Consecutive Optimizers for a Partitioning Problem

-

Consecutive Optimizers for a Partitioning Problem with Applications to Optimal Inventory Groupings for Joint Replenishment

i Ov

A. K. CHAKRAVARTY C'niversity of Wisconsin, Miltaukee, Wisconsin .t

J. B. ORLIN

·.··-·.·.- ..·-·'·:.' ,.......... . ·"

Massachusetts Institute of Technology, Cambridge, Massachusetts

,

I

U. G. ROTHBLUM

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Technion, Israel Institute of Technology, Haifa, Israel (Received November 1982; revised July 1983; accepted February 1984) We consider several subclasses of the problem of grouping n items (indexed 1, 2, .., n) into m subsets so as to minimize the function g(S 1, .. , S,). In general, these problems are very difficult to solve to optimality, even for the case m = 2. We provide several sufficient conditions on g(') that guarantee that there is an optimum partition in which each subset consists of consecutive integers (or else the partition S,, -, S,, satisfies a more general condition called semiconsecutiveness"). Moreover, by restricting attention to 'consecutive" (or serniconsecutive ") partitions, we can solve the partition problem in polynomial time for small values of m. If, in addition, g is symmetric, then the partition problem is solvable in purely polynomial time. We apply these results to generalizations of a problem in inventory groupings considered by the authors in a previous paper. We also relate the results to the Neyman-Pearson lemma in statistical hypothesis testing and to a graph partitioning problem of Barnes and Hoffman.

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tbi

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ar

b

and

ar+l br+i

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- bn

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For b, = 0, we consider adb, to be +cc or - according to a > 0 or a, < 0. If ai= bi = 0, al/b 1 is defined arbitrarily so that inequality (1) holds. As

usual, we let a and b denote the vectors whose coordinates are a, and bi, respectively. Subject clasification: 334 partitioning items into subgroups, 625 optimal inventory groupings.

820 Operations Research Vol. 33, No. 4, July-August 1985

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0030-364X/85/3304-0820 $01.25 c 1985 Operations Research Society of America

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Consecutive Optimizers for a PartitioningProblem

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821

Let N = , , n$. For each subset S of N, let as = Zies ai and let bs = is b. We consider the problem of partitioning the set N into m sets,say Si, , Sm, including possibly some empty sets, so as to minimize a real-valued function g(S 1 , , Sm), where g,() is assumed to have the form gm(S, ..

'""'''''' I'" ''"""''

'' "'

r

i

c

J

- --p----_--~--·--

··· ...

__

.1~.~

Sm) = hm(as,1 bs,

as, bs);

...

where h,() is a real-valued function of 2m variables. We permit the function hm,() to be nonsymmetric, in which case the cost of a partition depends on the order in which the sets are selected. A subset S C N is called consecutive if its elements are consecutive integers, e.g., 4, 5, 6. In particular, the empty set is considered to be consecutive. A partition P = S,, ., S } is called consecutive if Si is consecutive for each i. A partition P = {Sj, -*, S} is called semiconsecutive if there is a permutation r of the set of integers , ... , rnm} such that the sets S,(f) U U S(j) are consecutive for j = i, ... , m. In particular, the set S,(1) is consecutive. For example, the partition IS1, S, S2, = {f3, 7, 81, 1, 2, 9$, 14, 5, 61} is semiconsecutive because the three subsets S3, S3 U Si, 83 U SI U S2 are all consecutive. The purpose of this paper is to provide sufficient conditions that the optimization problem defined above has an optimal consecutive or semiconsecutive partition. Specifically, we show that if h, is concave in its 2m variables, then there is an optimal semiconsecutive partition. Moreover, if in addition, b > 0 or a > 0, then there is an optimal consecutive partition. Also, we show that if bi = for each i and, in addition, h is concave in the "a variables" for each fixed value of the "b variables," then a consecutive optimal partition exists. Of course, when b = 1 for each i, for any set S C {1, . . n}, bs equals I S I, the cardinality of S. In this case, for a given partition P = {Si, .. , Smt, (bs,, ... , bsm) is called the shape of the partition P, as per Hwang [1981]. When we wish to specify the number m of sets in a partition P, we call P an m-partition. As above, we allow some of the m sets to be empty. We observe that the number of ordered m-partitions that are consecutive is O(m!nm-l) and the number of semiconsecutive partitions is O(m!nm-2). Thus, for a fixed value of m, we can determine an optimal onsecutive or semiconsecutive m-partition in polynomial time via complete enumeration. Although complete enumeration is quite impractical even for moderate sizes of m, this fact contrasts sharply with the NP-completeness of the optimal partition problem for any fixed m > 2, since the knapsack problem is a special case of the partition problem for n = 2. (Of course, the total number of ordered m-partitions is mrn.) We note that the sensitivity of optimal consecutive partitions to changes in mn can be studied by using the results of Denardo et al. [19821.

