A separation decomposition for orders

A Separation Decomposition for Orders

Thomas Kampke

Forschungsinstitut fur anwendungsorientierte Wissensverarbeitung (FAW) , Helmholtzstr. 16, 89081 Ulm, Germany

Several decomposition types of orders and related discrete structures have been investigated so far. In this paper, we present a decomposition for orders based on partial “ignoring” of the order. Structures that are in a certain sense “prime” turn out to be decomposable. The relation to other decomposition methods is analyzed. Bounds for activity networks and reliability functions are obtained. 0 7994 John Wiley & Sons, h c .

1. INTRODUCTION

Decomposing structures can lead to the reduction of complexity of a “decomposable” problem both in a formal and informal way. It can also lead to approximate solu- tions of decomposable problems and decomposition yields interest in itself. Decomposition techniques have been studied for a variety of areas including orders, graphs, or k-ary relations, with k 2 3, and switching functions; see [ 11 and [ 71 for an overview. The present paper gives some results on the decomposition of finite orders, par- tially from a unifying point of view.

The decomposition introduced here is based on the deviation property. This property allows, e.g., a minimum representation of particular functions. The representation is based on redistributions of Mobius inverses [ 61. Here, we investigate order theoretic consequences related to the deviation property. If an ordered set does not have this property, then it can always be coarsened to have it. A coarsening technique is stated in terms of equivalence classes.

The structure of this paper is as follows: In Section 2, we introduce various forms of equivalence classes that lead to a decomposition of orden. In Section 3, we discuss the relation of this decomposition to two well-known decomposition principles: the split decomposition and the substitution decomposition. In Section 4, we apply the decomposition principle to obtain bounds for problems originating from orders. Upper bounds will be established

for functions of activity networks and lower bounds will be derived for reliability networks.

:= denotes a defining equation and :- denotes a defining equivalence, 0 marks the end of a proof or example, and 1 CI is the cardinality of set C. We stipulate the read- er’s familiarity with Hasse diagrams.

2. A DECOMPOSITION PRINCIPLE

Let an order ( U , I ) be defined-as usual-as a set U endowed with an order relation 1. An order relation is reflexive, antisymmetric, and transitive. We will use the term order also for the order relation I .

The decomposition of an order introduced here is based in the first place on the idea of ignoring a part of the order that is located “below” some other part of it. The ignored part is separated from the rest by an antichain. The antichain is supposed to consist of all remaining smallest elements giving rise to particular emphasis on antichains.

Definition 1. A set P s U is called pseudominimal if and only i f P is a c-maximal antichain in U , i.e., P is a E- maximal subset of U with each two elements being incomparable.

Elements of a pseudominimal set are generally not minimal with respect to the order 1. For example, all singletons of a power set form a pseudominimal set, but

NETWORKS, VOI. 24 (1994) 185-194 0 1994 John Wiley & Sons, Inc. CCC 0028-3045/94/030185-10

185

186 KAMPKE

@ is the unique smallest element if the power set is ordered by set inclusion.

A pseudominimal set is covered (not partitioned except in trivial cases) by subsets induced by arbitrary elements of the order. For all elements a in U , let

P ( a ) := { x E P and x I a }

denote the set of all pseudominimal elements below a. Furthermore, let

U ( P ) : = {xlxE U a n d P ( x ) # 0}

denote the set of all elements that have at least one pseudominimal element below (the “upper” of P) . U ( P ) is that part of the order that is not ignored.

Definition 2. An order ( U , I) has the deviation property (with respect to P ) , i f for arbitrary elements a , b in U ( P ) with b & a there exists an element c in P( b ) - P( a) .

Example 1. U1 has the deviation property with respect to P = { u l , u 2 } . Uzhasno t , s ince fora=u3andb=u4 ,b $ a but P ( u ~ ) - P ( u ~ ) = P - P = @:

1 u4

Every strict intree, i.e., every order in which each element possesses at most one immediate successor and each nonminimal element possesses at least two immediate predecessors, has the deviation property with respect to the set of all minimal elements. The converse is not true. However, each nonminimal element of ( U( P ) , I ) of an order with deviation property has at least two immediate predecessors in U( P ) .

