1. Complexity
6-1
The Class P
Complexity
Andrei Bulatov

2. The Class of “Easy” Problems
Complexity
6-2
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
Must be closed under substitutions

3. centuries 3855 6.5 years 58 minutes .059 seconds 366 35.7 12.7 days
Complexity
6-3
centuries
3855
6.5
years 58
minutes
.059
seconds
366
35.7
12.7
days
17.9
1.0
.001
13.0
5.2
1.7
24.3
3.2 .1
.216
.125
.064
.027
.008
.0036
.0025
.0016
.0009
.0004
.0001
.00006
.00005
.00004
.00003
.00002
.00001 n 60 50 40 30 20 10
Size n
Time
comp-
lexity

4. Different Models of Computation
Complexity
6-4
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!

5. Different Encodings Lemma 3.
Complexity
6-5
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!

6. The Class P Definition Note:
Complexity
6-6
The Class P
Definition
The class P contains all classes TIME[t(n)], where t(n) is a
polynomial of finite degree
Note:

7. Complexity
6-7
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)

8. Complexity
6-8
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)

9. Proving a Problem is in P
Complexity
6-9
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)

10. A Problem in P Greatest Common Divisor (GCD)
Complexity
6-10
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)

11. Proof: GCD(a,b) can be computed by Euclid’s algorithm
Complexity
6-11
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)

12. The Satisfiability Problem
Complexity
6-12
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

13. Resolution Axioms: a set of clauses Proof rules: resolution
Complexity
6-13
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

14. A 2-Satisfiability instance, , is unsatisfiable if and only if   
Complexity
6-14
Theorem
A 2-Satisfiability instance, , is unsatisfiable if and
only if    R
Example:
(1)
(2)
(3)
(4)
(5)
Proof:
This algorithm works in , because the number of all possible 2-clauses
is related by a quadratic factor to the number of variables

15. A Better Algorithm for 2-Satisfiability
Complexity
6-15
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 1

16. Polynomial-time reducibility
Complexity
6-16
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

17. Key Property of Polynomial-time Reducibility
Complexity
6-17
Key Property of Polynomial-time Reducibility
Lemma
A  B and B  P implies A  P
Proof:
The composition of polynomials is a polynomial

18. Reductions in P Almost all members of P can be reduced to each other
Complexity
6-18
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

19. Colourability Let G  (V,E) be a graph, with nodes V and edges E
Complexity
6-19
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)
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)

20. 2-Colourability  2-Satisfiability
Complexity
6-20
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

21. An Aside Some problems have been shown to be in P by non-constructive
Complexity
6-21
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)