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Chakravarty et al.

We formally state our main results concerning the existence of consecutive and semiconsecutive optimal partitions in Section 1, where we also survey a number of places where special cases of our results are used. In Section 2, we discuss the special case where g(*) is separable and symmetric, in which case optimal consecutive partitions can be determined in O(mn2 ) steps and optimal semiconsecutive partitions can be determined in O(mn4 ) steps. Next, in Section 3, we show how our results

apply to problems of inventories. In location problem. that apply to our

that involve optimal groupings for joint replenishment Section 4, we show how our results apply to a certain In Section 5, we discuss issues of NP-completeness results. Finally, in Section 6, we prove the results of

Sections 2 and 5. 1. THE MAIN RESULTS We next state our main results concerning optimal partitions that are either consecutive or semiconsecutive. The proofs are deferred to Section 6. THEOREM 1. Suppose h, is concave in its 2m variables and in addition

b

Ob

0, a

or a

0. Then there exists a consecutive optimal

partition. THEOREM 2. Suppose that h is concave in its 2m variables. Then there

exists a semiconsecutive optimal partition. The arguments used by Chakravarty et al. [1982] can be used to establish the validity of the following variant of Theorem 1. i

THEOREM 3. Suppose bi = for i = 1,

, n and in addition suppose that

for every fixed value of the even variables, the function h is concave in the remaining variables. Then there is a consecutive optimal partition. Although the partition problems considered in these theorems are

quite restricted, there have been several realms where special cases of .......

these results have been used. We next give a brief survey of such applications. One classical partitioning problem occurs in hypothesis testing in statistics. Outcomes of a random phenomenon are partitioned into two sets, one of which corresponds to the region where the null hypothesis is accepted and the other to the region where it is rejected. In this case the

.. U. ...

. c ..............,.

"'avariables" correspond to probability of type

error, the "b variables"

correspond to probability of type 2 error and the corresponding objective function is linear. Consequently, Theorem 1 applies and an optimal consecutive partition exists. This fact is the well-known NeymanPearson lemma. For more details on the application to statistical testing, i

ii

_I

see DeGroot [1975, p. 374].

Consecutive Optimizers for a Partitioning Problem

823

We next give a number of examples of studies where special cases of the results of Theorem 3 were obtained. In particular, the special case of Theorem 3 where h is separable and symmetric is discussed in Chakravarty et al. They apply the corresponding result to a problem of inventory grouping with joint replenishment (see Section 3 for more complex such problems). Also, Hwang, and Hwang et al. [1985] consider restricted classes of functions for which a corresponding optimization problem has consecutive optimizers. Their conditions entail monotonicity and additivity requirements, and some of their results are special cases of Theorem 3. These papers provide applications of their results to problems in storage and problems in group testing. Finally, Barnes and Hoffman [19841 considered a special instance of the partitioning problem studied in Theorem 3. Their work was motivated by a graph theoretic partitioning problem. They used the theory of submodular set functions to establish the existence of consecutive optimal partitions. Their results were developed independently of those in Chakravarty et al., and their proof is an interesting application of the theory of submodular set functions. In Section 5, we show that the Barnes-Hoffman partitioning problem is NP-complete. I

4

2. THE SYMMETRIC SEPARABLE CASE We next consider the case where g is symmetric and separable, i.e., for some function h: R 2 -- R

t

g,(S, ¶

a,

T

''''''''''''''''' '' ' '''

6

...

, Sm) = j> h(ass, bs,).

