In this section, motivated by dual greedy algorithms, we introduce the concept of a greedy fan. We need some basic notation.
Let V be a (nonempty) finite set. R and R+denote the set of real numbers and nonnegative real numbers, respectively. For a function f : →R on a set , the nonzero support supp f is defined by{A∈ | f(A)6=0}. For a subset A⊆V , the characteristic vectorχA∈RV is defined as
χA(e) =
1 if e∈A, 0 otherwise.
We need to recall the basic definitions about polyhedral subdivisions (see [17] for details). A set of polyhedral cones∆is said to be a polyhedral fan if every face of any P∈∆is in∆, and the intersection of any two member P,Q∈∆is the common face of P and Q. We denote by|∆|the union of all member of∆. ∆is also called a polyhedral cone subdivision of|∆|. If every member of∆is a simplicial cone, we call∆a simplicial fan or a simplicial cone subdivision.
2.1 Greedy Fans, Dual Greedy Algorithms, and Submodular Functions
We consider a simplicial cone subdivisions∆of RV+with the following additional property, where we call a 1-dimensional cone a vertex.
Each vertex of∆can be expressed by R+χAfor some nonempty set A⊆V . Let = ∆⊆2V be a nonempty family defined as
={A⊆V | R+χAis a vertex of∆}.
In particular,∆can be regarded as an abstract simplicial complex on the vertex set which is denoted by ˆ∆. We shall often identify∆with ˆ∆. For a nonempty X⊆V , we define a restriction∆Xto be{ ∈∆| ⊆RX+}. Note that∆Xis a simplicial cone subdivision of RX+.
We define greedy fans recursively. If #V=1, trivial simplicial cone subdivision of RV+is defined to be greedy. Now we suppose that we have already defined the set of all greedy fans of RU+with #U<#V . A simplicial cone subdivision∆of RV+ is said to be greedy if there exists a nonempty A⊆V such that
(G1) every maximal cone of∆contains R+χAas a vertex and (G2) for any e∈A, a restriction∆V\{e}is a greedy fan of RV+\{e}.
We call a vertex satisfying (G1) and (G2) of a greedy fan∆a center vertex. First, we see that every restriction of a greedy fan is a greedy.
Lemma 1 For a greedy fan∆of RV+and a nonempty X⊆V , a restriction∆Xis also a greedy fan.
From this lemma, we can define a functionΦ∆: 2V→2 as
Φ∆(X) ={A⊆V |A is a center vertex of∆X} (X⊆V,X6=/0) (2) andΦ∆(/0) =/0 for convenience.
Next we consider the membership problem of finding a member of∆which contains a given vector w∈RV+. For this, we try to find a conical expression of w of a member of∆. Consider the following procedure, whereΦ=Φ∆.
Procedure: Dual Greedy Input: A vector w∈RV+.
Output: λ∈R+ with w=∑A∈ λ(A)χA.
Initialization: w0←w, X←V ,λ(A)←0(∀A∈ ).
step1: If X=/0, then stop.
step2: Pick arbitrary A∗∈Φ(X)and e∗∈Argmin{w0(e)|e∈A∗}. step3: Putλ(X)←w0(e∗)and w0←w0−w0(e∗)χA∗.
step4: Put X←X\ {e∗}and go to step1.
Variants of this procedure have been considered by several authors [3], [4], [10], [1], [9], [6] to obtain an optimum of linear program [D]. This formulation usingΦ: 2V →2 is essentially due to Fujishige [6]. In fact, this procedure solves the membership problem of∆as follows.
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Proposition 2 For a greedy fan∆with vertex set ⊆2V\ {/0}and a nonnegative vector w∈RV+, letλ : →R be an output ofDual Greedyfor input vector w. Then we have
cone{χA|A∈suppλ} ∈∆.
Next we try to construct a polyhedron whose normal fan coincides with a given greedy fan∆. For a function f : →R, we define a polyhedron P(f)as
P(f) ={x∈RV|
∑
e∈A
x(e)≤f(A) (A∈ )} (3)
Note that P(f)is the feasible region of [P] in 1. A greedy fan∆with vertex set ⊆2Vis regular if there exists a function f : →R such that∆coincides with the normal fan of polyhedron P(f). Let(∆)ˆ ∗⊆2 be a family defined as
A function f : →R is said to be submodular if it satisfies submodularity inequalities. Let submodular cone ∆⊆R be defined as the set of all submodular functions on . We denote int ∆by the set of all interior point of ∆.
Theorem 3 Suppose the submodular cone ∆has interior points. Consider linear program [D] for a function f : →R.
