DOI 10.1007/s10898-006-9025-0 O R I G I NA L PA P E R
The mean-partition problem
Fei-Hwang Chang · Frank K. Hwang · Uriel G. Rothblum
Received: 4 July 2005 / Accepted: 13 March 2006 / Published online: 4 July 2006
© Springer Science+Business Media B.V. 2006
Abstract In mean-partition problems the goal is to partition a finite set of elements, each associated with a d-vector, into p disjoint parts so as to optimize an objective, which depends on the averages of the vectors that are assigned to each of the parts. Each partition is then associated with a d× p matrix whose columns are the corre-sponding averages and a useful approach in studying the problem is to explore the mean-partition polytope, defined as the convex hull of the set of matrices associated with feasible partitions.
Keywords Partition Problems· Combinatorial Optimization · Means
1 Introduction
Consider a finite set N = {1, . . . , n} where each element i in N is associated with a vector Ai∈ Rd. A partition of N is a finite ordered collectionπ = (π1,. . . , πp) where π1,. . . , πpare disjoint sets whose union is N. In this case, p is called the size ofπ, π1,. . . , πpare called the parts ofπ and π ≡ (|π1|, . . . , |πp|) is called the shape of π.
A p-partition is a partition of size p. Throughout, we assume that p, n and A1,. . . , An are given.
In a constrained-shape partition problem, one is to select a partition π of N whose shape is in a given set of integer p-vectors with coordinate-sum n so as to maximize an objective function F(.) that is defined over partitions. Special cases
F.-H. Chang· F. K. Hwang
Department of Applied Mathematics, National Chiaotung University, Hsinchu, Taiwan, ROC 300.
U. G. Rothblum (
B
)Faculty of Industrial Engineering and Management, Technion—Israel Institute of Technology, Haifa 32000, Israel
Fax: 972-4829-5688
include single-shape, bounded-shape and size partition problems in which the set consists of a single vector, is defined by lower and upper bounds and is unrestricted.
Partition problems are further classified by their objective function F(.). For a subset S of{1, . . . , n}, let
AS= i∈S
Ai∈ Rd (1.1)
and for a partitionπ = (π1,. . . , πp) , let
Aπ = (Aπ1,. . . , Aπp) ∈ Rd×p. (1.2)
A predominant class of partition problems is the sum-partition problem in which
F(π) = f (Aπ), (1.3)
where f(.) is a real-valued function on Rd×p. A useful approach in addressing partition problems in this class is to study the corresponding sum-partition polytope defined to be the convex hull of the Aπs, withπ ranging over the set of feasible partitions. The sum-partition polytope corresponding to a set of partitions is denoted P. If f is guaranteed to attain an optimum over Pat a vertex of that polytope then there must exist an optimal partitionπ with Aπ being a vertex of the P(see [10] for a sufficient condition for the optimality of vertices, which generalizes the classic conditions of convexity and quasi-convexity). In such cases, it is useful to identify properties of partitionsπ for which Aπis a vertex of the sum-partition polytope. For d= 1, Hwang et al. [9] gave an explicit solution of the bounded-shape sum-partition problem when
f is Schur convex and a majorization shape exists. Chang et al. [5] extended the result
to the general case that no majorizing shape exists; see, Sect. 4.
We next introduce the mean-partition problem, which is the subject of the current paper. For a nonempty subset S of{1, . . . , n}, let
¯AS= 1
|S|
i∈S
Ai∈ Rd (1.4)
and for a partitionπ = (π1,. . . , πp) (with nonempty parts) let
¯Aπ = ( ¯Aπ1,. . . , ¯Aπp). (1.5)
We next consider the mean-partition problem, which is the class of partition problems with
F(π) = g( ¯Aπ), (1.6)
where g(.) is a real-valued function on Rd×p; with d = 1, this problem was first explored by Anily and Federgruen [2]; see, Sect. 2 for details about their results. Also, the study of the mean-partition problem motivated Chang and Hwang [4] to study the supermodularity property of a function related to the mean-partition polytope; see, Sect. 3.