In this case, the order of the subsets in a consecutive partition does not matter. In particular, without loss of generality, we can assume that in an (optimal) consecutive partition, the indices in Si precede the indices in Si for i < j. Chakravarty et al. showed that the problem of determining an optimal consecutive partition reduces to the problem of finding the shortest path between two nodes in a graph where the number of edges in the path is at most m. This problem can be solved in O(mn 2) steps using a standard dynamic programming recursion. In the symmetric and separable case, we can determine an optimal semiconsecutive partition as follows. Let Sij = i + 1, ..- , j} for all i, j with 0 i j n. Let f be the optimal value of a semiconsecutive partitioning of Sp into t subsets. Finally, let h'(S) = h(as, bs) for each S N. Then f= h'(S) for 0 i j n. fii = miniq,,ish'(Sik U Sqj) +

At IP

4 L.

(Sq).

4 ) steps for It is easy to see that this recursion can be computed in 0(n each value of t, and thus the total number of computations is O(mn4 ).

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Chakravarty et al.

824

3. APPLICATIONS TO INVENTORY GROUPING Consider an economic order quantity model involving n items, where the ith item has (deterministic) demand rate Di, a unit inventory holding cost hi per unit time, and a fixed cost K for placing an order. The problem is to partition the n items into m subgroups and choose order cycles for the groups out of a given set of allowable (oint) order cycles, so as to

minimize the net average cost per unit time. Chakravarty [1982a, 1982b] gives motivation and more detailed explanation of special cases of this model. This model is a generalization of the joint replenishment models considered by Goyal [1974], Silver 1976], and Nocturne [1973], who study the problem in which a unit of order time X is determined for the group of n items, and the order cycles for each item is an integer multiple of X. They gve heuristic solutions to this problem. Chakravarty et al.

'"' ' ''

''

have studied a related model. Let a = 2-1hiDi and bi = K where the items are labeled so that (1) holds. As in the ordinary EOQ model (e.g., Wagner [1969, pp. 18-19]), if item i has order cycle r, then its order quantity is rDi. In particular, the net cost over an order cycle of a group S of items having the same order

cycle r is c(S, r)

= lies

2 1 rDihir +

ies Ki =

T2

as +

bs.

So, if the items are partitioned into groups S, ..., Sm having order cycles t, ... , ti, respectively, the total average cost per unit time is c,,(St,

.. ·, S,,M t,

.. ,

tn) =

E= t

'c(Si, tj) =

2Esy2 (tjas, + tbs).

The mn-vector of order cycles is assumed to be extracted from a set T of

allowable m-vectors of order cycles. This formulation allows one to impose joint restrictions on the order cycles, e.g., the requirement that all order cycles are integer multiples of the smallest one, or individual restrictions, e.g., that each order cycle be an integer in the set {1, 7, 301. Therefore, for a fixed partition {S,, ... , Sm} of the items, the minimum

average cost per unit time is gm(S,

C')

I'

I

· . .

Sm) = inffTCm(Sl,

, Sm,Yt,

(2)

... , t).

Evidently, since cm(S, ... S, t, ... , t) > 0, the infimum defining g,( ) is finite for each partition. The problem is to find a partition of the' items into subsets so as to minimize g,(,,), where gm(s) is defined as in (2). Now, since c(S, ) is linear in as and bs (for fixed r), we conclude

'I ·' i

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that c,(Sl, , S, t, , tin) is linear in as, bsl, *", as, bsm and therefore gm(SL, ', Sm) as the infimum of linear functions is concave

.f l .

in as,, bs,,

. .,

as, bs,. Thus, the assumptions of Theorem 1 are satisfied.

In particular, our results apply and there exists an optimal partition consisting of consecutive sets. A special case of this model allows the decision maker to

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Consecutive Optimizers for a Partitioning Problem

