On the structure of combinatorial problems and ... - Springer Link

ON THE S T R U C T U R E

OF C O M B I N A T O R I A L

STRUCTUP~

PI~SERVING

Giorgio Ausiello

- Centro

trolio e C a l c o l o

Automatici

00184

PROBLEMS

AND

REDUCTIONS

di Studio

dei Sistemi

del CNR,

di Con-

Via Eudossiana,

18

Roma.

Alessandro

D'Atri

- Istituto

di A u t o m a t i c a

sit~ di Roma,

Via Eudossiana,

Marco

- Istituto

Protasi

di Roma,

Piazzale

delle

18 - 00184

di M a t e m a t i c a Scienze

dell'Univer Roma.

dell'Universit~

- 00185

Roma.

ABSTRACT In this paper an o p t i m i z a t i o n zation p r o b l e m problem number

the c o n c e p t

problem

is defined.

is then introduced:

are c l a s s i f i e d of a p p r o x i m a t e

according solutions

of elements

are p a r t i a l l y

structures.

In this w a y

plete

problems

preserve

of c o m b i n a t o r i a l

ordering

all input elements

to their

ordered

according

structure

(based on the then the c l a s s e s

to the richness reductions

and some c o n d i t i o n s problem

to

over an optimi-

measure)and

preserving

of a c o m b i n a t o r i a l

associated

of an o p t i m i z a t i o n

"structure"

of d i f f e r e n t

can be i n t r o d u c e d

the s t r u c t u r e

A partial

problem

of their

among N P - c o m -

for a r e d u c t i o n to

are given.

I. I N T R O D U C T I O N In the r e c e n t (1972)

years,

after

a g r e a t deal of r e s e a r c h

of f i n d i n g

reductions

the basic w o r k efforts

among N P - c o m p l e t e

well k n o w n open p r o b l e m of e s t a b l i s h i n g if this p r o b l e m has b r o u g h t properties

still remains

In H a r t m a n i s

computable reduced

and B e r m a n

problems

are,

isomorphism.

I-I one into

(1975)

these r e d u c t i o n s These

(1976)

results

interesting

the

to NP.

Even

problems

combinatorial

the same up to a p o l y n o m i a l

In other words

all of the known which

(deterministic) and B e r m a n

the m u l t i p l i c i t y

seem to suggest

and of solving P is equal

of our work.

the other by a m a p p i n g

preserve

and Karp

to the p r o b l e m

it is shown that all of the k n o w n

so to say,

and in H a r t m a n i s

(1971)

the study of N P - c o m p l e t e

is also the d i r e c t i o n

in the size of the input on a in Simon

problems whether

a n e w light on some of their m o s t and this

NP-complete

open

of Cook

have b e e n d e v o t e d

a deep

problems

time can be

takes p o l y n o m i a l

time

TM. At the same time b o t h (1976)

it is shown that

of the solutions. structural

similarity

among

46

(all?) N P - c o m p l e t e problems but on the other side if we consider an o p t i m i z a t i o n p r o b l e m related to a c o m b i n a t o r i a l p r o b l e m we may notice a d i f f e r e n t i a t i o n among classes of N P - c o m p l e t e o p t i m i z a t i o n problems. Sahni

(1976), Sahni and G o n z a l e s

(1974) and J o h n s o n

(1976), in fact,

show that w h i l e in some p r o b l e m s like "job sequencl'n g " we can achieve a r b i t r a r i l y good a p p r o x i m a t e solutions in p o l y n o m i a l time, families of p r o b l e m s

(e.g. graph colouring)

a p p r o x i m a t i o n a l g o r i t h m can exist.

for other

no p o l y n o m i a l time good

This seems to suggest an intrinsic

d i f f e r e n t i a t i o n among o p t i m i z a t i o n problems and among their a s s o c i a t e d c o m b i n a t o r i a l p r o b l e m s and also the fact that not n e c e s s a r i l y reductions among c o m b i n a t o r i a l p r o b l e m s can be taken as reductions among optimization problems. In our w o r k we started from the idea of studying the d i f f e r e n c e s among N P - c o m p l e t e p r o b l e m s and of i n v e s t i g a t i n g such issues as i)

under w h a t conditions,

given a c o m b i n a t o r i a l p r o b l e m we can speak

of its subproblems and we can define its a p p r o x i m a t e solutions; ii)

w h a t reductions among c o m b i n a t o r i a l p r o b l e m s are such that they

map subproblems into subproblems and how t h e s e r e d u c t i o n s r e s p e c t to the a p p r o x i m a t e solutions

