What is a hard instance of a computational problem? - Springer Link

H H A T ()F

A

IS

A

H A R D

Ker-I

Ko 1

Dept.

of C o m p u t e r

P R O B L E M ?

Pekka O r p o n e n Dept.

Science

Texas

of C o m p u t e r

SF-00250

77004

Science

Helsinki

U.S.A.

Finland

Uwe S c h ~ n i n g

Osamu Watanabe 3

EWH R h e i n l a n d - P f a l z

Dept.

of I n f o r m a t i o n S c i e n c e

Seminar

Tokyo

I n s t i t u t e of T e c h n o l o g y

f~r

Informatik

D-5400 K o b l e n z

Tokyo

West G e r m a n y

Japan

Abstract.

In this paper

instances with

a measure

for the c o m p l e x i t y

respect to a g i v e n d e c i s i o n p r o b l e m

investigated.

Intuitively,

a problem

if every a l g o r i t h m that d e c i d e s

needs

2

U n i v e r s i t y of H e l s i n k i

U n i v e r s i t y of H o u s t o n Houston,

I N S T A N C E

C ( ) M P U T A T I O N A L

A

to look up

(a description of)

that all p r o b l e m s not instances.

an i n s t a n c e

Further,

in

P

have

x

x

is c o n s i d e r e d to be h a r d for A

in a table.

infinitely many

there exist p r o b l e m s

i n s t a n c e s b e i n g hard,

of p a r t i c u l a r

is i n t r o d u c e d and

in

The b e h a v i o r of h a r d

and runs A main

"fast" on result

(polynomially)

EXPTIME

x

states hard

w i t h all their

i n s t a n c e s under

polynomial

r e d u c t i o n s and the c o n n e c t i o n s w i t h c o m p l e x i t y cores a n d c i r c u i t s are studied.

1 R e s e a r c h of this author was s u p p o r t e d 12472 and D C R - 8 5 0 1 2 2 6 ;

Current

U n i v e r s i t y of C a l i f o r n i a ,

in part by N S F g r a n t s D C R 83-

address:

S a n t a Barbara,

Department

of M a t h e m a t i c s ,

CA 93106.

2 R e s e a r c h of this author was

s u p p o r t e d by the A c a d e m y of Finland.

3 Current

of M a t h e m a t i c s ,

address:

Santa Barbara,

Department

CA 93106.

U n i v e r s i t y of C a l i f o r n i a ,

198

I.

Introduction

There

are

putational view

(at least)

intractability

suggests

that

are

distributed

can

only

emphasized. also

Another

of

Hartmanis One

approach core

to s t u d y i n g introduced

core

for a p r o b l e m

that

every

hard"

A

algorithm

everywhere collection

in t h e

class

P

C.

of p r o b l e m has

such

function

[OS84] . R e c e n t l y ,

extensive

study

does

not

instances:

look-up.

algorithms

common

view

of a l g o r i t h m s

inherently

has been

for

in

is

intuitive

to d e c i d e

issues

feeling i.e.

hard

the p r o b l e m .

hard,

Such

instance,

the

"almost finite

This

difficulty

needs

by

complexity

approach

alteration everywhere"

s e t of

possibility

in f o r m u l a t i n g

C

cores Ko85,

about

complexity

iDstances

have

such time

is a " u n i f o r m l y

that any problem

every been

not

NP-

polynomial

a subject

of

OS 86]. the n o t i o n

of a c o m p l e x i t y

the c o m p l e x i t y still

cannot

removed

be

leads

algorithms for

of s i n g l e to a core.

f r o m the d e f i n i t i o n

c a n be d e c i d e d

it m e a n s

of a

[Lyn75] , a n d t h a t

of a c o r e

of p a t c h i n g

of

core

majorizes

is t h a t

instances

what

core

notion

than polynomial

It is k n o w n

density

ORS85,

more

a complexity

a complexity whose

the

(polynomial)

collection

A

instances.