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choose the order cycles of the groups independently out of a given set (which is independent of the enumeration of the group), i.e., T = xU U x ... x U for some set U C R. In this case, g, is symmetric and separable and the results of Section 3 apply. In particular, the problem of finding an optimal grouping can be reduced to a shortest path problem. We next consider modifications of this model in which costs are discounted, based on monetary volume. Specifically, we consider various discounting functions that transform undiscounted costs into discounted costs. A reasonable property of such functions is concavity, which represents higher discounting (on marginal cost) as the monetary volume increases. We consider two main forms of discounting. In the first, costs of each group are discounted independently, based on the undiscounted costs associated with each particular group. In the second, discounting is based on total cost at each particular period. Throughout, as before, for each item i, we use the notation a = 2'hl-Di and bi = Ki, where hi is the undiscounted unit inventory holding cost per period, Di is the (deterministic) demand rate and Ki is the undiscounted cost of placing an order. We first consider the case in which costs associated with each particular group are discounted independently. Let f() be the discounting function of the jth group for the cost of placing an order. Then, if Si is the jth group, the corresponding discounted cost for placing an order is f(EEs K) = f(bsj). We next consider two forms of discounting the holding costs: contractual and periodic. Under contractual discounting, the undiscounted holding cost for the entire order cycle is discounted. Let e() be the discounting function of the jth group for the holding cost for the entire order cycle. Then, if Sj is the jth group and r is the order cycle, the corresponding discounted holding cost is

e'If [is,

(

)Dhi] d

-

= er

2asj).

Under periodic discounting, the

undiscounted holding cost at each period is discounted separately. Let e'(*) be the corresponding discounting function of the jth group. Then, if Si is the jth group and r is its order cycle, the corresponding discounted holding cost is :p Iv-"

w0

'9t

j ei[Esj (r - )Dih d

=

e[2(T -

)as;] d.

Consequently, if the items are partitioned into groups SI, .., S, having order cycles t, ., ti, respectively, the total average cost per unit time under contractual discounting of holding cost is cm (S, , SM, ti, .., t) = Y= t [e(t 2as) + fi(s)l, and under periodic discounting of holding cost is Cm(S,,

', Sm, t,

..

tin) = E

tl'

{f

e[2(Tr

-

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Chakravartyet al.

For a fixed partition {S, ., Sm} of the items, the minimum average cost per unit time, gm(S5, Sm), is given by (2). Assume that each function e'(®) and fj(-) is nonnegative and concave. Then, the infimum defining g(*) is finite for each partition. Also, cm(Si, , Sm, tl, ', tm) is concave in as1, bs,, , * as,,, bs, and therefore so is gm(S 1, ., Sm) (as the infimum of concave functions). Thus, the assumptions of Theorem 1 are satisfied. In particular, our results apply and there exists an optimal partition consisting of consecutive sets. We note that if Ki is independent of i, Theorem 3 assures that these conclusions hold even when the functions fi(.) are not concave. We next consider the case in which costs are discounted at each period, based on the total corresponding undiscounted costs at that period. In this case, we cannot compute discounted cost for each particular group. Alternatively, discounted costs must be calculated with respect to a given partition, say I{S, , Sm,, and corresponding ordering cycles, say tl, .. , t. For simplicity, we assume that t, ., t are integers. Let t = (t, ... , t,) and let q(t) be the least common multiple of t, . , t,. We will next calculate the average cost per unit time over a partition cycle consisting of q consecutive time periods. Without loss of generality, we assume that the partition cycle starts in period zero. For p = 0, ... , q(t) - 1, let J(p, t) be the set of all indices j E {1, .. , ml such that an order for the items in group j takes place in period p. Let f(*) be the discounting function of the cost of placing an order. Then, the discounted costs of placing all orders that are due in period p is '',

i

i

r j

i 7 ii

-··

I

f[IEJ(pt (iESj

Ki)] = f(jEi(p,t) b).

We will next consider, as before, both contractual and periodic discounting of holding costs. Let e be the corresponding discounting function. Under contractual discounting of holding costs, the corresponding discounted holding cost in period p is

·· i

s

2-'t,2 Dih)] = e(IEJ(p.,t) ti 2as).

e[jEj(p,t) (iEs

Alternatively, under periodic discounting of holding costs, the corresponding discounted holding cost in period p is ..... ...,.....,

'· .i

etEJ

1

es[tjt -

(p)]Dhj = e=

1

2[tj -

:t p, t)]as},

where ,(p, t) is last ordering period of items in the jth group at or before periodp. Therefore, the total average cost per unit time under contractual discounting of holding cost is Cm(S 1 , ,

i

Sm, tl,

.