As in the case of the sum-partition problem, given a set of p-partitions , the
mean-partition polytope ¯Pis defined as the convex hull of{ ¯Aπ :π ∈ }. And when
g is guaranteed to attain an optimum over ¯P at a vertex, there exists an optimal partition with ¯Abeing a vertex of the ¯P. It is then useful to identify properties of partitionsπ for which ¯Aπis a vertex of the mean-partition polytope.
The purpose of the current paper is to study the mean-partition problem. Our first result (in Sect. 2) is the observation that the single-shape mean-partition prob-lem can be reduced to a corresponding sum-partition probprob-lem. We use the reduction as a tool for deducing properties of optimal partitions of mean-partition problems from corresponding results about sum-partition problems. We also observe that every single-shape mean-partition polytope is the image of a corresponding sum-partition polytope under a one-to-one linear transformation with the sets of partitions corre-sponding to the vertices of the two polytopes coinciding. Consequently, properties of partitionsπ with Aπ being a vertex of Phold for partitions with ¯Aπ being a vertex of ¯P. While the above tools do not extend to bounded-shape problems, the exis-tence of properties of optimal partitions of single-shape partition problems extends to bounded-shape problems.
In Sect. 3, we explore the single-shape mean-partition polytopes with d= 1. We derive an explicit representation of these polytopes and review alternative approaches to address the mean partition problem.
In Sect. 4, we continue the examination of mean-partition problems with d = 1 and establish geometric properties (reverse size-consecutiveness) for optimal parti-tions of single-shape problems, providing more structure of optimal partiparti-tions than is obtainable from the transformation approach. For the bounded-shape problem, we are able to shrink the set of consecutive partitions and still preserve the existence of an optimal partition in the shrunk set.
2 Reduction of single-shape mean-partition problems to sum-partition problems
We start by recording the observation that single-shape mean-partition problems are reducible to corresponding sum-partition problems.
Lemma 2.1 Let n1,. . . , np be positive integers whose coordinate-sum is n. Then the single-shape mean-partition problem with prescribed-shape(n1,. . . , np) and objective function given by(1.6) coincides with the corresponding sum-partition problem with objective function given by(1.3) where f satisfies
f(x1,. . . , xp) = g( x1 n1 ,. . . ,xp np) for x ∈ R p. (2.1) Lemma 2.1 implies that properties of optimal solutions for single-shape mean-partition problems are deducible from properties of optimal solutions of correspond-ing sum-partition problems. For example, it is known that when the Ais are distinct,
every single-shape sum-partition problem with f (quasi-) convex has at least one
disjoint optimal partition, that is, an optimal partition for which the convex hulls of
the vectors Ai corresponding to distinct parts are disjoint (see [3]); further, the set of disjoint partitions has at most O[nd(p2)] partitions and these can be enumerated in
polynomial time (see [1] or [8]). These results establish the polynomial solvability of the single-shape sum-partition problem when the function f is (quasi-) convex. (For the relaxation of the assumption that the Ais are distinct see [8].) Now, as a function g is (quasi-) convex if and only if so is the function f that is defined through (2.1), we
Corollary 2.2 Suppose the Ais are distinct. Then every single-shape mean-partition problem with objective function given by(1.6) where g is (quasi-) convex has at least one disjoint optimal solution, and such problems are solvable in polynomial time.
The following standard observation allows one to extend conclusions about the presence of (geometric and combinatorial) properties in optimal partitions from single-shape to constrained-shape problems [see, Lemma 1 of Golany et al. (2005, submitted)]: Consider a cost function F over p-partitions and a property Q of p-par-titions such that for each single-shape mean-partition problem with cost function F,
Q is satisfied by some (every) optimal partition. Then, for every constrained-shape
partition problem with cost function F, Q is satisfied by some (every) optimal partition. The above arguments combine with Lemma 2.1 to show that any property that is present in optimal solutions of single-shape sum-partition problem, is present in optimal solutions of corresponding constrained-shape mean-partition problems. But, these conclusions cannot be reached by using (2.1) to map constrained-shape mean-partition problems onto corresponding sum-mean-partition problems. The above observa-tion is demonstrated in the next corollary.
