Slide 1
Submitted By:GROUP MEMBERS : GAURI RAJPAL.(41319) SHIVANI
SINGH.(41324)INTRACTABLE PROBLEMSSubmitted To: Mr. Shriprakash
Dwivedi.INDEXIntroduction 1Classification 2Decision Problems
3Complexity Class P 4The Class NP 5Relation Between P and NP
6Polynomial Reductions 7NP Complete Problems 8Some Recent Advances
9-10Conclusion 11References 12
Page No.IntroductionA Turing machine M is said to be of time
complexity T(n) if whenever M is given an input w of length n , M
halts after making atmost T(n) moves, regardless of whether or not
M accepts.In computational time complexity theory, we have two
kinds of problems Problem is tractable if there exists at least one
polynomial bound algorithm that solves it.Problem is intractable if
there does not exist any polynomial bound algorithm that solves
it.
1classificationTractable : An algorithm is polynomial bound if
its worst case growth rate can be bound by a polynomial p(n) in the
size n of the problem
Intractable : All algorithms that solve the problem are not
polynomial bound. It has a worst case growth rate f(n) which cannot
be bound by a polynomial p(n) in the size n of the problem. For
intractable problems the bounds are:
2Complexity Class PTractableClass P: problems that can be solved
in polynomial timeO (nk) for some constant k.Easy problems like
sorting, making a photo mosaic using duplicate tiles, simulating
the universe are all in P.4 .
The class NP NP is the class of decision problems for which
there is a polynomial bounded verification algorithm
It can be shown that: all decision problems in P, and decision
problems such as traveling salesman, knapsack, bin pack, are also
in NP
5The relation between P and NP It is not known whether P =
NP.
Problems in P can be solved quickly
Problems in NP can be verified quickly.
It is easier to verify a solution than to solve a problem.
Some researchers believe that P and NP are not the same
class.6Polynomial ReductionsThe definition of NP-completeness uses
the notion of polynomial reductions of one problem A to another
problem B, written as A BWe need the notion of a polytime
transducer a TM that:Takes an input of length n.Operates
deterministically for some polynomial time p(n).Produces an output
on a separate output tape.
stateninputscratchtapesoutput< p(n)Remember: important
requirementis that time < p(n).7Let L and M be langauges.Say L
is polytime reducible to M if there is a polytime transducer T such
that for every input w to T, the output x = T(w) is in M if and
only if w is in L.
Tin L
notin Lin Mnot in MNP Complete Problems
A problem/language M is said to be NP-complete if for every
language L in NP, there is a polytime reduction from L to
M.Fundamental property: if M has a polytime algorithm, then L also
has a polytime algorithm.I.e., if M is in P, then every L in NP is
also in P, or P = NP.8Some Recent Advances The Class coNP and NP
Complete
coNP = { { 0,1 }* \ L : L does not belong to NP }
It is widely believed that NP is not closed under
complementation i.e. NP is not equal to coNP.This conjecture
implies that P is not equal to NP ( because P is closed under
complementation.
Optimal Search Algorithms For NP Relations
Theorem: For every NP relation R there exists an algorithm A
that satisfies that A correctly solves the search problem of R.
9Promise Problems are natural generalizations of decision
problems. Refer to a 3- way partition of the set of strings into
yes-instances, no- instances and instances that violate the
promise.Many natural problems are actually promise problems. For
instance, consider the following problem: Given adirected acyclic
graph, determine if the graph has apathof length 10.
Theyesinstances are directed acyclic graphs with a path of length
10, whereas thenoinstances are directed acyclic graphs with no path
of length 10. The promise is the set of directed acyclic graphs.
10Conclusion Is P = NP??After decades of research, everyone
believes that P NP, i.e. that no polynomial-time solutions for very
hard problems exist. But no one has proven it. If you do, you will
be very famous, and moderately wealthy.. 11References Basics of
Complexity Theory by Oded Goldreich
David Evans - www.cs.virginia.edu/evans
12Thank You!!