.,

tm)

q(t)-{1$p(to [e(XyCJ(p.t) t 2as) + f(XJEJ(p,t) bsj)]}

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Consecutive Optimizers for a Partitioning Problem

t I-1 II

827

and under periodic discounting of holding costs, it is Cn(Si,

Sm, t

',

', tin)

= q(t)t-'jq;t'

e{_- 2[tj- gi(p, t)Jasj +

f(JEJ(p,t)

bS)}l.

t.r

If the functions e(') and f(') are nonnegative and concave, c(Sl, * X

i.

4' "

Sm, tl,

, asm, bs,,

tm) is concave in as,,

',

,

,

bs, and our previous

arguments assure that there exists an optimal partition consisting of consecutive sets. We note that there are many variants of these models under which our conclusions still hold. For example, the discounting functions can be

allowed to be time dependent over each cycle. Also, the holding costs and

...

the ordering costs need not be discounted independently, i.e., the total cost need not be the sum of the discounted holding cost and ordering

cost, but can be given by alternative (concave) functions of the undis. .

,.

4 4

.

counted holding costs and ordering costs. Finally, the discounting func-

tion of ordering cost can be used to introduce a group ordering cost that is independent of the individual items' ordering costs (and is imposed on to the top of them). Chakravarty (1983) considered an alternative model in which purchasing costs of the different items are discounted, and the holding cost per dollar value per unit time is the same for all items. Let Vi be the

undiscounted unit price of the ith item, let Di be the (deterministic) demand rate of item i, let Ai be the ordering cost associated with item i, and let h be the holding cost per dollar value per unit time. Also, let ai = V1D and b = K, where the items are labeled so that (1) holds. Then,

the net cost (consisting of purchasing costs, holding costs, and ordering costs) over an order cycle of a group S of items having the same order cycle c(S,

is ) =

2 .+

ies (ViDir + 2-'rVDir

Ai) = as + 2-Ihr 2 as + bs.

Discounting the purchasing costs, holding costs and ordering costs can now be introduced correspondingly. In particular, our earlier analysis can be used to argue the existence of optimal grouping in which each set is consecutive. ·:

-·

~~ ·

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4

4. APPLICATION OF SEMICONSECUTIVENESS TO A LOCATION PROBLEM Consider the problem in which each of n points, say (a, bl), ... , (a,, b), must be assigned to one of the lines in a set L of n lines all passing through the origin, so as to minimize the sum of weighted distances between the points and the lines to which they are assigned.

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Chakravarty et al.

828 F; i i

We assume that the set L of lines is to be selected from a collection I of sets of lines. One motivation for this problem is the assignment of customers to subway lines, all of which pass through a central location, e.g., downtown. This model is a variant of the one considered by Megiddo and Tamir (1983) in which the decision maker selects only a single line. Let the points (a, bl), ., - (a,, b,) be ordered so that (1) holds. Also, for j = . m,, let the jth line in L be given by the equation ajx + jy 0, where aj2 + fij2 0, and let uj be the weight corresponding to distances from line j. Without loss of generality, assume that a12 + fif = 1. The distance between a point (a, b) and the line ax + y = 0 is known to be (a2 + l2)-1 /2 1aa + b I Among the points assigned to line j are points that lie on or above the line, i.e., atai + fjbi < 0, and others which lie below it, i.e., ajai + Fbi < 0. Consider a partition S,, T 1, .. ., Sk, T] of the points where for J = 1, ... , m, Sj and Tj are, respectively, the points that are assigned to the jth line and lie on or above it and those that lie below it. The sum of the corresponding weighted distances from lines in L associated with such a partition is

·I

i :1 1 1

I ···--······ ··''" """ '· ······-' · ···· `"I f·· r ·j r

i .'

b j r ·I 3 j ·ii b ;. t i' 1 j r

gL(S, T 1, ... , Sk, T) = X-_= wi[E s, (a/ai + fjb)t- Xi8r (aja¢ + flbi)]

= X-= wi[(a,as, + flbs) - (aar+ fbbTrj). Evidently, this function is linear in as1, bs,, a T ,) bT, ... , a, bs, Thus, the minimum distance to any of the lines in I is

i f i I i 1

inf(gL(S 1 , T 1, ...

which is concave in As, bsl, aT 1 , bTr,

...