(do they p r e s e r v e

behave w i t h

"good" approxi-

mations?); iii) w h a t o r d e r i n g can we define among the subproblems of a c o m b i n a torial p r o b l e m and w h a t classes of c o m b i n a t o r i a l problems can be m a p p e d one into the other by o r d e r p r e s e r v i n g reductions. Even if most of the q u e s t i o n s are still far from being answered, in this paper we p r e s e n t some p r o m i s i n g results and a f r a m e w o r k for further research. First of all we r e s t r i c t ourselves to c o n s i d e r i n g c o m b i n a t o r i a l p r o b l e m s w h i c h are a s s o c i a t e d to o p t i m i z a t i o n p r o b l e m s w i t h integer measure

. Then,

is introduced:

in ~3 a partial o r d e r i n g over an o p t i m i z a t i o n p r o b l e m all input elements

(instances)

b l e m are c l a s s i f i c a l according to their

of an o p t i m i z a t i o n pro-

"structure"

of a p p r o x i m a t e solutions of d i f f e r e n t measure)

(based on the number

and then the classes of

elements are p a r t i a l l y ordered a c c o r d i n g to the richness of their stru~ tures. We next define

(in ~4) r e d u c t i o n s w h i c h p r e s e r v e the o r d e r i n g

and, more particularly,

reductions w h i c h p r e s e r v e the structure.

Some

c o n d i t i o n s for a r e d u c t i o n to be structure p r e s e r v i n g are then given and m a n y examples of c o m b i n a t o r i a l problems w h i c h can be m a p p e d one into the other by s t r u c t u r e p r e s e r v i n g r e d u c t i o n s are shown.

47 2. O P T I M I Z A T I O N AND C O M B I N A T O R I A L P R O B L E M S In the following parts of this paper we will r e s t r i c t o u r s e l v e s to c o n s i d e r i n g o p t i m i z a t i o n p r o b l e m s w h i c h are "strictly r e l a t e d " t o N P - c o m p l e t e c o m b i n a t o r i a l problems. A n y w a y we will give first some d e f i n i t i o n s w h i c h are m o r e g e n e r a l than r e q u i r e d and w h i c h are derived from similar d e f i n i t i o n s given by J o h n s o n

(1973).

In these d e f i n i t i o n s we try to

capture the basic objects w h i c h c o n s t i t u t e an o p t i m i z a t i o n problem: e s s e n t i a l l y input objects,

proximate solutions,

output objects among w h i c h we find the ap-

a measure over the solutions, an o r d e r i n g on the

values of the m e a s u r e w h i c h c h a r a c t e r i z e s an o p t i m i z a t i o n p r o b l e m as a

maximization or a minimization problem. D E F I N I T I O N I. An optimization problem A is a 5-tuple A ~ where:

INPUT, O U T P U T are c o u n t a b l e domains, SOL:

INPUT ~ {AIA is a finite subset of OUTPUT} is a r e c u r s i v e m a p p i n g that p r o v i d e s a p p r o x i m a t e solutions for any given element of INPUT

Q : is a totally o r d e r e d set m

: OUTPUT ~ Q

D E F I N I T I O N 2. i) The optimal value m~(x)

of an input x of an o p t i m i z a -

tion p r o b l e m A is m~(x)

= b e s t {m(y) ly C SOL(x)}

ii)The optimal solutions A~(x) A~(x)

= {y E SOL(x)Im(y)

of an input x are

= m~(x)}.

