“Complexity”

1. “Complexity”

2. 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…

3. 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

4. Classes P and NP Class P 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? • 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 

5. 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?

6. 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.

7. 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

8. Academic Integrity • What it means • Typical ASU policy • Homeworks • Exams • Take-Home Exams • Term papers

9. Scholarship Opportunities • General Scholarships • The FURI program • NSF REU program

10. Is exponential complexity the worst? 2n • 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]

11. 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 + PeanoArithmatic, inference is undecidable • There may be theorems that are true but you can’t prove them [Godel]

12. Reducing Problems… Mathematician reduces “mattress on fire” problem Make Rao Happy General NP-problem Make Everyone in ASU Happy Make Little Tommy Happy Boolean Satisfiability Problem 3-SAT Make his entire family happy

13. 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?

14. 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 P7VP9 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

15. 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?

16. This is what happens! You would expect this p=0.5 Hardness of 3-sat as a function of #clauses/#variables Probability that there is a satisfying assignment Cost of solving (either by finding a solution or showing there ain’t one) ~4.3 Phase Transition! #clauses/#variables

17. 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) Phase Transition in SAT