computability and complexity

Computability and Complexity 12-1

Computability and ComplexityAndrei Bulatov

The Class P


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


Timecomp-lexity

Size n

10 20 30 40 50 60

n.00001

seconds.00002

seconds.00003

seconds.00004

seconds.00005

seconds.00006

seconds

.0001

seconds.0004

seconds.0009

seconds.0016

seconds.0025

seconds.0036

seconds

.001

seconds.008

seconds.027

seconds.064

seconds.125

seconds.216

seconds

.1

seconds3.2

seconds24.3

seconds1.7

minutes5.2

minutes13.0

minutes

.001

seconds1.0

seconds17.9

minutes12.7

days35.7

years366

centuries

.059

seconds58

minutes6.5

years3855

centuries centuries centuries

2n

3n

5n

n2

n38102 13103.1


Different Models of Computation

Lemma 1. We can simulate t steps of a k-tape TM with an equivalent one-tape TM in steps.)( 2tO

Lemma 2. We can simulate t steps of a RAM computation with a 3-tape TM in steps.)( 3tO

It seems that improving our machines we can obtain any polynomial speed-up!


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 )

1b

2b1, 21 bb

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!

Definition Definition

0

]TIME[

k

knP


The Class P

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

Note:


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 objectthat does not depend on the exact details of the computationalmodel 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)


P doable?

The class P is a reasonable mathematical model of the class ofproblems 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 constantsare not too large)

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


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)


A Problem in P

Instance: 3 integers, a, b and g.

Question: Does GCD(a,b)=g?

Greatest Common Divisor (GCD)

(When g=1 this is equivalent to the problem Relatively Prime)


Proof: GCD(a,b) can be computed by Euclid’s algorithm

Function GCD(a,b) : integer

Begin

if b=0 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)


The Satisfiability Problem

Instance: A conjunctive normal form .

Question: Is satisfiable?

Satisfiability

Instance: A conjunctive normal form every clause of which contains exactly k literals.

Question: Is satisfiable?

k-Satisfiability


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

Theorem A 2-Satisfiability instance, , is unsatisfiable if and only if

Theorem A 2-Satisfiability instance, , is unsatisfiable if and only if R


Example:

YXZXZYZXYX , , , ,(1) (2) (3) (4) (5)

,ZX

Proof:

,ZY ,YX ,YX ,ZX

,XXX ,YX ,XXX ,ZX

This algorithm works in , because the number of all possible 2-clausesis related by a quadratic factor to the number of variables

)( 2nO


A Better Algorithm for 2-Satisfiability

1. Declare all clauses unsatisfied and literals unassigned2. Select an arbitrary unassigned variable X and assign X true,

X false3. Select an unsatisfied clause

4. Repeat step 3, until either

a) if neither literal is assigned, ignore;b) if either literal is assigned true, declare clause satisfied;c) if both are false, restart the algorithm setting X false,

and X true. If a conflict occurs again, declare unsatisfiable

d) if one literal false, other unsatisfied, set the other to true, its negation to false, and declare the clause satisfied

a) all clauses are satisfied, return satisfiableb) All clauses remaining unsatisfied have no variables assigned.

In this case return to step 2


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 *x

BA )(xfx

This fundamental tool of complexity theory was introduced by Karp in 1972


Key Property of Polynomial-time Reducibility

Lemma

A B and B P implies A P

Proof:

The composition of polynomials is a polynomial


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

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

A

A

xw

xwxf

if'

if)(

Since A P, f is computable in polynomial-time, and is a reductionfrom A to B


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)

Instance: A graph G.

Question: Is there a colouring of G using at most k colours?

k-Colourability

(When k = 2 this is equivalent to Bipartite)


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

iv iX

),( ji vv

)()( jiji XXXX and

This translation of a 2-COLOURABILITY problem to a 2-SATISFIABILITYproblem 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


An Aside

Some problems have been shown to be in P by non-constructivearguments (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 exponentialalgorithm!

(See Bovet and Cresceni for more discussion of this)

computability and complexity

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