Corollary 2.3 Suppose the Ais are distinct. Then every constrained-shape mean-partition problem with objective function given by(1.6) where g is (quasi-) convex has at least one disjoint optimal solution. Further, assuming efficient (that is, polynomial) verifiability of the shape-constraints, such problems are solvable in polynomial time. Proof The existence of disjoint optimal partitions follows from the observation
pre-ceding the statement of the corollary and Corollary 2.2, and the polynomial solv-ability follows from the polynomial enumersolv-ability of disjoint partitions (established
in [8]).
Anily and Federgruen [2] studied the bounded-shape mean-partition problem for
d= 1 under the objective function f (π) = pi=1h( ¯Aπ, ni). They proved that if for each ni, h(X, ni) is convex and nondecreasing in X, then there exists a disjoint optimal
partition. Their result follows from the above discussion with d= 1 and with f (π) as a special type of (quasi-) convex function. We note that with stronger assumptions on h(X, y), Anily and Federgruen obtained additional, tighter, results, which are not available from our approach.
We next go back to single-shape problems and record an isomorphism between single-shape mean-partition polytopes and the corresponding sum-partition poly-topes. With n1,. . . , np as the given positive integers, let Pbe the set of partitions
with shape(n1,. . . , np) and let Dn1,...,npbe the p×p diagonal matrix whose diagonal
ele-ments are, respectively, n1,. . . , np. For every partitionπ ∈ , ¯Aπ = (Anπ11 ,. . . ,A
πp np ) = (Dn1,...,np)−1Aπ, and therefore
¯P= conv{(Dn1,...,np)−1Aπ :π ∈ } = {(Dn1,...,np)−1X : X∈ P}. (2.2)
Thus, the one-to-one linear transformation
X= (X1,. . . , Xp) → (Dn1,...,np)−1X= X1 n1 ,. . . ,X p np (2.3) maps Ponto ¯P, that is, the single-shape mean-partition polytope is the one-to-one linear image of the corresponding single-shape sum-partition polytope. A virtue of
this transformation is that it preserves vertices. Consequently, any bound on the num-ber of vertices of Pis a bound on the number of vertices of ¯Pand any algorithm for generating the vertices of Pcan be used to generate the vertices of ¯P.
3 Mean-partition polytopes with d= 1
In this section, we consider the mean-partition polytope with d= 1. We review known results about single-shape sum-partition polytopes and show how these are transform-able to mean-partition polytopes by the one-to-one transformation outlined in Sect. 2. We also review difficulties in the direct simulation to mean-partition problems of the approach that has been implemented successfully for the sum-partition problem. Fol-lowing standard notation, we use the notationθ1,. . . , θnfor the scalars A1,. . . , An, respectively.
Gao et al. [6] developed an effective approach to study sum-partition polytopes with d= 1 by deriving explicit representations for the corresponding sum-partition polytopes through systems of linear inequalities (in fact, [6] considers only single-shape problems and the general case is developed in Hwang and Rothblum [11]). To present the approach, let be a set of p-partitions and consider the real-valued functionλ∗ on subsets of{1, . . . , p}, where for each nonempty I ⊆ {1, . . . , p}
λ∗(I) = min π=(π1,...,πp)∈ j∈I θπj (3.1) andλ∗(∅) = 0; in particular, λ∗({1, . . . , p}) =nj=1θj.
A real-valued functionλ over subset of {1, . . . , p} is used to define two polytopes. First, Cλis defined to be the set of vectors x in Rpsatisfying
i∈I
xi≥ λ(I) for all I ⊆ {1, . . . , p} (3.2) and
p
i=1
xi= λ({1. . . . , p}). (3.3)
Also, for each permutationσ = (σ1,. . . , σp) of {1, . . . , p} and k ∈ {1, . . . , p}, let jσ(k)
denote the index for whichσjσ(k)= k and let λσbe the vector((λσ)1,. . . , (λσ)p) where
for each k= 1, . . . , p, (λσ)k= λ({σ1,. . . , σjσ(k)}) − λ({σ1,. . . , σjσ(k−1)}). With as the
set of permutations of{1, . . . , p}, the second polytope corresponding to λ, denoted Hλ, is defined as the convex hull of{λσ:σ ∈ }. The function λ is called supermodular if for every pair of subsets I and J of{1, . . . , p}, λ(I ∪ J) + λ(I ∩ J) ≥ λ(I) + λ(J). Shapley [13] proved that when λ is supermodular, Hλ = Cλ and the λσs are the vertices of this polytope.