Let the height hf( a ) of an element a in U( P ) be defined to be maximum cardinality of a chain u1 I - - - I a of distinct elements ui E U with uI E P ( a ) . By induction on I P( ui ) I, the height of an element can be bounded by the “width” I PI :

Lemma 1. Let an order ( U , I) have the deviation property with respect to P. Then, for all a in U ( P ) :

For intrees, the inequality of Lemma 1 may hold with equality. If I U(P)I is odd, then there is a unique order of maximal height with deviation property; this order has a single element a. of maximum height hf( ao) = I PI :

Partitions of U( P ) can be obtained from sets that are dual to P( u ) . These are the sets of all successors of fixed elements u :

S ( u ) : = { v l u E U a n d u I u } .

Let the elements of the pseudominimal set be labeled according to a fixed numbering P = { u I , . . . , u,} and let T be a permutation of { I , . . . , in } ; notation: K E S,,, . A partition of U( P ) into m sets depending on K is given by

Each C: is not empty, since for pseudominimal element u we have { u } = P( u ) E S( u ) , so different pseudominimal elements belong to different classes.

For partitions CT, . . . , C;, we introduce a binary relation re1 on U( p ) , formulating the idea that elements are “related” if they belong to the same class irrespective of the permutation:

are1 b :o V K E S, 3k(7r) E (1,. . . , m } such that ( a , b } G C;(=).

Moreover, we define function F on the power set f7 U ( P ) ) by

for A E U( P ) . F gives the cardinality of the set of pseudominimal elements that are below some element of A . F is a submodular function:

F ( A U B ) + F ( A n B ) = I { x I x E P and 3a E A

with x I a jlb E B withx I b) 1 + I { xIx E P and 3b E B with x I b J a E A

(=: a)

w i t h x I a } l ( = : p ) + 1 {xlx E P and 3a E A with x I a and 3b E B

w i t h x I b } l (=:y)

+ J ( x l x E P a n d 3 c E A n B w i t h x I c ) l (=:y’).

A SEPARATION DECOMPOSITION FOR ORDERS 187

y > 0 is possible even in the case of .4 n B = 0. From y’ 5 y follows

For two-element sets A = { a , b } and B = { a, c } , with a I /I I c, the deviation property makes the submodularity inequality strict, it makes re1 the finest relation possible, and it has several other consequences. Lemma 2.

1. I f ( Ll, 5 ) has the deviation property with respect to P, then,fiwction B : U ( P ) --* P ( P ) with B ( u ) = P ( a ) is one-to-one.

- . 7 I f ( L ; , . I ) has the deviation propert?’ with respect to PI and ( U z . I) has the deviation property with respect ru P2, then ( UI X U 2 , 5 ) has the deviation property with respect to PI X P?: ( i i , , u 2 ) I ( v I , V Z ) :- l i l I v1 and I I ? I v2

3. I f ( L’, I) has the deviation property bt’ith respect to P , then re1 is the identity on L1( P ) .

4 . I f ( L‘, 2 ) has /he deviation property with respect to P , then we obtain V a E U ( P ) : a = sztp( P( a ) ) .

The converses of Parts 1 and 3 do not hold. Part 4 does not mean that the supremum of arbitrary elements from P exists. Thus, ( U( P ) , I ) generally is not an upper semilattice (and. hence, not a lattice), even if it has the deviation property and even if it has a largest element.

Proof of Lemma 2 Parts 1 and 2 are clear. Part 3. Let a 5 b with a # b. Then, there exists

c E P( h ) ~ P( a). Choose H E S,,, with c = i iT, I ). Hence, h E CY. but a @ C; = S( c).

Let a and b be incomparable. Analog. Part 4. Vzi E P( a ) , we have ii I a (clear). Assume that there exists a minimal upper bound

: # N of P( a ) . Then, Vzi E P( a ) : 11 I z and, hence. P( a ) s P(z).

Let 2 and u be comparable. Since a is an upper bound for P ( a ) . z I a . Hence,

P ( z ) E P ( u ) . Altogether, P ( a ) = P ( z ) , which is by : # L( a contradiction to B being one-to-one.

Let 2 and CI be incomparable. Choose a’ : = 2 and b‘ : = a. a‘ and 6‘ are not comparable.

Hence, by the deviation property # P( b’) - P( a’) . On the other hand, P( b’ ) - P( a’) = P( a ) - P( z ) = 0 [since

0 P( a ) L P( z ) ] . a contradiction.

The deviation property is a ‘‘local’’ property. It is, hence, intuitively appealing that recognizing this property is of polynomial worst-case complexity [ 5 1. If the order

does not have the deviation property, then this can be obtained by coarsening. Coarsening is performed by taking equivalence classes according to pseudominimal elements. By such a construction order U2 of Example 1 results- up to an isomorphism-in L $ . Lemma 3 (“Quotients with respect to a pseitdomini- ma1 .set”).