W i t h the n o t a t i o n SOL(~ ] we denote the set of elements of O U T P U T w h i c h are a p p r o x i m a t e s o l u t i o n s to the input x and the o p t i m i z a t i o n p r o b l e m is the p r o b l e m of looking for the b e s t s o l u t i o n in SOL(x). N o t e that w i t h the n o t a t i o n w h i c h is optimal in SOL(x)

"best {...}" we m e a n the e l e m e n t of Q

a c c o r d i n g to the o r d e r i n g of Q. Since SOL(x)

is finite such an element always exists. our examples,

under increasing order

(maximization problems)

(minimization problems).

x. For example,

in the futurer

c o m b i n a t o r i a l problems, Besides,

most polynomial EXAMPLE

ordered

or d e c r e a s i n g order

In order to be meaningful,

O U T P U T has to be " c o n s i d e r a b l y simpler"

time at most.

In m o s t cases, and in all of

the m e a s u r e will have values over the integers,

to decide the set

than to compute SOL of a g i v e n

since we are i n t e r e s t e d in N P - c o m p l e t e

we will ask O U T P U T to be d e c i d a b l e in p o l y n o m i a l also to compute the f u n c t i o n m should require at

time.

I. MAX-CLIQUE:

The p r o b l e m consists in looking for the m a x i m u m

48

complete subgraph in a given graph x. INPUT

set of finite graphs

:

OUTPUT

: set of c o m p l e t e finite graphs

SOL(x)

: set S of c o m p l e t e subgraphs of a g i v e n graph x

Q : integers in increasing order m : number of nodes of a complete graph. EXAMPLE 2. M A X - S A T I S F I A B I L I T Y :

This p r o b l e m consists in looking for

the m a x i m u m set of clauses w in a given formula of p r o p o s i t i o n a l calculus in CNF over {xl,...,x r} (Cn V ...VC, nl)V

(where Cij = x£ or x£)

(Ca V . . . V C 2 n 2 ) A . . . A ( C m I V . . . V Cmn m )

such that w is satisfiable,

and in e x h i b i t i n g

the s a t i s f y i n g truth assignment. INPUT

:

set of formulae of p r o p o s i t i o n a l calculus in CNF

OUTPUT

:

{ ) = number of clauses in w. In A p p e n d i x we give a list of o p t i m i z a t i o n problems w h i c h we will refer to t h r o u g h o u t the paper. Given an o p t i m i z a t i o n p r o b l e m A, any total function f such that (VxE

INPUT)If(x)

@ SOL(x)]

and that is polynomially computable,

can

be c o n s i d e r e d a p o l y n o m i a l a p p r o x i m a t i o n of A. We w i l l be m a i n l y i n t e r e s t e d in p r o b l e m s that a d m i t at least one p o l y n o m i a l approximation.

For example, while M A X - C L I Q U E and ~ X - S A T I S

F I A B I L I T Y admit p o l y n o m i a l approximations, MIN-EXACT-COVER

(see Appendix),

this is not the case w i t h

except, of course,

if P = NP.

Strictly related to an o p t i m i z a t i o n p r o b l e m we have a r e c o g n i t i o n p r o b l e m that we will call "associated c o m b i n a t o r i a l problem".

The m o r e

natural r e c o g n i t i o n p r o b l e m a s s o c i a t e d to an o p t i m i z a t i o n problem, derives from the q u e s t i o n of w h e t h e r an element x of INPUT has an app r o x i m a t e s o l u t i o n of m e a s u r e at least

(at most)

K or not. M o s t of the

c o m b i n a t o r i a l p r o b l e m s a p p e a r i n g in the literature are of this type. D E F I N I T I O N 3. Let A = be an o p t i m i z a t i o n

The combinatorial problem associated to A is the set

A c C INPUT x Q where A c = { < x , K } I K ~ m ~ ( x )

under the o r d e r i n g of Q}.

For example the c o m b i n a t o r i a l p r o b l e m CLIQUE a s s o c i a t e d to the o p t i m i z a

49

tion p r o b l e m M A X - C L I Q U E c o m p l e t e s u b g r a p h of

consists in d e c i d i n g w h e t h e r a graph x ihas a

(at leastl

K nodes or not. Besides~

the combina o,o

torial p r o b l e m P A R T I A L - S A T I S F ! A B I L I T Y w h i c h is the set {{w,K>lw has at least K s a t i s f i a b l e clauses}

is clearly the r e c o g n i t i o n p r o b l e m as-

s o c i a t e d to M A X - S A T I S F I A B I L I T Y .