,

bT.

aT,

Sa, Ta))

, as.,

bs,. Therefore, by Theorem

2, there is an optimal partition that is semiconsecutive. 5. THE NP-COMPLETENESS OF AN OPTIMAL PARTITION PROBLEM Barnes and Hoffman considered an optimization problem which can be cast as the following recognition problem.

r

Barnes-Hoffman Partition Problem INPUT: Nonnegative integers m, n, dj,

·;

,.:·,...-

:·

··.· I..·.

= n, and integers a,

...

,

... ,

dm satisfying d, +

+ dm

..-

an and K.

QUESTION: Is there a partition of {1, ... , n} into m subsets SI, so that IS I = di for i = 1, - m and so that

...

, Sm

X 1 d 1 (ies, a)2 > K? Barnes and Hoffman demonstrated that, for every "yes" answer to this question, there is always a feasible partition consisting of consecutive subsets. Moreover, since they were interested in problems for which

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829

Consecutive Optimizers for a Partitioning Problem

m = 3, enumeration of all feasible consecutive partitions was quite practical. The Barnes-Hoffman partition problem satisfies the conditions of Theorem 3 (except for the fact that they maximize a convex function

I

rather than minimize a concave function). Thus, the optimality of con-

-t

secutive partitions is a special case of Theorem 3. Although the Barnes-Hoffman problem is polynomially solvable for fixed m, the corresponding recognition problem is NP-complete if m is allowed to vary jointly with n. We next prove the NP-completeness via a transformation from 3-partition. -r

3-Partition INPUT: Nonnegative integers bl,

3k, and such that b, +

..

.. ,

b3k satisfying bi < k 2 for i

1,

+ b3k is divisible by k.

QUESTION: Is it possible to partition {1,

... ,

3k} into k subsets SI,

...

Sk so that Zies b = (l/k)

1,

2% bi- for ji

.,k?

We consider the version of 3-partition in which the subsets are not specified to have three elements each. Garey and Johnson 1978] showed that the above variant of the 3-partition problem is NP-complete.

I

THEOREM 4. The Barnes-Hoffman partitionproblem is NP-complete. 6. PROOFS

.*

Proof of Theorem 1. We consider only the case where b > 0, since the remaining cases follow from analogous arguments. Our proof follows by induction on the number m of subsets in the partitions. The case m = 1 is trivial. We next consider the case m = 2. We first remark that the case in which a = b = 0Ofor some i can be ignored. Noting that each 2partition can be written as IS, N\S} for some set S C N and that for such -r

, aN\s = an - as and

bNS = bN -

b

we can write the optimal

partitioning problem with m = 2 as the following optimization problem: bs, aNvs, bNs) minscNvg 2 (S, N\S) = minscNh (as, 2 = minscNh 2 (as, bs, a

bs).

- a s , b-

(3)

Since the set of ordered pairs (as, b): S C NJ is finite, its convex hull, which will be denoted by C, is a convex polyhedron. It now follows from (3) that .*

minscNg2(S, N\S) = minScNh 2(as, bs,

aN-

->min(xy)ech 2(x, Y, aN-

'9

as, bN -

x, bN - Y).

..

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830

Chakravarty et al.

Since the function

h 2 (x,

y,

AN -

x, bN -

) is concave in the variables

(x, y), the latter minimization problem attains a minimum at an extreme point of C. It therefore suffices to show that, for each extreme point (x*, y*) of C, there exists a set S* C N such that (x*, y*) = (as, bse) and [S*, N\S* is a consecutive partition. Let (x*, y*) be an extreme point of C. Since C is a polyhedron, it is

well known that there exists a linear function f on C that attains its

unique minimum over C at (x*, y*). Since f is linear, it has a represen-

tation f(x, y) = ax +

By

defined by real numbers a and F. The uniqueness

of (x*, y*) as the minimizer of f over C assures that we do not have

''''"''''L''''

a

=

f = 0. Next observe that as C is the convex hull of f(as, bs): S C Nt, ax* + fy* = minx,y)Eccax +

= minscNaas +

y

bs = minscN Xies (aai + fbi).

The set S* = i: aai + bi < 01 is clearly optimal for the last optimization problem. Hence, we conclude from the uniqueness of (x*, y*) as the minimizer off over C, that (x*, y*) = (as., bs.).