Let (U I I) be an order and let P be psc.udominirna1. Then:

1. Relation - with l i l - 1i2 giver? hj, P( i i I ) = P( i i 2 1. is an equivalence relarion on U( P ) .

2. The set qf equivalence classes U ( P ) / - := { [ u ] / [ u ] G Ll and V u , . it2 E [ 1 4 1 : l i I - zi2 } is ordered by 5 - ,

where

3. ( I/( P)/ -, 5 , ) has the deviation property with respect

4 . The order I - preserves the original order I~ meaning (hat Vu, v E U ( P ) , 11 I v implies [ u ] I- [ v ] .

t o p / - := . ( [ 1 r ] J 1 r E P } .

Prouf Part 1 is clear. Part 2. Relation I- is well defined and inherits the

Part 3. Let for arbitrary a € U ( P ) be defined P( [ a ] ) / - order properties from the set inclusion.

:= { [ z i ] l [ u ] E P / - and [ u ] I, [ a ] } .

1. Case [ a ] 5 , [ b ] with [ a ] # [ h ] . It will be shown that there exists c E P with [ c ]

E P ( [ b ] ) / - ~ P ( [ a ] ) / - . From [ a ] I, [ h ] follows P ( a ) c P ( b ) . If P(a ) = P ( h ) , then [ a ] = [ b ] , meaning that exists c E P( b ) - P( a ) . By definition o f I-, this yields

2. Cases [ a ] and [ h ] are incomparable with respect [c l € P ( [ b I ) / - - - P ( [ a l ) / - - .

to I-. It will be shown that there exists c E P with [ c l

then for all zi in Pwith [ u ] E P ( [ b ] ) / - : ii I a , which means that P ( b ) c P(a ) . This, in turn, means that [ h ] I - [ a ] : a contradiction.

E P ( [ b l ) / - - P ( [ a I ) / - . I f f T [ b l ) / - c P ( [ a I ) / - ,

Part 4 is clear. i j

Remark 1. Zf( U , I) has the deviation property, then U( P ) is isomorphic to U( P ) / -. meaning that taking quotients /- consists of u mere relabeling ofelements.

Equivalence classes with respect to relation - leave P unchanged while U( P ) - P may be affected. Equivalence

188 KAMPKE

classes [ u ] are trivial if the order has the deviation property.

Another equivalence relation that leaves U( P) - Pun- changed and possibly affects P can also be established on ( U , I). Even if U has the deviation property, this other quotient structure generally is not isomorphic to ( U( P), I ) . The quotient is induced by relation x, which is com- plementary to - and which requires mild additional con- straints when its classes ought to maintain the deviation property. For u E P, let, therefore,

T ( u ) : = { w J w E U ( P ) a n d w z u } - { u }

= S ( u ) - { u } ;

for pseudominimal element u, set T ( u ) consists of all successors except u itself. It can be seen that

is an equivalence relation on P. Its equivalence classes will be denoted by [ u],, and P‘ := { [ u ] , 1 u E P} denotes the set of all these classes. We now may impose a structure on the set

U’ := ( U( P) - P) u P‘,

which is a union of “elements” of the original set U and “classes” of that set. U’ can be ordered by a mixed relation that coincides with the original order between two elements of U ( P ) - P. If one of the items in a painvise comparison is a class in P‘, we simply require the same relation to hold for all class members:

a s b , if a , b € V ( P ) - P

V w E [ u ] , : w ~ b , if a = [ u ] , E P ‘

and b E U ( P ) - P.

Relation I’ is a well-defined order, because T as function P -P P( U ( P ) ) is trivially constant on the equivalence classes of relation =. Thus, for u E P and a E U( P) - P , we have

M I a - a € T(u)=, [ u ] , E P ( a ) .

The last inclusion can be inverted and it can even be strict if both elements a and u are pseudominimal; P( a ) C [ u] , is then feasible. All classes [ u ] , in P’ are incomparable with respect to 5’. Lemma 4 (“Preserving the deviation property under coarsening the pseudominimal set ”) .

Let ( U , I) have the deviation property with respect to P.