3. E Q U I V A L E N C E CLASSES AND OILDER!NGS O V E R C O M B I N A T O R I A L PROBLEMS~ As we p o i n t e d out in the i n t r o d u c t i o n our p u r p o s e is to find w h a t are the r e l a t i o n s among d i f f e r e n t N P - c o m p l e t e c o m b i n a t o r i a l p r o b l e m s not o n l y in terms of m a p p i n g s of one p r o b l e m into the other, but also by taking into a c c o u n t the a p p r o x i m a t e solutions.

For e x a m p l e we w o u l d

like to find a way of m a p p i n g a p p r o x i m a t e solutions of one p r o b l e m into c o r r e s p o n d i n g a p p r o x i m a t e solutions of another problem. The first i n t u i t i v e a p p r o a c h w o u l d be to i n t r o d u c e a r e l a t i o n over a c o m b i n a t o r i a l p r o b l e m such that any input is greater or equal than all of its subproblems,

but it is easy to see that this a p p r o a c h w o u l d

not a l l o w to a c h i e v e a partial order. It is instead n e c e s s a r y to i n t r o d u c e a c l a s s i f i c a t i o n of the elements of an o p t i m i z a t i o n p r o b l e m on the base of the c o m b i n a t o r i a l s t r u c t u r e of the elements and then to i n t r o d u c e an o r d e r i n g among the e q u i v a l e n c e classes of elements.

For a w i d e family of c o m b i n a t o r i a l

p r o b l e m s we n o t i c e d that it is s u f f i c i e n t l y m e a n i n g f u l to c h a r a c t e r i z e the c o m b i n a t o r i a l

structure of an e l e m e n t by the number of a p p r o x i m a t e

s o l u t i o n s at any level of a p p r o x i m a t i o n and to c l a s s i f y the e l e m e n t s of a c o m b i n a t o r i a l p r o b l e m a c c o r d i n g to this c o n c e p t of c o m b i n a t o r i a l structure. N o t e that this c h a r a c t e r i z a t i o n is stronger than c o n s i d e r i n g only the number of optimal solutions but w e a k e r than c o n s i d e r i n g the c o m p l e t e and-or g r a p h of the s u b p r o b l e m s of an input of a c o m b i n a t o r i a l problem. DEFINITION

4. Let A ~ be an o p t i m i z a t i o n

p r o b l e m and let x E INPUT; we define structure

of x (denoted ~x ) the

list of the c a r d i n a l i t i e s of the a p p r o x i m a t e s o l u t i o n s of x. F o r m a l l y ~x ~ = < a l , . . . , a n)

, n _> 0 such that there exist u l , . . . , u n @ Q w i t h

u i < ui+ I (for every i < n) w i t h r e s p e c t to the o r d e r i n g of Q and

i) ii)

(Vi) lai = Ai(x)

IAi(x) 1?

= {YlY e SOL(x)

and m(y)

= u i}

(approximate solutions at level u i )

iii)

U A i ( x ) = SOL(x) i

50

Obviously

A

n

(x) = A~(x)

as d e f i n e d

in d e f i n i t i o n

DEFINITION

5. An o p t i m i z a t i o n

problem

x E INPUT,

for every e l e m e n t

a i in £x

the measure) empty

the set of a p p r o x i m a t e

is said to be convex if for every (corresponding

solutions

to a v a l u e

of x of m e a s u r e

u i of

q is non

for every q such that u i ! q ~ m~(x)"

For example,

such p r o b l e m s

as MAX-CLIQUE,

C O V E R are convex w h i l e ~ X - S U B S E T - S U M At this point we m a y d e f i n e DEFINITION

valence

equivalence

class of x under

the i n t r i n s i c

(i.e.

combinatorial

multiset

(x ~ Ay)

this c o n c e p t

property

we

if and only if Ix] A the equi-

of s t r u c t u r e

of a p r o b l e m

captures

also in the case of

the p r o b l e m M A X - S U B S E T - S U M .