We next show that the fact that b > 0 assures that both S* and N\S* are consecutive. We consider a number of cases separately. If a > 0, then S* = {i: ai/bi < -/aI and if a < 0, then S = i: a/bi > -fi/a}. In either case, (1) assures that both S* and N\S* are consecutive. Finally, if a

=

0, then f

0, and S* = i: bi < 0} = 0 if

> 0 and S* = i: b> 0}

if f < 0. In either case, we trivially have that both S* and N\S* are consecutive. This completes our proof of Theorem 1 when m = 2. We next consider the case where m 3. For each subset S of N, let max S = max{i: i E S, min S = minfi: i S and d(S) = max S - min S. Let P = S1, .. , S,) be an optimal m-partition of N that minimizes E=1 d(Sj) with respect to all such (optimal) partitions. We claim that for each j = , .·. , m, Sj is consecutive. We see this result as follows. Suppose that not all subsets are consecutive. Then we may determine two distinct sets of P, say S, and S2, such that for some i E S2, min S < i < max SI. For j = 1, 2, let mj = min Sj and M = max S. Observe that minlM, M 2} > i and maxm,, m2} i. Hence, minf{M, M2} - maxIm1 , m2)

.. .

>

0.

(4)

Let {S,', S2'} be an optimal partition of S U S into two subsets that are consecutive with respect to S U S2, where S3, , Sm remain fixed (as is possible by the established result of Theorem I when m = 2 and the observation that h is concave in the variables corresponding to S, and S2 when the remaining variables are fixed). Then

d(S,') + d(S21) s maxIM1, M

~ ~ ~ ~ ~ ~~ ~ ~ ~ ~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~1-

_~~~~~~~ __II_ CI_I __

l

2

- min{ml, m2} - 1

831

Consecutive Optimizers for a PartitioningProblem

and, therefore, using (4), we conclude that

d(S )1 + d(S 2) = M - ml + M2 - m2

I

= max{M1, M21 + minlMi, M2t - minIm1 , nm2

-

max{ml,

mn2

> d(S1') + d(S2') + 1.

This conclusion contradicts the choice of S, proof.

Proof of Theorem 2. Our proof follows by induction on the number m of subsets in the partitions. The case m = 1 is trivial. We next consider the case m = 2. The argument used in the proof of Theorem 1 shows that it suffices to prove that for every pair of real numbers a and A,the set

..

.

, Sm and completes our

+

,~~~~

S* = i: aaj +

Obi

< 0 has the property that S*, N\S*1 is a semicon-

secutive partition of N, or equivalently, that either S* or N\S* is a consecutive set. We consider a number of cases separately. If a > O,then S* = i = 1, . ., r: ai/bi > -/a U i r + 1, . , n ailbi < -/a} and if a < , then S* = i I, ..., r: alb < -/a} U i = r + 1, ., n: ail/bi > -/a). In the former case, (1) assures that S* is consecutive, and in the latter case, () assures that N\S* is consecutive. Hence, in either case, the partition S*, N\S*} is semiconsecutive. Finally, if i

a =0,

7

and S* = {i: bi > 0I if < O0. In either case, we have that S* is consecutive (though in the latter case N\S* is not consecutive if r > and for some i, a > = bi) and, therefore, the partition S*, NJ\S*l is semiconsecutive. Next assume that the conclusions of Theorem 2 hold whenever the number of sets in the partitioning problem is less than m ( 3), and consider partitioning problems with m sets. We prove the existence of a semiconsecutive optimal partition for the corresponding partitioning problems by induction on the number of elements in the partitioned set N. The cases where the number of elements in N is 1, 2, or 3 are straightforward. Next assume that each partitioning problem of a set consisting of less than n elements into mn sets, where the assumptions of Theorem 2 are satisfied,.has a semiconsecutive optimal partition. We next consider such partitioning problems of a set N with n elements. Let r be defined by (1). If r = Oor r n, then Theorem I1assures the existence of a consecutive optimal partition. Henceforth, assume that 1 r c n. For each subset S C N, let S- = S n 1, -, r and S + = S n r - 1, · , n}. Let S,, ... , Smn} be a given optimal partition of N. We continue our inductive step by first considering the following two

4

then

0. Moreover, S = i: b < 0} = 1, ...,

cases. (a) IS?I > 2 or ISj- I - 2 for some A* 51

(b) S = I)} or S = In} or S i = 11, n} for some j

E 1, E

_

...,

> 0

m, and

11, .. , m. We will

·dB

__

r} if

______________L_--

-

i .. 4 I .J id ,I

i

1I

.