1. For each equivalence class [ u ] , with 1 [ u ] , I 2 2, let for all a E T ( u ) exist v E P ( a ) - [u],(l[u],I = 1 admitted), meaning that P ( a ) contains at least one element of another class [u] , . (U‘, I’) then has the deviation property with respect to P‘ = { [ u ] , I u E P} .

2. x induces the coarsest partition P” of P that is agreeable with I such that the deviation property of ( U( P) - P) U PI’ is preserved, i f a condition similar to that of Part 1 is valid for P”.

ProoJ Part 1

1. Case a I ‘ b with a # band a , b E U ( P ) - P.

c E P ( b ) - P ( a ) , implying [ c ] , n P ( a ) = @. In this case, a I b and a # b. Hence, there exists

Assume: 3uo E [ c] , n P( a ) . uo E [ c ] , =+ T(uo) = T ( c ) . uo E P ( a ) * uo I a =+

( a # uo due to a E U( P ) - P ) : a E T( uo). Thus, overall, a E T ( c ) - c I a - c E P ( a ) ; hence, a contradiction. From [ c ] , n P(a ) = 0 follows (since c I b ) : [ c ] , I’ b and [ c ] , $’ a . 2. Case a I’ b with a f b, a E P’ and b E U(P) - P.

If I a I = I [ u ] , I 2 2, then for b exists (by the condition of the lemma) v E P( b ) - [ u ] , . This means that [ u ] I ’ b and [ u ] , is incomparable to [ u ] , with respect to 5’; hence, [ u ] , $ ‘a .

For la1 = I[u]- l = 1 there exists c E P ( b ) - P ( a ) with [ c ] , s ’ b and [ c ] , n [ u ] , = @. Thus, [ c ] , < ’ a . 3. Case a and b incomparable with respect to 5’.

a, b E U ( P ) - P: The same argument as in Case 1 applies.

a E P‘, b E U( P) - P or vice versa: The same argument as in Case 2 applies.

a , b E P’: This case is trivial, since a n b = /21 allowing the choice [c], = b.

Part 2. Each partition P” of P that is constant with respect to I induces an order I“ with

a i b , if a , b E U ( P ) - P

V w E k : w I b , if a = k E P ”

and b E U ( P ) - P.

Thus, Vk E P” 3u E P with k E [ u],. 0

Example 2. Let ( U , I) be an order with U = { u l , . . . ,

the deviation property with respect to P = { u l , . . . , us } .

= T( u4) = { u7, 248 } , T( u s ) = { 2.48 } . This results in classes [u l l - = {u,}, [u21= = ( ~ 2 1 , [ u d , = (243, u4}, and [ u 5 ] , = (24). Hcassediagramsof(U, ~ ) a n d ( U ’ , 5’) appear below:

Us} and M I , U2 I U6; 2.42, U 3 , U4 5 2 4 7 5 U s ; 245 I Ug. U has

Now T(ul) = { u 6 } , T ( U 2 ) = {%, u77 U S } , T(U3)


The condition of Lemma 4.1 is not redundant as is shown by an example with U = { u I , u2, u 3 } ordered by u I I u3 and u2 I u3. P = { u I , u 2 } leads to [ ul ] - = [ u2];r = { uI , u2 } , which leads to U‘ being the two-element set U’ = { [u l l - , 243) . Since [ u l ] % I’ u3, U’does not have the deviation property with respect to P’ = { [ u l ] = } .

Quotients according to / - can be formed iteratively by selecting a pseudominimal set in each iteration. The outcome of such an iteration can equivalently be obtained in a single step if the pseudominimal sets behave in a proper way. Let P and Q be two &-maximal antichains of ( U , I ) such that whenever u E Q and v E P are comparable then v I u. Such pseudominimal sets will be called agreeable.

Let the set of all successors of u that also lie in Q be denoted by

Q ( u ) : = { w l w E Q a n d w 2 u } = Q n S ( u ) .

Q( u ) consists of all pseudominimal elements above u (with respect to Q) while P( u ) consists of all those below (with respect to P ) .

Let - I be the equivalence relation with respect to P and let -2 be the equivalence relation with respect to Q as in Lemma 3. Elements u and 0 are not necessarily equivalent with respect to - 2 if they have the same sets Q( u ) and Q( v), but they are equivalent if they have the same elements of Q below; the latter sets are not denoted explicitly here.

Lemma 5 (“Iterated quotients with respect to two pseudominimal sets”).