Dual Greedyproduces an optimal dual solution of [D] for any nonnegative cost vector if and only if f is submodular for∆.
In particular, f∈int ∆if and only if∆coincides with the normal fan of P(f).
PROOF: Only-if-part of the first statement follows from the definition of the submodularity inequalities (5). We show if-part.
We can take g∈int ∆. Consider linear program [D] with a submodular function f and a cost vector w∈RV+. Since both [P] and [D] are feasible, [D] has an optimal solution. We take an optimal solutionλ∗of [D] which minimizes the value
∑A∈ λ∗(A)g(A). We claim suppλ∗∈∆. If so,ˆ λ∗must be the output ofDual Greedy. Suppose that suppλ∗6∈∆. Thenˆ by submodularity of f , the objective value of [D] for ˜λ is given by
A
∑
∈Hence ˜λ is also optimal to [D]. Similarly, we have
A
∑
∈ the optimal solution of [D] is unique. From this, we obtain the latter statement of this theorem. 2Corollary 4 A greedy fan is regular if and only if its submodular cone has interior points.
2.2 Greedy Multiple-Choice Functions
In this subsection, we discuss properties of the mapΦ∆associated with greedy fan∆and try to define greedy fans by means of a certain mapΦ: 2V→22V. Our purpose is to derive the conditions ofΦwhich determine a greedy fan. First, we see that Φ∆has the following properties.
Proposition 5 For a greedy fan∆with vertex set , a mapΦ=Φ∆: 2V→2 has the following properties.
(C1) for a nonempty X⊆V ,Φ(X)is nonempty and any A∈Φ(X)is nonempty.
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(C2) for X⊆V and A∈Φ(X), we have A⊆X .
(M1) for X,Y ⊆V and A∈Φ(X), if A⊆Y⊆X , then we have A∈Φ(Y).
(M2) for X⊆V and A,B∈Φ(X), we have A=B or A∩B=/0.
If a functionΦ: 2V→22V satisfies (C1) and (C2), we callΦ: 2V →22V a choice function. Given a multiple-choice functionΦ: 2V→22V, we can applyDual Greedyfor any nonnegative input vector. However, the outputλdepends on the choices of A∗and e∗in step 1. Next we discuss the uniqueness of outputs ofDual Greedyfor general multiple-choice functions. For a multiple-choice functionΦand an input vector y∈RV+, a sequence{(Ai,ei)}ni=1⊆2V×V is said to be feasible if Ai=A∗and ei=e∗can be chosen by the step 1 of the ith iteration step for input vector y. We see the following.
Lemma 6 If a sequence{(Ai,ei)}ni=1is feasible to some input vector, then it is also feasible to any vector in cone{χAi}ni=1. Let∆Φbe a set of simplicial cones defined as
∆Φ={cone{χA}A∈suppλ |λ is an output for some w∈RV+}. (7) Lemma 6 above implies that any face of any member of∆Φ is also contained by ∆Φ. Hence, if the outputλ is uniquely determined for any w∈RV+,∆Φ forms a simplicial cone subdivision of RV+. In particular, this fan is greedy, and its center vertices are determined by the multiple-choice functionΦ. In fact, the conditions (M1) and (M2) are sufficient for this uniqueness as follows, where for a functionΦ: 2V→22V, we define the image ImΦ⊆2Vas
ImΦ={A⊆V| ∃nonempty X⊆V,A∈Φ(X)}. (8)
Theorem 7 If a multiple-choice functionΦ: 2V→22V satisfies the conditions (M1) and (M2), the solution ofDual Greedy is determined independently of the choices A∗and e∗in step 1 for any nonnegative input vector. Furthermore, the vertex set of∆Φis given by ImΦ.
PRO OF: We use induction on # supp w for a nonnegative input vector w. In the case of # supp w≤1, the statement clearly holds. Consider a nonnegative input vector w with # supp w>1. Let{Ai,ei}ni=1and{Bi,di}ni=1be two feasible sequences for w. Letλ andµbe outputs of w using feasible sequences{Ai,ei}ni=1and{Bi,di}ni=1, respectively. We define Xiand Yias
X1=Y1=V, Xi+1=Xi\ {ei},Yi+1=Yi\ {di} (9) for i=1, . . . ,n. We show λ =µ. Let i be the smallest number satisfying λ(Ai)>0. Similarly, let j be the smallest number satisfyingµ(Bj)>0. Then we have Ai⊆supp w⊆Xiand Bj⊆supp w⊆Yj. By (M1), we have Ai,Bj∈Φ(supp w).