If we

inputs

x = ([3,5,5,7,2] ,I0)

(where y is o b t a i n e d

INPUT:

p r o b l e m and let x,y E INPUT;

~x = ~y)" We d e n o t e

of the fact that

the f o l l o w i n g

over

~A

non c o n v e x p r o b l e m we m a y c o n s i d e r consider

non c o n v e x problem.

classes

to y in the p r o b l e m J

the same s t r u c t u r e

As an e x a m p l e

MAX-SATISFIABILITY,MIN-NODE-

is a typical

6. Let A be an o p t i m i z a t i o n

say that x is equivalent they have

y = ([6,10,I0,14,4],20)

from x by simply d o u b l i n g

all elements

of the

and the constraint)

we have

~

and the list lists

2.

x

= £

y

= (1,1,2,2~I,1,3>

Z

turns out to be equal to Z even y x of m e a s u r e s h a p p e n to be different.

In the f o l l o w i n g among classes

definition

of elements

we b e g i n

if the c o r r e s p o n d i n g

to i n t r o d u c e

of INPUT and,

hence,

a concept

indirectly

of ordering

a more

formal

corucgpt of subproblem. DEFINITION

7. Let

Zr and Zs be the structures

if Zr = < a l ' ' ' ' ' a n > ii < ... < i n_< m and Clearly over

of r , s @ I N P U T we say Zr~Zs

and there

exist i ...., ~

with

(¥j < n)[aj ! b i j ] -

this r e l a t i o n

is a p a r t i a l

order

and induces

a partial

order

INPUT/= -A

DEFINITION A

and £s = < b 1 ' ' ' ' ' b m >

8. Let r,s E INPUT;

we say that r is a s u b p r o b l e m

(denoted r ~ A s) if the s t r u c t u r e

of r is a s u b s t r u c t u r e

of s in

of s , f o r m a l l y

~r ! k sDEFINITION Note

that

9. Let[x]AJ[S] A E I N P U T / { A l w e the c o n c e p t

of subproblem,

say that

instead,

[r] A ! A [ s ] A

if r ! A

does not give rise

s-

to a

51 partial hold

order

over

INPUT b e c a u s e

the a n t i s y m m e t r i c

property

does not

in that case. For e x a m p l e

beside

if we again

the a l r e a d y

consider

considered

the p r o b l e m M A X - S U B S E T - S U M

elements

and

x and y we ex~m.ine the s t r u c t u r e

of the input

~, Z =

z =

we

find out the f o l l o w i n g

Another

if x is the g r a p h

relations:

Ix]

: [yl

[x]

< [ z]

example

is o b t a i n e d

~

/

B ~

< 1,1,,2,2,1,1,3~1~3)

~

if we c o n s i d e r

the p r o b l e m ~.~X-CLIQUE

the list is (1,5,8,5,1 >

D

A

E the list is

if y is the graph

E the list is < 1 , 5 , 6 , 2 >

and if z is the graph

E so that c l e a r l y

[y]

= [z]

4. O R D E R P R E S E R V I N G In K a r p

REDUCTIONS

(1972)

orial p r o b l e m s

was

the c o n c e p t

(associated

we will

which

least the o r d e r i n g

defined

(i.e. dified

and the value

definition

DEFINITION

reductions

problems)

the

according

reduction

among c o ~ i n a t - -

studied.

consider

preserve

both constituents

the i n p u t

(many-one)

to o p t i m i z a t i o n

of those r e d u c t i o n s

considering

of

first e x t e n s i v e l y

In this p a r a g r a p h problems

< [x] .

among

combinatorial

and we will

~'combinatorial

be c o n c e r n e d

structure '~ or at

to it. Since we are i n t e r e s t e d

of an e l e m e n t

of a c o m b i n a t o r i a l

of the measure)

we give a

in

problem

slightly

mo-

of a reduction.

10. Let A c and B c be two c o m b i n a t o r i a l

to the o p t i m i z a t i o n

problems

A and B; we say that

problems

associated

ACiB c (A c is poly-

52

nomial by reducible fl

to B c) if t h e r e

: INPUT A ~ INPUT B and