Suppose P and Q be agreeable pseudominimal sets where in case of comparable elements the larger element belongs to Q. Let the following condition be true:

Forul I 242 with P ( u l ) f 0, Q ( u l ) f 0, and Q(u2) = 0 (meaning that uI lies between P and Q and u2 lies above Q), there exists s E Q( uI ) with u1 I s I u2.

Then,

Proof Similar to the proof of the subsequent and more important theorem. 0

Equivalence classes formed so far can be refined by a modification of taking two pseudominimal sets. Instead of forming classes “above” agreeable pseudominimal sets, we now restrict the formation of classes to elements “between” pseudominimal sets. The difficulty with this is not the definition of reasonable classes but the formulation of an order relation that is somehow induced by the original order 1. The latter is achieved by a vague analogy to the mixing order I’ on U’ (compare Lemma 4 and its preceding construction). The intended order will be derived by set inclusion for elements that are eligible for the formation of proper classes and by sticking to the original order in all other cases.

We next form classes, focusing on a part of the U where an analog of / - can be established. For an element u of U , consider

{ W I P( w ) f 0, Q( w ) f 0 and P( w) = P( u ) } ,

if P ( u ) # 0 and Q ( u ) f @

:= [ { u } , otherwise.

Class H( u ) consists in nontrivial situations ( H ( u ) # { u } ) of those elements that have the same pseudominimal elements of P below and that have pseudominimal elements of Q above.

A binary relation sH on the range H ( U ) = ran,( U ) can be defined by the mentioned mixing mode:

Without further assumptions, sff is not an order on H( U ) . The assumptions guaranteeing the order properties of sH

190 KAMPKE

can intuitively be described as “smoothing” conditions for “entering” and “leaving” the area between P and Q.

Theorem 1 (“Simultaneous quotients with respect to two pseudominimal sets’y .

Let U be an order, P and Q be two agreeable pseudominimal sets as above, and let also H and sH be as above. U( P ) C U is admitted. Let two conditions hold:

1. For u1 I u2 with P ( u l ) = 0, P ( u 2 ) f 0, and Q ( u 2 ) # 0, there exists s E P( u2) with u I 5 s.

2. For u2 I u3 with P( u2) f 0, Q( u2) f 0, and Q( u3) = 0, we have Vu with P ( u ) and Q ( u ) f 0: u I u3.

Then, sH is an order on H ( U ) .

terparts:

1. If there is a chain from uI below P to u2 between P and Q , then there is such a chain entering the area between P and Q at the lowest possible “level” P.

2. If some element u2 between P and Q is smaller than u3 above Q , then all elements between P and Q are smaller than u3.

Proof of Theorem 1. We list only those cases for dem- onstrating the transitivity in which the two conditions matter. Let H ( U I ) IH H(u2) and H(u2) I H H ( u 3 ) .

1. Case P ( u I ) = 0, P ( u z ) , Q(u2) f 0, Q(u3) = 0. H ( u , ) sH H ( u 2 ) means that 3u; E H ( u 2 ) with uI I u; ; hence, by Condition 1, 3s E P( u2) with uI I s.

H( u2) sH H ( u 3 ) means that 3u’; E H( u2) with u‘; I u3 and s I u’;. Altogether, u1 I s 5 u’l I u3 3 H ( u l )

Conditions 1 and 2 of Theorem 1 have intuitive coun-

I H H(U3).

211 u2 113

2. Case P(uI>, Q < u l ) , P ( u z ) , Q ( u d f 0, Q(u3) = 0. H ( u l ) sH H( 242) means that P( u l ) c P( 242).

H ( u2) IH H( u3) means that 3u; E H( u2) with u; I u3. 0 3 (by condition 2) uI I ug - H ( u l ) I H H(u3).

This allows us to introduce our basic notion:

Definition 3. The transition from ( U , I) to (H( U ) , 5”) under the conditions of Theorem I is called separation decomposition of ( U , I) with respect to P and Q.

The name separation decomposition is obvious, since forming proper classes takes place in that part of U that is separated from the rest by the sets P and Q.

Remark 2. I f Q is chosen to be the set of all I-maximal elements of U , then the separation decomposition with respect to P and Q is the same [on U( P)] as taking quotients /- with respect to P.

Under the circumstances of Remark 2, Condition 2 of Theorem 1 is satisfied since there does not exist any u3 with Q(u3) = 0.