Given a set of p-partitions and the corresponding function λ∗ defined by (3.1), it is immediately verified that P⊆ Cλ∗. The set of partitions is called consistent if for every permutationσ of {1, . . . , p} there exists a partition π in with θπ = (λ∗)σ. It follows immediately from the definition that Hλ∗ ⊆ Pwhenever is consistent; further, it is proved in [11] that consistency of implies that λ∗ is supermodular, implying that in this case Hλ∗ = P = Cλ∗. The equality P = Cλ∗ provides a
representation of the sum-partition polytope Pas the feasible set of a correspond-ing system of linear inequalities. And the equality P= Hλ∗, together with Shapley characterization of vertices of the polytope corresponding to a supermodular function, provide a characterization of the vertices of the sum-partition polytope Pthrough the p! (λ
∗)σs. Examples of consistent sets of partitions include single-shape problems and bounded-shape problems where theθis are one-sided, that is, either nonnega-tive or nonposinonnega-tive; see [11]. We recall from [7] that for any bounded-shape set of partitions, λ∗ is supermodular, implying that Hλ∗ = Cλ∗(⊇ P); but, for
arbi-trary bounded-shape problems, without the assumption that theθis are one-sided, the supermodularity ofλ∗ does not imply the representation P= Cλ∗.
We next combine the representation of single-shape sum-partition polytopes through (3.2)–(3.3) with the transformation (2.3) to obtain a representation of sin-gle-shape mean-partition polytopes. We also obtain a representation of the vertices of such polytopes. For the latter, recall that for each permutationσ = (σ1,. . . , σp) of
{1, . . . , p} and k ∈ {1, . . . , p}, jσ(k) denotes the index for which σjσ(k)= k.
Lemma 3.1 Let n1,. . . , npbe positive integers whose coordinate-sum is n and let be the set of partitions with shape(n1,. . . , np). Then ¯Pis the set of vectors y∈ Rpthat satisfy
i∈I
niyi≥ λ(I) for all I ⊆ {1, . . . , p} (3.4)
and
p
i=1
niyi= λ({1, . . . , p}). (3.5)
Further, the vertices of ¯P are available from the p! permutations of {1, . . . , p} with permutationσ corresponding to the vector vσ having
(vσ)k= 1
nk[(λ
∗)({σ1,. . . , σjσ(k)}) − (λ∗)({σ1,. . . , σjσ(k−1)})] for k = 1, . . . , p.(3.6) Proof Let Dn1,...,npbe the p× p diagonal matrix whose diagonal elements are,
respec-tively, n1,. . . , np. We observe from (2.2) that
¯P= {(Dn1,...,np)−1x : x∈ P} = {y : (Dn1,...,np)y ∈ P}. (3.7)
Using the representation of Pthrough (3.2)–(3.3) we get the representation of ¯P
as the set of vectors y ∈ Rp that satisfy (3.4)–(3.5). Finally, the representation of the vertices of ¯Pfollows from the representation of the vertices of the single-shape sum-partition polytope P mentioned earlier, and that observation made in Sect. 2 that the transformation x→ (Dn1,...,np)−1x (described in (2.3)) maps vertices of P
onto vertices of ¯P.
Chang and Hwang [4] tried to develop a direct approach for studying the mean-partition problem with d = 1. Given a set of partitions , with no partition in having empty parts, consider the real-valued function ¯λ on subsets of {1, . . . , p} where for each nonempty I⊆ {1, . . . , p}
¯λ ∗(I) =π=(πmin 1,...,πp)∈ j∈I ¯θπj (3.8)
and ¯λ∗(∅) = 0. Chang and Hwang [4] proved that for a single-shape set of partitions
, ¯λ
∗ is supermodular; it then follows from the result of [13] mentioned in the
Intro-duction that H¯λ∗ = Cλ¯∗. But, the following example demonstrates that this polytope can be different from ¯P.
