“Complexity”

P=NP? Who knows? Who cares?

• Let’s revisit some questions from last time– How many pairwise comparisons do I

need to do to check if a sequence of n-numbers is sorted?

– If I have a procedure for checking whether a sequence is sorted, is it reasonable to sort a sequence of numbers by generating permutations and testing if any of them are sorted?

– What major CS theorem did someone claimed to have proved recently?

Intelligence is putting the “test” part of Generate&Test into generate part…

Exactly when do we say an algorithm is “slow”?

• We kind of felt that O(N! * N) is a bit much complexity

• How about O(N2)? O(N10)? Where do we draw the line?– Meet the Computer Scientist Nightmare– So “Polynomial” ~ “easy” & “exponential” ~ “hard”– 2n eventually overtakes any nk however large k is..

• How do we know if a problem is “really” hard to solve or it is just that I am dumb and didn’t know how to do better?

2n

Classes P and NP

Class P• If a problem can be solved

in time polynomial in the size of the input it is considered an “easy” problem– Note that your failure to solve

a problem in polynomial time doesn’t mean it is not polynomial (you could come up with O(N* N!) algorithm for sorting, after all

Class NP• Technically “if a problem can

be solved in polynomial time by a non-deterministic turing machine, then it is in class NP”

• Informally, if you can check the correctness of a solution in polynomial time, then it is in class NP– Are there problems where even

checking the solution is hard?

Tower of Hanoi (or Brahma)• Shift the disks from the left

peg to the right peg– You can lift one disk at a time – You can use the middle peg to

“park” disks– You can never ever have a

larger disk on top of a smaller disk (or KABOOM)

• How many moves to solve a 2-disk version? A 3-disk one? An n-disk one?– How long does it take (in

terms of input size), to check if you have a correct solution?

How to explain to your boss as to why your program is so slow…

I can't find an efficient algorithm, I guess I'm just too dumb.

I can't find an efficient algorithm, because no such algorithm is possible.

I can't find an efficient algorithm, but neither can all these famous people.

The P=NP question• Clearly, all polynomial problems are

also NP problems• Do we know for sure that there are

NP problems that are not polynomial?

• If we assume this, then we are assuming P != NP

• If P = NP, then some smarter person can still solve a problem that we thought can’t be solved in polytime– Can imply more than a loss of face…

For example, factorization is known to be an NP-Complete problem; and forms the basis for all of cryptography.. If P=NP, then all the cryptography standards can be broken!

NP-Complete: “hardest” problems in class NP

EVERY problem in class NP can be reduced to an NP-Complete problem in polynomial time --So you can solve that problem by using an algorithm that solves the NP-complete problem

Academic Integrity

• What it means• Typical ASU policy– Homeworks– Exams• Take-Home Exams

– Term papers

Scholarship Opportunities

• General Scholarships• The FURI program• NSF REU program

Is exponential complexity the worst?

• After all, we can have 22

More fundamental question:Can every computational

problem be solved in finite time?

“Decidability” --Unfortunately not

guaranteed [and Hilbert turns in his grave]

2n

Some Decidability Challenges

• In First Order Logic, inference (proving theorems) is semi-decidable – If the theorem is true, you can show

that in finite time; if it is false you may never be able to show it

• In First Order Logic + Peano Arithmatic, inference is undecidable– There may be theorems that are true

but you can’t prove them [Godel]

Reducing Problems…

Mathematician reduces “mattress on fire” problem

Make Rao Happy

Make Everyone in ASU Happy Make Little

Tommy Happy

Make his entire family happy

General NP-problem

Boolean Satisfiability

Problem

3-SAT

Practical Implications of Intractability

• A class of problems is said to be NP-hard as long as the class contains at least one instance that will take exponential time..

• What if 99% of the instances are actually easy to solved?--Where then are the wild things?

Satisfiability problem

• Given a set of propositions P1 P2 … Pn • ..and a set of “clauses” that the propositions must satisfy

– Clauses are of the form P1 V P7 VP9 V P12 V ~P35

– Size of a clause is the number of propositions it mentions; size can be anywhere from 1 to n

• Find a T/F assignment to the propositions that respects all clauses

• Is it in class NP? How many “potential” solutions?

Canonical NP-Complete Problem.• 3-SAT is where all clauses are of length 3

Example of a SAT problem

• P,Q,R are propositions• Clauses– P V ~Q V R– Q V ~R V ~P

• Is P=False, Q=True, R=False as solution?• Is Boolean SAT in NP?

Hardness of 3-sat as a function of #clauses/#variables

#clauses/#variables

Probability that there is a satisfying assignment

Cost of solving (either by finding a solution or showing there ain’t one)

p=0.5You would expect this

This is what happens!

~4.3Phase Transition!

Phase Transition in SAT

Theoretically we only know that phase transition ratio occurs between 3.26 and 4.596.

Experimentally, it seems to be close to 4.3(We also have a proof that 3-SAT has sharp threshold)