Even if all classes H( u ) are of cardinality 1, the separation decomposition may be different from merely relabeling since the induced order sH may differ from the original order I:

Example 3. Let U = { u l , . . . , u s } with u l , 242, u3 I US and u2, u3 I u4. Selecting P = { u I , u2, u 3 } and Q = { u4, u 5 } results in H ( u i ) = { u i } , i = 1, . . . , 5 , but ( U , I) and ( H ( U ) , sH) are not isomorphic:

0

3. RELATION TO OTHER DECOMPOSITIONS

The separation decomposition turns out to be a gener- alization of the intuitively appealing substitution decomposition (with respect to c-maximal autonomous sets, see below) and it generalizes the slightly less appealing simple split decomposition for distributive lattices (see, e.g., [ 41). As the latter is only of formal interest here, we will consider it first and then turn to the substitution decomposition.

Theorem 2. The simple split decomposition for distributive lattices that are subsets of a power set is a case of relation /-.

Prooj (Concerning definitions and notations compare

Let II = { A l , . . . , A k } be apartition ofM. LetA be [ 4 , P . 381).

compatible with II, meaning that

V A j E II with Ai f l A f $3 : Ai c A;


.-l is thus representable as a union of suitable A , E n. This results in a distributive lattice:

D ( I I ) := i.41.4 E ‘P(.M) and ‘4 is compatible with II},

which will be decomposed by the split decomposition. Therefore, we generate a further lattice D ( I I ) ( ( A by the partition II and another partition II 11 A :

1 i i ~ . - l : = ~ - 3 ~ ~ ~ , ~ n . ~ , n ~ = 0 } u { ~ } aggregation of II by A

and Zl(II ) l lA := D(II l lA) aggregation of D ( I I ) by A .

A dissection S = { TI . T I 1 of M is called compatible with II if TI and T2 are compatible with II. Under the conditions stated, each dissection compatible with II is good, meaning that if V A E 2)(II) at least one of the following conditions holds:

A good dissection S = { T I , T2 } of M is called a split of D ( II ) if for T [ II] : = { A , 1.4, E II and A , c T ) :

each of the sets TI and T2 contains at least two sets from TI. 3J(II)liTI and D(I I ) l (T , are then called simple split decompositions of D II ).

Turning to the substitution decomposition with respect to autonomous sets, we repeat their definition. A suborder A of U is called autonomous [ 71, if for all ZI E U - 1 exactly one of the three following conditions is valid:

1 . All elements u of A satisfy 11 5 v .

2 . All elements v of A satisfy u I 1 1

3 . All elements u of A are incomparable with 11.

All singletons of Uand U itself are autonomous (trivial autonomous sets) ; subsets of autonomous sets generally are not autonomous. Autonomous sets and the substitution decomposition also arise in other structures than orders, e g , in function decomposition. Such other areas will not be considered here.

The set .4( U ) of all autonomous sets of an order ( U , 5) endowed with set inclusion E is itself an order. It may itself possess nontrivial autonomous subsets, so ( A ( U ) , E ) may be decomposable by the substitution decomposition. The autonomy of the singletons. however, makes ( A ( U ) , G ) indecomposable in the sense of forming equivalence classes with respect to the pseudominimal set of singletons.

Lemma 6. Let ( U ? I ) be an order. ( A ( U ) , E) has the deviation propert!’ with respect to { .x- 1 I ,Y E U } . Hence, it is invariant with respect to,forming quotients /-; it is, hence, indecomposable.

Statement ( A ) . Proof: { { x } I x E L’] having the deviation property is shown in a straightforward manner. The rest is an immediate consequence Of Remark

IIll TI is pseudominimal in 23(II). Each .4 E D ( II ) 1 1 Ti = 23 ( IIll TI) is a superset of T ,

orofA, E II withrl, n TI = $3.- EachA E D(II l lTI ) - ( II 11 Ti ) lies strictly above (II 1 1 TI ), meaning that (11 (1 TI ) Note that the sense of indecomposability in Lemma 6 is pseudominimal in D ( II ) . is (by Remark 2 ) that of the separation decomposition Statement (B) . with the upper pseudominimal set being the set of all Forming the quotient / - in 2)( II) with respect to P

maximal elements. := IIllT, leadsto B(II\lT,) . Taking quotients / - with respect to pseudominimal Forming equivalence classes - is done only with re-

spect to pseudominimal sets in 11 ; sets from - 11 sets and the substitution decomposition generally lead to = { A , 1 A , E II and A , G TI } are irrelevant for this. All different structures: .‘I E D ( II), which are of the’form