Example 3.1 Let n = 3, θ1 = 1, θ2 = 2, θ3 = 3, p = 2 and consider the
mean-partition problem corresponding to the set of partitions with shape (1, 2). The set contains the three partitions ({1}, {2, 3}), ({2}, {1, 3}) and ({3}, {1, 2}) whose cor-responding vectors are, respectively, (1, 2.5), (2, 2) and (3, 1.5). The mean-partition polytope is then the line-segment connecting(1, 2.5) and (3, 1.5). Also, we have that ¯λ
∗({1}) = 1/1 = 1, ¯λ∗({2}) = 1 + 2/2 = 1.5 and ¯λ∗({1, 2}) = min{1/1 + 2 + 3/2 =
3.5, 2/1 + 1 + 3/2 = 4, 3/1 + 1 + 2/2 = 4.5} = 3.5. So, C¯λ∗ is the polytope defined by
the inequalities x1≥ 1 , x2≥ 1.5 , x1+x2= 3.5, that is, it is the line-segment connecting (1, 2.5) and (2, 1.5). Finally, the two permutations (1, 2) and (2, 1) of {1, 2} correspond,
respectively, to the vectors(¯λ∗)(1,2) = (¯λ∗({1}), ¯λ∗({1, 2}) − ¯λ∗({1}) = (1, 2.5) and
(¯λ
∗)(2,1) = (¯λ∗({1, 2}) − ¯λ∗({2}), ¯λ∗({2})) = (2, 1.5), and H¯λ∗ is the line-segment
connecting these points (see Fig. 1 for an example of ¯P, C¯λ∗ and H¯λ∗). Notice that the equality C¯λ∗ = H¯λ∗ is consistent with the conclusion of [4].
One explanation for C¯λ∗ being different from ¯Pis the fact that the
coordinate-sums of the points in the mean-partition polytope need not be constant, hence, it seems natural to relax the equality constraint (3.3) in the definition of C¯λ∗. But, such a relax-ation will typically result in an unbounded polyhedron (consider Example 3.1). With the goal of augmenting the constraints of (3.2) with upper bounds on the variables, consider the real-valued function on subsets of{1, . . . , p} defined by
(¯λ∗)(I) = max
π=(π1,...,πp)∈
j∈I
¯θπj for each I⊆ {1, . . . , p} (3.9)
(with(¯λ∗)(∅) = 0). Further, let Kbe the set of vectors x in Rpthat satisfy
(¯λ∗)(I) ≤ i∈I
xi≤ (¯λ∗)(I) for all I ⊆ {1, . . . , p}. (3.10)
Evidently, Kcontains the corresponding mean-partition polytope ¯P. But, the fol-lowing continuation of Example 3.1 demostrates that the inclusion may be strict.
Example 3.1 (Continued) Reconsider the data of Example 3.1. The modification of C¯λ∗ obtained through the relaxation of (3.3) is the (unbounded) polyhedron, which Fig. 1 ¯P and C¯λ∗ = H¯λ∗ in Example 3.1 0 0.5 1 1.5 2 2.5 3 y 0.5 1 1.5 2 2.5 3 x 0 0.5 1 1.5 2 2.5 3 y 0.5 1 1.5 2 2.5 3 x
is defined by the constraints x1 ≥ 1, x2 ≥ 1.5 and x1 + x2 ≥ 3.5; see Fig. 2. Also, (¯λ∗)({1}) = 3
1 = 3, (¯λ∗)({2}) = 2 + 3/2 = 2.5 and (¯λ∗)({1, 2}) = 3/1 + 1 + 2/2 =
4.5. So, Kis the polytope defined by the inequalities 1≤ x1≤ 3, 1.5 ≤ x2≤ 2.5 and
3.5≤ x1+ x2 ≤ 4.5 , which equals the convex hull of {1, 2.5}, {2, 1.5}, {3, 1.5}, {2, 2.5};
see Fig. 2. As is always the case, the polytope K includes the mean-partition
polytope ¯P.