say anything

finite

[Lyn75] . A

infinite

OS84,

of t h i s

really

any

any

cores

[ESY85,

A shortcoming

However,

is the

discussed,

In a sense,

sets

because

used

problem

feasible

been

by Lynch

is an

on

have

these

that decides

complete

core

"distributional"

strong

c a n be

algorithm have

This

by the

instances

complexity"

but

behavior

is s u g g e s t e d

the c o m -

of a d i f f i c u l t

manner,

distributions,

problem

The

causes

[Har83b] ,

complexity

almost

irregular

of a n y p a r t i c u l a r

"instance

problems.

the a s y m p t o t i c

view

individual

independent ideas

where

of w h a t

and no-instances

very

"smooth"

theory,

views

of d e c i s i o n

the y e s -

in s o m e

determine

complexity

that

two principal

trivially

with

a single

by a table

tables

is a b a s i c

instance

to be

hard. This into

shows

account

consideration. tion

t,

that

the

a measure

sizes

Here we

define

the

of

instance

of the d e c i s i o n take

algorithms

the f o l l o w i n g

t-bounded

complexity

complexity

for

approach. of a n

should

also

take

the p r o b l e m

For a g i v e n

instance

x

under

func-

with

199

respect

to a p r o b l e m

decides

A

This

and

idea

complexity

The

that

issue

with

complexity given

measure

for

amount

in s i z e

stance

smallest

to

different.

difficulty

notion

the

size

a certain The

as

algorithm

of

Kolmogorov

of

the

time

Kolmogorov

such, of

which

x.

the

(within

a string

the

the

instance

measures

string

(generalized)

is s u f f i c i e n t

complexity

description

of

We

introduce

vestigate

some

linear

notion. cores

it

way

whereas

deciding

smallest

bound). measure

the

is

instance

a problem

on a

our

speed-up

concept

set

it

is s h o w n

problem

A

either

in

results

of

show

P

has

Section that

provides

an u p p e r

because

increase

of

the

an

a description instance

Kolmogorov

of

to be

an

fast

than

this

instance

hard

complexity,

instance

in t h e

of b o t h

are that

the

3 gives exist

sets

many

if

i.e.

its

in

in-

if t h e r e

including

in

the

P

of

formally

an

a

and

of

the

complexity

complexity

complexity.

that

hard

for

for

A.

a complexity

are core to

hard

instances.

One

answer

EXPTIME

in-

instance

is n a t u r a l

in

and

"explanation" instance

it

an a f f i r m a t i v e sets

of

instances

forms

instances,

infinitely

in t e r m s

of

more

example,

in t e r m s

set or

section

For

the

presented

hard

there

next

is g i v e n

is f i n i t e

considering

not

its

properties.

theorem

Further,

When

to

an

complexity

in a table.

elementary

a given

include

consider

to d e c i d e

Characterizations

of

essentially to

we

is c l o s e

better

Kolmogorov

complexity,

Therefore,

is n o m u c h

are

which

of

of

on

related

rather

addresses

instance

an a l g o r i t h m .

also

size

instance,

bound

set

is

complexity

Nevertheless,

the

Ko84]

a given

however, the

the

t(Ixl)

closely

Har83a,

produces

here,

concerned

to b e

in t i m e

is o b v i o u s l y

[Ko165,

algorithm

A

runs

for

to

ask

whether

of

this

which

a

our

each

main

question.

all

We

instances

hard. The

behavior

reductions

are

then

polynomial-time again

to h a r d

of

hard

instances

investigated

reductions, instances

between

instance

P/Poly

from

(w.r.t.

complexity

[KLS0]

are

hard

and

under

polynomial-time

in S e c t i o n instances B) . F i n a l l y , the

demonstrated.

4.

It t u r n s

(w.r.t.

A)

out

have

in S e c t i o n

"nonuniform"

computable

classes

5,

that to b e

under mapped

connections P/log

and

200

2.

Measurin q

First, machine number M

Instance

some

of

steps

not

on a l l

whose

ordering

2.1.

that

M

on

x),

ones.

with

respect

ICt(x~A)

The

t-bounded

By

= min

any A

{ ~M~

K(x)

we

between

following these

Proposition.

