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Decision Problems

To keep things simple, we will mainly concern ourselves withdecision problems. These problems only require a single bit output:

``yes'' and ``no''.

ow would you solve the !ollowing decision problems"

#s this directed graph acyclic"

#s there a spanning tree o! this undirected graph with total

weight less than w"

Does this bipartite graph have a per!ect $all nodes matched%

matching"Does the pattern p appear as a substring in te&t t"


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P

P is the set o! decision problems that can be solved in worstcase

polynomial time:

#! the input is o! si(e n, the running time must be )$nk%.

*ote that k can depend on the problem class, but not the

particular instance.

+ll the decision problems mentioned above are in P.


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*ice Pu((le

The class *P $meaning nondeterministic polynomial time% is the

set o! problems that might appear in a pu((le maga(ine: ``*icepu((le.''

hat makes these problems special is that they might be hard to

solve, but a short answer can always be printed in the back, and it

is easy to see that the answer iscorrect once you see it.

-&ample... Does matri& + have an / decomposition"

*o guarantee i! answer is ``no''.


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*P

Technically speaking:

+ problem is in *P i! it has a short accepting certi!icate.

+n accepting certi!icate is something that we can use to

quickly show that the answer is ``yes'' $i! it is yes%.

0uickly means in polynomial time.1hort means polynomial si(e.

This means that all problems in P are in *P $since we don't even

need a certi!icate to quickly show the answer is ``yes''%.

2ut other problems in *P may not be in P. 3iven an integer &, is

it composite" ow do we know this is in *P"


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3ood 3uessing

+nother way o! thinking o! *P is it is the set o! problems that can

solved e!!iciently by a really good guesser.

The guesser essentially picks the accepting certi!icate out o! the air

$*ondeterministic Polynomial time%. #t can then convince itsel! that

it is correct using a polynomial

time algorithm. $ike a rightbrain, le!tbrain sort o! thing.%

4learly this isn't a practically use!ul characteri(ation: how could

we build such a machine"


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-&ponential /pperbound

+nother use!ul property o! the class *P is that all *P problems can

be solved in e&ponential time $-5P%.

This is because we can always list out all short certi!icates in

e&ponential time and check all )$6nk% o! them.

Thus, P is in *P, and *P is in -5P. +lthough we know that P is

not equal to -5P, it is possible that *P 7 P, or -5P, or neither.

8rustrating9


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*Phardness

+s we will see, some problems are at least as hard to solve as any

problem in *P. e call such problems *Phard.

ow might we argue that problem 5 is at least as hard $to within

a polynomial !actor% as problem "

#! 5 is at least as hard as , how would we e&pect an algorithm

that is able to solve 5 to behave"


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+;D +*D -+1 P;)2-


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+;D +*D -+1 P;)2->?

3arey and @ohnson: Computers and Intractability, 8reeman =>A>

1chriBver: Theory of Linear and Integer Programming, iley, =>C, $4hapter 6%

P: 4ollection E o! problems is in P $polynomialtime solvable% i!

there e&ists a polynomialtime algorithm that solves any problem in E, i.e, the

algorithm that requires at most

f(s)

basic steps where s7si(e o! the input and f is a polynomial.

NP:4ollection E o! decision problems is in *P $nondeterministic polynomial

time solvable% i! there e&ists a polynomial time algorithm to check thecorrectness o! the -1 answer.

co-NP:replace -1 by *) in the above de!inition.


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co*P


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)ne o! the central $and widely and intensively studied FG years% problems o!

$theoretical% computer science is to prove that

$a%P

NP $b% NP co-NP.

+ll evidence indicates that these conBectures are true.Disproving any o! these two conBectures would not only be considered truly

spectacular, but would also come as a tremendous surprise $with a variety o! !a

reaching counterintuitive consequences%.

NP-complete: 4ollection E o! problems is *Pcomplete i! $a% it is *P and $b% polynomialtime algorithm e&isted !or solving problems in E, then P7*P.

P

*Pco *P


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Theorem.$HarpPapadimitriou, ICG%

#! collection E o! membership problems is *Pcomplete, then either

$=% *P7co*P

or

$6% 4Ds !or E cannot be described $in terms o! giving a !inite number o!

classesJrules !or de!ining hal!spaces !rom 4D%.

Precise de!initions o! what !orm o! description is ruled out $unless *P7co*P% by the theorem

can be !ound in

Harp and Papadimitriou: K)n linear characteri(ation o! combinatorialoptimi(ation problemsL, SIAM ournal on Computing11$=>C6%, 6GF6.

+lso, 4hapter =C o! 1chriBverMs Theory of Linear and Integer Programming$iley, =>C% has

discussion o! the result.

$The result is o!ten stated in terms o! the class o! all possible KrationalL polyhedra. owever,!ocusing on any *Pcomplete class is equivalent to studying the general class.%


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-5+


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-5+


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p MC( ) %

#*P/T: w Rd

Find max{ f(s)= w s | s PMC(n) }.

ma& attained at some e&treme point o!PMC(n), i.e., at some &i

$that represents ;

=

==

OSQO

%,$

%,$

%,$

%,$ %OSQ$O%$

a$bbaC$a

$a

$a

C$a

$a

i$aii

.i

/

a$bbaT/x/.f

#%n$mic Pro!r$mmin!:


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8or every a 4 do 0alue('a-)1 & /(a'a-)

8or i76Rn, do

8or every 2 such that Q2Q7i, do

8ind

Value(C )is the sol)tion to the line$r optimi($tion pro*lem.

#nside loop requires at most Hi6i steps !or some const. H.

There are 6nrepetitions o! the inside loop.

Thus the algorithm needs at most HM$n6n%6 steps.

#n terms o! the input si(e d7n6n= :

The algorithm requires at most d6

steps !or some const. . $Theare very crude estimates, but we don

have to be precise !or our purpose here

"ine$r Optimi($tion onP (n)c$n *e done in pol%nomi$l time

+= ','

%',$P%OS$ma&:%$$b$$

$b

$b

/b$0alue$0alue

The prob2distr2 o3er the set of all n4 ran5ings explains the data


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p f g p

3iven the set o! queries D:

#! the number o! queries d !$n% !or some polynomial !,

!inding 4D !or the corresponding polytope is likely to be hopeless.

#! the number o! queries d U Han, there is hope.

E+ercise:1how that the linear optimi(ation problem corresponding to the

membership problem !or the +pproval Noting Polytope is easy $i.e., can be

solved in polynomial time%.

int: hat is the input si(e"

int: Devise a dynamic programming algorithm !or !inding a

ma&. weight ma&imal chain.

*ote that the rankings played no conceptual role.


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,ener$l &odel

4laim:

6!i3en any data set there is a probability distribution o3er thefiniteset of

alternati3es (states of nature) that explains the data27

The initeset o! queriesJquestions:

Then, !or any0, the probability that 0 holds is

,et pol%top$l represent$tion o the model.

Ho to choose #

s*

s*

ppp

SSS

,,,

,,,

=

=s

i

ii p8p ,

===other/ise8

Sintrue9ifp9P9T

m

mm %Q$:%$

%$%$9

9Tp9Pn

i

ii=

=

,,,O d* 999: =


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Possi*le /t$tes o N$t)re

1=,R,1s

0)eries$d o! them%

Pol%tope$in VG,=Wd%

&odel

Ch$r$cteri($tion

"ine$r"ine$r

Optimi($tionOptimi($tion

NP C l P bl


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NP-Complete Problems


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