Example 4. Let ( U , I) be an order with U = { zi,, . . . ~

u 5 } and u i I 142 I u3 I us and u2 I u4 I us. The set A .- { u2, . . . , u5 1 is autonomous and U does not have the

such that A, - 11 for Jo and A, ~ for deviation propert!’ wirh respect to { zil } . The .szibstitution decomposition with respect to A leads to a two-element order U l ; forming the quotient with respect to { uI } leads to a singleton U2. Hasse diagrams qf U , UI , and CJZ appear

A=UA, , with J = J o U J , s { l , . . . , k } , .- J E J

J , . will be identified with u A,, All A a(n) / - are, hence (in a natural way), compatible with II 11 TI , meaning that 2)(IIl lT,) = B(I I ) / - . C below:

J E J O

192 KAMPKE

0 li U?

Things change, if Q may be different from the set of <-maximal elements:

Theorem 3. Let A be a s-maximal autonomous subset of a jinite order ( U , I ) . Then, there exist P and Q such that the resulting separation decomposition equals the substitution decomposition of ( U , 5) with respect to A (up to isomorphism).

ProoJ If A does not contain‘a (unique) smallest element uo, i.e., if I { u 1 u E A and u <-minimal in A } I 2 2, then introduce uo 4 U such that 1. V u E A : u o I u .

2. V u E U - A for which exist v E A with u I v: u I u,, I 0.

Since A is autonomous in ( U , I), A’ := A U { u o } is autonomous in U’ := U U { uo } . (In the case u,, E A, choose A’ := A and U’ := U.)

Let U* := { u I u EA’and u s-maximal in A ’ } . Choose a s-maximal set Uo E U’ - A’ = U - A such that all u E Uo are incomparable among each other and incomparable to v E A ’. Such a set Uo exists since we admit Uo = 0. Define two agreeable pseudominimal sets:

P := { u o } U Uo and Q := U* U Uo:

All elements u in A’ satisfy P( u ) = { uo } , i.e., all elements of the enlarged autonomous set have exactly one pseudominimal element in P below. The G-maximality of A implies

V u E A’ : H ( u ) = A’ and

v u E V - A’ : H ( u ) = { u } .

Since A is autonomous, all conditions of Theorem 1 are satisfied with s : = u,,. ( H ( V ) , sH) obviously is isomorphic to the substitution decomposition of ( U , I) with respect to autonomous set A . 0

The separation decomposition does not generalize the substitution decomposition with respect to an arbitrary autonomous set. This can be seen by an autonomous set consisting of exactly three incomparable elements; the

P resp. Q

separation decomposition cannot separate a two-element subset from it. The substitution decomposition however can. Nevertheless, the construction from the last proof does also apply to suitable, not necessarily s-maximal autonomous sets.

The separation decomposition with respect to agreeable pseudominimal sets P and Q may also be nontrivially applicable in case of an order that is indecomposable with respect to the substitution decomposition. In this sense, the separation decomposition is more general than the substitution decomposition. We give an example of a decomposition that cannot be performed by the substitution decomposition.

Example 5. Let ( U , I ) be an order with U = { u l , . . . ,

I u6, u7; u7 I 24 I ulo; u9 I uIo. { u2, . . . , u9> is not autonomous; nevertheless, U can be decomposed in the

U l o } and U I 5 U2 5 U5; U4 5 2.45; U1 I U3 I Ug I Us,’ U5


sense of the separation decomposition with respect to P := { u2, u3, u4, u 9 } and Q:= { u s , u 9 } . Thedecomposition results in ( H ( U ) , I ~ ) with H ( ui) = { ui } , i E { 1, 2, 3, 4, 9, lo}, H(U5) = { U s , u,} and H(us) = { u6, u s } .

( H ( U ) , I ~ ) cannot be generated by the substitution decompositionfrom ( U , I), since H( u 5 ) as wellas H ( u6)

are not autonomous. Hasse diagrams of U and H ( U ) appear below:

Q < P

U

4. BOUNDS

The separation decomposition is monotone, meaning that u I u implies H( u ) sH H ( v ) . This property leads to bounds for activity and reliability networks.