We next consider the variant of Kthat corresponds to the sum-partition problem and show that it coincides with C. Specifically, let(λ∗)(I) be defined by the right-hand side of (3.9) withθ replacing ¯θ, and consider the system of linear inequalities given by
(λ∗)(I) ≤
i∈I
xi≤ (λ∗)(I) for all I ⊆ {1, . . . , p}. (3.11)
Evidently,(λ∗)({1, . . . , p}) = (λ∗)({1, . . . , p}) =nj=1θjand the pair of constraints
of (3.11) corresponding to I = {1, . . . , p}, together, coincide with the constraint p
i=1xi =
n
j=1θj, that is, with (3.3). Also, for each I ⊂ {1, . . . , p}, (λ∗)(I) =
n
j=1θj− (λ∗)(Ic), implying that the constraint
i∈Ixi ≤ (λ∗)(I) coincides with
the constrainti∈Icxi≥ (λ∗)(Ic). Thus, indeed, the set of vectors in Rpthat satisfy
(3.11) coincides with C.
4 Mean-partition problems with d= 1
In this section, we give some results about single-, bounded- and constrained-shape mean-partition problems with d= 1 that are not obtainable through the techniques described in Sect. 2 and 3.
Throughout this section we continue to let d= 1 and use the notation θ1,. . . , θnfor the scalars A1,. . . , An, respectively. Further, for simplicity, we assume that these sca-lars are distinct and that (by possibly reindexing them)θ1< θ2< · · · < θn. Also, let
n1,. . . , npbe positive integers, which sum to n and consider the single-shape partition
problems with prescribed shape(n1,. . . , np) . A reverse-size-consecutive partition is
a consecutive partition with the smallest elements being assigned to the larger parts. Henceforth, we assume that the parts are labelled so that n1 ≤ n2 ≤ · · · ≤ np, in
this case, up to index-permutation of parts having the same size, there is a unique
Fig. 2 The Modification of C¯λ∗ by relaxing (3.3) and Kin Example 3.1 0 1 2 3 4 5 y 1 2 3 4 5 x 0 0.5 1 1.5 2 2.5 3 y 0.5 1 1.5 2 2.5 3 x
reverse-size-consecutive partitionπ∗with shape(n1,. . . , np) and it is given by πp∗=
{1, . . . , np}, πp∗−1= {np+ 1, . . . , np+ np−1}, . . . , π1∗= {n − n1+ 1, . . . , n} .
For a vector a in Rpand i= 1, . . . , p, let a[i]be the ith largest member of{a1,. . . , ap}.
Given vectors a and b in Rp, we say that a weakly submajorizes b, written a wb if k i=1 a[i]≥ k i=1 b[i] for k= 1, . . . , p − 1. (4.1) If further p i=1 ai= p i=1 bi, (4.2)
then a is said to majorize b, written a b.
A real-valued function f on Rpis Schur convex if f(a) ≥ f (b) whenever a major-izes b. A Schur convex function is known to be symmetric (that is, invariant under coordinate-premutation); see [12] for further details about majorization and Schur convexity. In particular, the following result is well-known
Proposition 4.1 If f is nondecreasing Schur convex on Rpand a and b are vectors in
Rpsatisfying awb, then f(a) ≥ f (b).
The next lemma establishes an important property of reverse size-consecutive par-titions.
Lemma 4.2 Consider the case with p = 2. Then for every partition π = (π1,π2), ( ¯θπ∗
1, ¯θπ2∗) w( ¯θπ1, ¯θπ2).
Proof Consider a partitionπ = (π1,π2). We first prove that
max ¯θπ∗ 1, ¯θπ2∗ ≥ max ¯θπ1, ¯θπ2 , by proving that ¯θπ∗ 1 ≥ max ¯θπ1, ¯θπ2 . It is trivial that ¯θπ1 ≤ ¯θπ∗ 1. Similarly, withπ 2as
the set of the n2largest indices, ¯θπ2 ≤ ¯θπ2. Further,θπ2 = θπ1∗ + a where a is the sum
of(n2− n1) θi’s, each of which is smaller then ¯θπ1∗. Consequently,
¯θπ2 ≤ ¯θπ2 = θπ∗ 1 + a n1+ (n2− n1) ≤ max θπ∗ 1 n1 , a n2− n1 = ¯θπ∗ 1
and therefore max{ ¯θπ∗
1, ¯θπ2∗} ≥ ¯θπ1∗ ≥ max ¯θπ1, ¯θπ2
,
We next consider the single-shape sum-partition problem with prescribed shape
(n1, n2) and objective given by (1.3) with the function f being the linear function
mapping x∈ R2into x
1/n1+ x2/n2. As 1/n1≥ 1/n2, it follows from Theorem 2.1 of [9]
that the consecutive partition under which the n1indices associated with the smallest θis are assigned to the second part and the n
2 indices associated with the largetθis
are assigned to the first part, that is, the reverse size-consecutive partition is opti-mal. This proves that ¯θπ∗
1 + ¯θπ2∗ ≥ ¯θπ1+ ¯θπ2 , completing the proof that( ¯θπ1∗, ¯θπ2∗) w
From Proposition 4.1 and Lemma 4.2 we have.
