NP Completeness Prof. Dr. Debora Weber-Wulff Winter term 2006/07 - - PowerPoint PPT Presentation

NP Completeness

• Prof. Dr. Debora Weber-Wulff

Winter term 2006/07

Inefficiency and Intractability

• From the book:

Algorithmics, the Spirit of Computing, David Harel. Addison-Wesley, 1992

Complexity Paradox?

• Complicated algorithms often have

a lower complexity than the simple, obvious solutions

• But we have fast computers, so

who cares, anyway?

Unfortunately,....

• There exist problems that cannot

be solved in even quadratic time.

• There are problems for which a

solution just does not exist.

• There are problems that also need

exponential space!

Towers of Hanoi

• What was the complexity?

2 N -1

Towers of Hanoi

• Even moving a million rings a

second, for 64 rings you need half a million years to complete the puzzle.

• If you need 10 sec / ring, you need

five trillion years!

• This is provably the lower bound.

Decision problems

• Don't do anything but answer

"yes" or "no"

• Purpose is to decide if a property

holds for the input

The Monkey Puzzle

• Can 9 of these be laid in a 3x3?

General Problem

• Given N cards, N= M*M, each edge

decorated with an upper or lower half of a figure.

• Can the cards be laid out in an

M*M pattern so that the figures of adjoining cards match?

Brute Force

• For all possible cards and all

possible orientations of the cards, lay them out in M*M

• Return "yes" if this is a valid

layout, otherwise continue

• When all cases have been tested,

return "no".

Complexity?

• Disregarding the orientation:

O(N!)

• With the orientation:

O(4*N!) which is still O (N!)

• O(9) = 362 800,

O(4*9) = 1 451 520

That's not so bad

• But what about N=25?
• 25! is a number with 26 digits, if

you could test a million arrangements a second, you would need over 490 billion years to check it!

Isn't there a clever solution?

• You can stop as soon as you find

that the pattern is illegal

• This does not really change the

complexity.

• There is no known way to solve

this problem in Polynomial time

Reasonable vs. Unreasonable Time

• Factorial grows faster than

exponential, even though N1000 is still greater than N! for many values of N.

• For example at N=1165, the

factorial oversteps the exponential function.

Good vs. Bad functions

• Good functions run in Polynomial

time: Nk + c1*Nk-1 + .... + ck-1

• We say they have an upper bound
• f Nk
• Everything else is

superpolynomial

Superpolynomial

• logarithmic, linear, quadratic,

cubic – all are polynomial

• 1.001N + N6, 5N, NN, N! are

exponential or worse

• N log

2 N are superpolynomial but not

exponential

Growth rates

Tractable

• Problems that are solvable in

polynomial time are called tractable

• Problems that are solvable in

superpolynomial time are called intractable

Intractable Problems

• Need impractically large amounts
• f time even on relatively small

inputs

• And there are lots of them.... close

to 1000 different problems.

Proof?

• Even worse, we have not been able

to prove that no polynomial solution exists!

• We just don't know!
• This class of problems is called

NPC, NP-complete problems, and it is growing

NP-Complete

• Strangely enough, have a linear

lower bound, but an exponential upper bound.

NP-Complete problems

• Two-dimensional Arrangment
• Path-finding (TSP)
• Scheduling and Matching
• Determining Logical Truth
• Coloring Maps and Graphs

Short Certificates

• Even though they are difficult to

find, once we have a solution "yes", we can easily prove that it solves the problem.

• We call this the short certificate or

magic coin

Short Certificates

• are often linear
• For example, it is hard to find a

Hamiltonian path, but easy to test that it is indeed one.

Magic Coin

• We begin with a partial solution

and extend it.

• If we could flip a magic coin to

determine whether the next step leads to a yes or no, we could win

• This is called non-determinism

NPC

• Non-deterministic Polynomial are

those problems which are intractable, but have a non- deterministic polynomial certificate

• C stands for Complete

Completeness

• NPC problems are those which can

be solved if we find a polynomial solution for even one!!

• Either all NPC problems are

tractable, or they are all intractable, but we just don't know!

Polynomial-time Reduction

• It is possible in Polynomial time to

reduce one NPC-Problem to another one.

• TSP can be formulated as

Hamiltonian Path.

P = NP?

• This is one of the big research

questions!

• Open since it was posed in 1971

Testing for Primeness

• Is a number not prime?
• If the answer is yes, we deliver a

short certificate, the two factors, which can be readily multiplied

• So this is NP, but we do not know

if it is NP complete, or even P!

Testing for Primeness

• Is this number prime?
• No short certificate, so not
• bvious, that it is in NP (but this

has since been proven)

• But we do not know if this is NP-

complete or not!!!

Intractable and not NPC

• Towers of Hanoi
• Roadblock
• Satisfiablity – reasoning about

propositional calculus

Can it get worse?

• Of course – double or triple

exponential!

• Satisfiability in Presburger

Arithmetic

Complexity Classes

Imperfect Solutions

• Sometimes we are happy with a

solution that is not optimal – an approximation

• Often heuristic (rule of thumb)

methods yield good results, for example with TSP.

No amount of cleverness can help when your problem is NP!