For

such

that

t

IC t(xlA)

Proof.

Let

t(Ixl) ,

i.e.

accepts

A.

input

produces

z, some

M,

if if

M

of TMS,

are

enumerated

following.

t

a function

on

the

t-bounded

instance

complexity

timeM(x)

! t(Ix])

}.

and

of

a string

x

timeM(:~)

version

points

to O O

by

as

and

unbounded

TMs

the

out

of

is

! t(Ixl)

Kolmogorov

}.

complexity.

a straightforward

relationship

measures.

2.2.

for

= x

is e q u a l

enumeration

is t h e

Turing

timeM(x)

accepted

shorter

set, ~

The

by

(it

set

standard

recursive

= A

M,

takes

that

string.

a deterministic

of

definition

a

proposition

be

the

some

complexity

= rain { ,M,' ' I MO,)

the

x

L(M)

property

I L(M)

Kolmoqorov

M

size

is d e f i n e d

Kt(x)

The

On

be

x to

denote

by

central

A

and

the

assume

the

Our

Let

instance

and We

Let

numbers,

[MI

on

satisfies

Definition.

o_~.f x

by

instances,

longsr

natural

is g i v e n .

denote

halt

halts

before

notation

(TM) . W e

does

Complexity

all

< K t(x)

Mx

be Kt(x)

Combine M

and

recursive

all

set

A

there

is a c o n s t a n t

c

x,

+ c.

a minimal =

machine

IMxl .

these

first

output

every

Let

two

operates

symbol,

it

MA

that be

machines like

produces some

fixed

to a n e w Mx

on

is c o m p a r e d

the

in t i m e

machine

machine

input with

x

~.

M Each

that as

follows.

time

corresponding

Mx

201

symbol z.

of

z.

In c a s e

whether

In

of

x

is

case

of

x = z,

M

disagreement, stops

in

A

or

not.

IM{


t(n)

for

timeM0(Y)

function n.

for

Let

each

y.

Define

j~(x)

=

each

y < x,

timeM(Y) and

MinTM(x)

=

the

minimum

and

M i n T M " (x)

= the

timeM,(X)

minimum

and

Note

(i)

that

3 ClYX

(Proof and

M0

as

IC t (xIA)

(i)) . G i v e n

if

we

such

= A(y)

that

L(M)

TM

M

such

that

M ~((x)

< dt*t(~xl)

have,

it d o e s

by

M(x) not

Assumption

+ dt*t(~y~)

}

= A

IMinTM(x) i •




then

the

M ~ ~',

M"

~ M i n T M " (z) ~ + c O

K ( e + l ) 21%(z)

In

~"

t i m e M- (Y)

because

constants

In

M ~

B =

follows.

ICP(xlA)

-- n,

all

is w e l l - d e f i n e d

> x0

table

= A,

set

a machine

finite

the

L(M)

the



be

This

IMI

for

Then

~ ~xl

that

a constant

that

has

{ x

respectively.

Since

and

the

x,

less

can

TM

be

Let

is

always

each

ICP(xtA)

x

the

polynomial

claim

So,

double

size

!

size

Easy(M)

x.

such

and

is a s s i g n e d

stage

prints

=

construction,

x g

TM

Easy(M)

M ~ B.

accepts

the

c0

finite

the

If

is

set

that

that

IS(M,k) L

Let

at

Bx+ 1

for

is

x

or

individual

implies Now,

x,

(2)).

doubled

"heavier"

}

From

M ~ B.

the

(i)) . A s s u m e

stage

(Proof

then

then

in

IcP(xIA)

>__ I C 2 n ( x I A )

.

= A

timeM(x)

2 n.

in

x ~ is

i) .

an

and this

case

Easy(M). integer

Easy(M),

and

IMI

Since k
_

Finally, x,

Ixl

x,

each

3.7.

we

= n,

stages

for

at

x

to

check

most

set

It

immune, [BS85]

it

y,

struction

can

stated



~,...,x

string

IMI

~MI

at

Remarks.

ties

we

!

conproper-

i.e.,

every

everywhere.

