Download presentation

Presentation is loading. Please wait.

Published byBarrett Delaney Modified over 3 years ago

1
Theory of Computing Lecture 16 MAS 714 Hartmut Klauck

2
Problems that are not in P EXP: class of all L that can be decided by Turing machines that run in time at most exp(p(n)) for some polynomial p We will show later that EXP is not equal to P – i.e., there are problems that can be solved in exponential time, but not in polynomial time – These problems tend to be less interesting

3
Problems that are not in P? Definition: – Given an undirected graph, a clique is a set of vertices S µ V, such that all u,v 2 S are connected – The language MaxClique is the set of all G,k, such that G has a clique of size k (or more) Simple Algorithm: enumerate all subsets of size k, test if they form a clique Time O(n k ¢ poly(k)) Since k is not constant this is not a polynomial time algorithm

4
Clique MaxClique is in P if and only if thousands of other interesting problems are We don’t know an efficient algorithm for any of them Widely believed: there is no such algorithm

5
Fun Piece of Information A random graph is a graph generated by putting each possible edge into G with probability ½ The largest clique in G has size d r(n) e or b r(n) c for an explicit r(n) that is around (2+o(1)) log(n) With probability approaching 1 Hence on random graphs the max clique can be found in sub-exponential time

6
The class NP NP: nondeterministic polynomial time NP is the class of problems where a `solution’ or proof can be verified in polynomial time Example: Given a set S of k vertices in G we can check in time O(k 2 ) if S is a clique

7
NP Guess and check definition: A language L is in NP, if there is a language R 2 P such that for all x: [There is a poly(|x|) length string y s.t. x#y 2 R], x 2 L y is called proof or witness

8
Example Max Flow: – Given a flow we can check that the flow is maximal in linear time – But then we can find the max flow in polynomial time anyway Clique: – We can check a clique easily – Don’t know how to find a maximum clique fast

9
Example Non-Primality: – Given a natural number x, is it NOT a prime? – Any a,b such that ab=x prove that x is no prime as long as a,b are not 1 How can we find a,b? – It is not known how to factor numbers efficiently Note: Checking primality/non-primality is even known to be in P (but not easy to see this)

10
Original definition of NP Definition: A nondeterministic Turing machine is defined like a Turing machine, but the transition function can now map to several successors – I.e., ± (q,a) is a subset of Q £ ¡ £ {left,stay,right} Interpretation: on input x there are many computations of the machine – At each step the computation can branch out – The machine accepts if there is at least one accepting computation/branch – The machine `guesses’ a good computation

11
Nondeterministic TM The language L M accepted by a nondeterministic TM is the set of inputs x for which there is an accepting computation of M on x A language L µ ¡ * is accepted by a nondeterministic TM M if L M =L

12
Time The time used by an NTM M on x is the minimum number of steps M performs on x during any accepting computation on x – Time on inputs x not in L is not defined Time complexity of M: as before Nondeterministic time complexity of L upper bounded by g(n): as before

13
NP Definition: NP is the set of languages that are accepted by some nondeterministic Turing machine with polynomial time complexity Note: Both definitions [guess and check/NTM] are equivalent

14
Notes on NP Nondeterministic Turing machines are not a realistic model of computation They are interesting because NP contains many interesting problems Furthermore they formalize proof systems – NP is the set of languages L for which the statement x 2 L can be verified efficiently – I.e., there is a powerful prover P, while the verifier V is a polynomial time Turing machine. The prover provides a proof y – x 2 L iff there is a proof y such that x#y is accepted by V

15
P vs. NP P µ NP by definition Are there problems in NP that are not in P? 10 6 dollar reward from the Clay Math Institute Most researchers believe P NP There is a large class of problems in NP that are believed to be hard – NP-complete problems

16
P, NP, EXP P is a subset of NP by definition NP is a subset of EXP: – Use guess and check definition – Enumerate all witnesses y and check – Time: There are exp(poly(n)) many witnesses. Checking each is time poly(n)

17
NP Completeness We will identify a class of problems that capture the hardness of NP P=NP if and only if one of these problems is in P – None are known to be in P These are the problems L in NP, such that every problem in NP can be solved by a deterministic polynomial time TM given a free subroutine that solves L

18
NP Completeness Informally, L reduces to S, if, given an `oracle’ that decides S for free, we can compute L in polynomial time (deterministically) NP-complete: L is in NP and all S in NP reduce to L (in polynomial time) – This means L is hardest in NP

19
Coping with hard problems Suppose we try to find an algorithm for a problem Maybe it is NP-complete? – Reduce some known NP-complete problem to it – Then we know it’s hopeless If so, we can try – approximation – heuristics Don’t have to waste our time searching for an algorithm that (probably) does not exist

Similar presentations

OK

CSCI 2670 Introduction to Theory of Computing November 29, 2005.

CSCI 2670 Introduction to Theory of Computing November 29, 2005.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To ensure the functioning of the site, we use **cookies**. We share information about your activities on the site with our partners and Google partners: social networks and companies engaged in advertising and web analytics. For more information, see the Privacy Policy and Google Privacy & Terms.
Your consent to our cookies if you continue to use this website.

Ads by Google

Ppt on different types of soil Ppt on ip addresses class and range Ppt on shell scripting language Ppt on power grid failure Ppt on australian continent Download seminar ppt on graphical password authentication Ppt on current trends in hrm Ppt on object-oriented programming concepts polymorphism Ppt on the road not taken robert Paper presentation ppt on nanotechnology