····:· ,.....,...,..I... L...·

.

.

..

_1

:t :r cf

Chakravarty et al.

832

show that, in either of these two cases, we can find an alternative optimal partition which is semiconsecutive. We will then complete our proof by showing that the remaining case can be transformed into one that satisfies either (a) or (b). We first consider case (a). We will assume that Sjl1 2 for some j E II, .. , m}; similar arguments apply to the case where j Sj- I - 2 for some j E {1, ... , mi. We first show that, without loss of generality, we can assume that Si, is consecutive. Assume that there are k nonempty sets among Si +, , S +. In particular, I N I > k + 1. Hold S1-, S,- fixed as well as each Si- which is empty, and apply Theorem I to repartition No into a corresponding consecutive optimal partition having k sets. We thereby obtain an alternative optimal partition of the original problem, say IS,, , Sm, such that S 1+t ... , Sr+ are consecutive, Si Si for i E {1, ... , m} and Si+ = Si+ whenever Si = 0. In particular, since IN+ _ k + 1, ISj+ > 2 for some j E {1, ... , m}. Let T Si+. Then all the elements in T can be combined into a single element to which we attach an "a" value aT and a "b" value br. Evidently, since T C N +, we have that if a s ai/bi s A for each i E T, then a S aT/br s ATConsequently, the corresponding partitioning problem where the elements in T are combined into a single element satisfies (I) and the other assumptions of Theorem 2. Since the number of elements in the partitioned set of this modified problem is less than n, we conclude from the induction assumption that a semiconsecutive optimal partition exists. As an immediate consequence, the original problem also has a semiconsecutive

..

·

· :· ·· . -· ··- ·. . ····- ·.

r · · ······

optimal solution. We next consider case (b) in which for some j E I1, ..- , m, either Sj= lI1, Sj-= Inl or S = 1, n}. In this case, we can hold Sj fixed and, by the induction assumption, repartition N\Sj into a corresponding semiconsecutive optimal partition having m - sets. Therefore, if we add S. to this partition, we obtain a semiconsecutive optimal partition of N into m sets. Finally, we consider the case in which neither (a) or (b) hold. Suppose that 1 E S and n E S. Since 1 r < n, we have that 1 E Sp- and n E Sq+. Moreover, since neither (a) nor (b) hold, we have that Sp- = I1}, Sq+ = n} and Sp and S,- consist of exactly a single element each. By the established conclusion of Theorem 2 for the case m = 2, we can find a semiconsecutive partition Sp', Sq'} of Sp U Sq, such that {S: i E I1, ... , m}\ p, q}} U Sp', Sq'} is optimal. It is easy to verify that this new partition must satisfy either (a) or (b). This result completes our inductive proof. Proof of Theorem 3. Chakravarty et al. established a special case of Theorem 3 in which g,(') is symmetric and separable. Their arguments apply to the general case considered in Theorem 3 and are omitted. To prove Theorem 4, we first state and prove the following lemma.

ii ----3--)"CC---9-··cl·14arraa9aaral--

sr_

__

Icgsp--s

-----

-I-------- ----1

Consecutive Optimizers for a Partitioning Problem

833

LEMMA . Let a, , a,, d, , d,, and K be data for the BarnesHoffman partitionproblem such that K = E (ai2 : 1 i n). Then, there is a feasible partition if and only if there is a partition S, ., Sm such that ai = at for all i, t in the same subset S'. Proof. By the Cauchy-Schwarz inequality (Xies, ai) 2 - I Sj I Zis, a?

and equality holds if and only if a = a for all i, t E S. Thus,

81 (lies, ai)2

O .

.." ._.,... _.... ""' ? ··--C

< X,

a,2

i, t

and equality holds if and only if a1 = at for every two indices

·I "

in the

same subset. Proof of Theorem 4. Let b, .. , b3K be the input for the 3-partition problem and let n = Y3k bhi, and let 7= k-'n. Let m = 3k and let di bi for i = 1, ..., 3k. Finally, let a+jg = j for 1 < i c and j