Reductions

efficient of

A

exactly

reduction should what

be

will

from

a

mapped be

shown

to

210

4.1. via

TheOrem. some

Let

A

reduction

polynomial

q

and

f.

such

ICq+P°q(x~A)

Then

that

Let

bounded

by a polynomial

let

M

Mf

obtain

p

= A

Let

q.

be

Let

4.2.

is

such

!

~M'~

!

~Mf~

=

~MI

c l, c

such

Let

+

by

B

and a

instances

x,

f

with

such

f(y)

~ B

and

p.

y

run t i m e

without

Given

that

input

its

loss

a string

x,

timeM(f(x))

!

first

Mf

using

runs

M.

of

to

Clearly,

+

~Mi + c 1 c

for

first

A

and

B

of

be

there

x

recursive

exists

for e a c h

constant

for

then

4.1,

A,

there

all p o l y n o m i a l s

>

+ c

independent

f. T h e n

that

By Theorem

ICP(f(x) IB)

Now,

A ~

that

reduction

that

B

that on

whether

+ p(q(Ixl)).

(q+poq,c-d)-hard

Proof.

computing

accepting

< q(Ixl)

Corollar~.

q

and

be a n y p o l y n o m i a l l

the m a c h i n e

+ p ( I f ( x ) l)

constants

nomial

p

< q(Ixl)

follows

some

c

to be n o n d e c r e a s i n g .

= I c P ( f (x) ~B)

via

p

that

+ c.

and then decides

ICq+P°q(x~A)

for

such

a constant

polynomials

fixed machine

machine

M"

exists

sets

and

timeM.(X)

It t h e n

for a l l

some

recursive

there

may be assumed

f (y) ,

L(M')

be

be a minimal

p ( I f ( x ) I) •

be

! IcP(f(x) IB)

Proof.

generality,

B

x,

c

p

such

that

d

is

(p,c)-hard

cI

and constants

A ~

B

and a poly-

and polynomial

is a c o n s t a n t

Icq+P°q(xIA)

generating

f(x)

sets

a constant

p, for

if

x

B.

and a polynomial

q

c,

- c I.

then

computing

f (x),

we

get

K(f(x))




in

Theorem.

reduction

_ c1

reductions

from

Letting

Icq+P°q(xIA)

"non-inJective"

"many"

answer

c 2.

>

investigate

very

that

of

we

constant

for

Hence,

also

strings

x

for

x

all

I,

IC p ( x ~ A )

(The

machine

Theorem

obtain

there in

by

this the

sets.

that

We

is

to

the

result say

e>0

similar

above,

4.2

following

is a c o n s t a n t

[BS85]

If (x) ~ + c = c o n s t a n t .

Corollary

the

EXPTIME-hard

+

witnessing

4.1) . But,

Applying we



Let

c

respect

density.

be

d,

IC r ( x ~ A )

(with

is a c o n s t a n t

I IcP(xIB)

property

a constant

EXPTIME-hard

set

exponential

Proof.

be

almost

density.

densities,

and

everywhere.

There-

This

implies

that

the

the

theorem

is p r o v e d .

213

5.

Connections

Although probably esting we

the

different

easily Let

A

Circuit

from

M(x)

= A(x)

for

5.1.

Definition.

for

that

between

extend be

Complexity

motivation

connections

can

sets.

with

the

a set.

the

all

x

For

each

of

instance

of

circuit

complexity,

both

notions

of

instance ~e

notion

say

on w h i c h

set

a TM M

A,

there

are

is

inter-

nonuniform

complexity.

notion

to n o n r e c u r s i v e

complexity

that

complexity

M

is c o m p a t i b l e

with

First,

A

if

halts.

each

function

t

and

A

and

each

string

x,

define

~t(xIA)

= min

{ IMI

I M

is c o m p a t i b l e

timeM(X)

This

gives

following class

us

a more

general

proposition

of

recursive

shows

Proposition.

For

such

that

t

~-~t(xZA)

Proof.

all