Theorem 4.3 Every single-shape mean-partition problem with objective function given by(1.6) where g is nondecreasing and Schur convex has a reverse size-consecutive optimal partition.
Proof By [5], (reverse) size-consecutiveness is strongly 2-shape-sortable, which
implies that in order to prove that the property exists in some optimal partition, it suffices to consider the case where p = 2. For that case, Lemma 4.2 implies that for every partitionπ = (π1,π2), ( ¯θπ∗
1, ¯θπ2∗) w( ¯θπ1, ¯θπ2), and therefore Proposition 4.1
assures that F(π∗) = g( ¯θπ∗) ≥ g( ¯θπ) = F(π), establishing the optimality of π∗. As there is essentially a unique reverse size-consecutive partition for each shape, the constrained-shape mean-partition problem corresponding to a set of shapes we need to compare only|| partitions, one for each shape in . For bounded-shape problems,|| is available through lower and upper bounds on the part-sizes. One can then use recent results of Chang et al. (2005, submitted) to observe that it suffices to consider only those shapes in, which are not majorized by any other shape in . Further a bound of 2p−1 has been established on the number of such unmajorized shapes of bounded-shape problem along with enumeration scheme. Thus we get the following result
Proposition 4.4 For every bounded-shape mean-partition problem with objective func-tion given by(1.6) where g is nondecreasing and Schur convex it is possible to construct efficiently a set of 2p−1 reverse size-consecutive partitions that contain an optimal
partition.
Although we do not know how to efficiently describe constrained-shape mean-partition polytopes, we can bound the number of vertices of such polytopes by the sum of the number of vertices of the corresponding single-shape polytopes for each shape in. Since there is a one-to-one mapping between the vertices of the single-shape mean-partition polytopes and the vertices of the corresponding single-single-shape sum-partition polyotpes which are generated by the disjoint partitons and correspond to the p! permutations of {1, . . . , p} (see Sect. 3), we obtain a bound of ||p! on the number of vertices of the constrained-shape mean-partition polytope corresponding to. Further, if objective function is given by (1.6) where g is (quasi-) convex, one can enumerate a list of||p! (disjoint) partitions which contains an optimal one.
Theorem 4.5 Suppose the Ai’s are distinct. Then the constrained-shape mean-partition problem corresponding to a set of shapes with objective function given by (1.6) where g is (quasi-) convex can be solved with effort that is proportional to||, with p considered fixed.
The upper bound on the number of vertices of constrained-shape mean-partition polytopes, derived in the paragraph preceding Theorem 4.5, is not tight, as the follow-ing example demonstrates.
Example 4.1 Let n = 4, θi = i for i = 1, . . . , 4, p = 2 and = {(1, 3), (2, 2), (3, 1)}.
Each shape defines two disjoint partitions with two associated vectors—the three shapes, contribute respectively the vectors(1, 3), (4, 2), (1.5, 3.5), (3.5, 1.5) and (3, 1),
(2, 4). The mean-partition polytope is then the convex hull of the above 6 points, but
Acknowledgements Fei-Hwang Chang and F.K. Hwang’s research was partially supported by ROC National Science Grant NSC 93-2115-M-009-013. U.G. Rothblum’s research was supported by a grant from ISF—the Israel Science Foundation, by a VPR grant at the Technion, and by the Fund for the Promotion of Research at the Technion.
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