Let an activity network (compare [ 2,7]) and a vector x E RL‘l of so-called job durations be given, where ( U , I) is an order and x has nonnegative, real coordinates xu, u E U. Every element of the order corresponds to a real-world activity or event (“job”). The order I indicates a natural or technological restriction for feasible sequences of job executions: All <-minimal jobs can be executed without condition. Each job that is not <-minimal requires all its predecessors to be completed before its ex- ecution can be started.

Consider the network function called makespan CmaX( U , I, x) := maxcE@(U) C x,, with @( U ) beingthe

set of all c-maximal chains (“paths”) of ( U , I). Cmu( U , I, x) is the completion time of the last job of U.

Let the separation decomposition be applicable to ( U , I ) with respect to two agreeable pseudominimal sets P and Q. A new activity network ( H ( U ) , sH) is generated; each element H ( u ) E H ( U ) is an idealized job. Job H ( u ) is assigned duration x ~ ( ~ ) : = 2 xu/. The coordinates

xH(,,) form the vector x ~ ( ~ ) E RLH(’)I. The makespan of the original network can now be bounded.

a€ C

u’EH( U )

Theorem 4. rfthe separation decomposition is applicable, then Vx E RL”“ :

Proof: If u I u with u, u E U , then H ( u ) I H ( u ) . A part or the whole of each sum C xu with C E @( U ) also

appears for C E @ ( H ( U) ) . The nonnegativity of the UE c

xvs gives the inequality. 0

The bound of the last theorem holds for more general cost functions than makespan like Markovian cost functions (see [ 91). The same intuition as in Theorem 4 applies: By the separation decomposition jobs of the same class H( u ) , which may be performed simultaneously in ( U , I), have to be performed in some linear order according to ( H ( U ) , I ~ ) . In all cases, it thus holds: Coarser classes H ( u ) result in more crude upper bounds.

For stochastic activity networks, associate random variables [ 31 provide upper bounds for the makespan. These bounds can be improved by substitution decomposition, which takes out some “variability” of the stochastic nature of the problem. A corresponding effect for the separation decomposition cannot be expected; since the inequality of Theorem 4 may be strict, Cma,( U , I, X) < Cmax(H( U ) , I H , xH(U)) is possible.

In reliability theory, the problem of calculating the so- called s-t reliability arises. The s-t reliability is the probability p ( G ; s, t ) of there being a path from node s to node t in a connected (directed acyclic) graph G. Intu- itively, this is the probability that the connection from s to t “works”. For each arc e E G, there is a given probability pe that e works and failures of arcs are assumed to be stochastically independent. The computation of p( G; s, t ) generally is NP-hard [ 81 and, in principle, the com-

194 KAMPKE

putation can be done by the method of inclusion-exclu- sion [ 101.

Applying the separation decomposition to G results in a suitable graph G'. Each s-t path in G corresponds to a subset of an s-t path in G'. This leads eventually to

where the calculation of p ( G ' ; s, t ) may be simplified compared to that ofp( G; s, t ) . The simplification is based on a simplified system of paths.

I would like to thank the referee whose detailed comments improved the presentation of the paper.

REFERENCES

[ 1 J W. H. Cunningham and J. Edmonds, A combinatorial decomposition theory. Can. J. Math. 32 (1980) 734- 765.

S . E. Elmaghraby, Activity Networks. Wiley, New York (1977). J. D. Esary, F. Proschan, and D. W. Walkup, Association of random variables. Ann. Math. Stat. 38 ( 1967) 1466- 1474.

[ 2 ]

[3]

[4]

[5]

S. Fujishige, A decomposition of distributive lattices. Dis- crete Math. 55 (1985) 35-55. T. Kampke, Additive functions and their application to uncertain information. Ann. Operations Res. 32 (1991)

T. Kampke, Lower probabilities and function representation. Submitted.

R. H. Mohring and F. J. Radermacher, Substitution decomposition for discrete structures and connections with combinatorial optimization. Ann. Discrete Math. 19

A. Rosenthal, Computing the reliability of complex networks. SIAM J. Appl. Math. 32 ( 1977) 384-393. G. Weiss and M. Pinedo, Scheduling tasks with exponen- tial service times on non-identical processors to minimize various cost functions. J. Appl. Prob. 17 (1980) 187- 202. D. E. Whited, D. R. Shier, and J. P. Jarvis, Reliability computations for planar networks. ORSA J. Comput. 2

5 1-66.

[6]

[7]

(1984) 257-356.

[ 8 ]

[9]

[ 101

( 1990) 46-60.

Received December 199 1 Accepted November 1993

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A separation decomposition for orders