not
  satisfied, hence must
• have 2-colouring

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Computability and Complexity
12-21
An Aside
Some problems have been shown to be in P by
non-constructive arguments (i.e. by proving a
polynomial time algorithm must exist).
E.g. the problem Knotlessness can a given
graph be embedded in 3-dimensional space without
knots
In these cases we may not even know how to find
an exponential algorithm!
(See Bovet and Cresceni for more discussion of
this)