Hence Ai=Bj or Ai∩Bj=/0 follows from (M2). If Ai=Bj, then we haveλ(Ai) =µ(Bj). If Ai∩Bj=/0, then we have Ai⊆supp w\ {dj} ⊆Xi and Bj+1⊆supp w\ {dj} ⊆Yj+1. Similarly we have Ai=Bj+1or Ai∩Bj+1= /0. Repeating this process, there exists some number k such that Ai=Bj+kor Ai=supp w\ {dj,dj+1, . . .dj+k−1}. In the latter case, we have Bj+k∈Φ(Ai). Hence Bj+k=Aiholds. Since Ai∩Bj+1=Ai∩Bj+2=···=Ai∩Bj+k−1=/0, we haveλ(Ai) =µ(Bj+k). We define a modified input vector w0=w−λ(Ai)χAi=w−µ(Bj+k)χBj+k. Then we have # supp w0<# supp w. Two sequences {(Ai,ei)}ni=1and{(Bi,di)}ni=1are also feasible to the modified input vector w0by Lemma 6. Letλ0andµ0be two outputs of Dual Greedyfor input vector w0using feasible sequences{(Ai,ei)}ni=1and{(Bi,di)}ni=1, respectively. Then we have
λ0(A) =
0 if A=Ai=Bj+k
λ(A) otherwise , µ0(A) =
0 if A=Ai=Bj+k
µ(A) otherwise (A⊆V).
By induction,λ0=µ0holds. This impliesλ=µ.
Finally, we show that the set of vertices of∆Φcoincides with ImΦ. It suffices to show that for any A∈ImΦ, the output λ∈R2V ofDual Greedyfor input vectorχAis given asχ{A}∈R2V. Let{(Ai,ei)}ni=1be a feasible sequence toχAandλbe the resulting output. Let j be the smallest number havingλ(Ai). Similarly to above discussions, we have Ai⊆suppχA=A⊆Xi, where Xiis defined by (9). Hence we have Ai∈Φ(A). This implies Ai=A andλ =χ{A}. 2
A multiple-choice function is said to be greedy if it satisfies (M1) and (M2). Hence, we obtain greedy fans by greedy multiple-choice functions. The proof of Theorem 7 gives the following criterion when two greedy multiple-choice functions produce same greedy fan.
Proposition 8 For two greedy multiple-choice functionsΦ1,Φ2: 2V→22V, the following conditions are equivalent.
(1) ∆Φ1=∆Φ2.
(2) for any X⊆V , A∈Φ1(X), and B∈Φ2(X), we have A=B or A∩B=/0.
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2.3 Acyclic Greedy Fans
Here, we introduce a certain class of greedy fans which can be represented by posets. This approach is motivated by dual greedy systems by Frank [4]. Throughout this subsection, denotes a subset of 2V\ {/0}. A pair of A,B⊆V is said to be intersecting if it satisfies A∩B6=/0, A6⊆B, and B6⊆A.
Let = ( ,≤)be a poset on . We define a functionΦ : 2V→2 associated with as
Φ (X) ={A∈ |A⊆X,∀B∈ ,B>A⇒B6⊆X} (X⊆V), (10) that is,Φ (X)is a set of maximal members of contained in X . Such a functionΦ was used by Frank [4] in his dual greedy algorithm.
A poset = ( ,≤)is said to be greedy ifΦ is a greedy multiple-choice function and satisfies ImΦ = . Hence, from a greedy poset , we obtain a greedy fan, which is denoted by∆ . A greedy fan∆is said to be acyclic if there exists a greedy poset such that∆=∆ . A greedy poset( ,≤)is a refinement of poset( ,⊆)as follows.
Lemma 9 Let = ( ,≤)be a greedy poset. For any A,B∈ , if A⊆B, then A≤B.
PROOF: ImΦ = and (M1) forΦ implies B∈Φ (B). Suppose A6≤B. Then A and B are noncomparable. Then there exists C∈ such that C6=B and C∈Φ (B). Hence we have C∩B6=/0. This is a contradiction to (M2). 2
For a poset = ( ,≤), a linear extension( ,≤∗)of is a totally ordered set which satisfies A≤B⇒A≤∗B (A,B∈ ).
Then we observe the following.
Lemma 10 If contains every singleton. then any linear extension of( ,⊆)is greedy.
Furthermore, from the condition of Proposition 8, we have the following.
Proposition 11 Let is a greedy poset. Then,∆ =∆ ∗ holds for any linear extension ∗of . Theorem 12 Acyclic greedy fans are regular.
PROOF: By Proposition 11, it suffices to show the present theorem when the greedy poset( ,≤)is a totally ordered set.
Suppose that ={A1,A2, . . . ,Am}is ordered by
A1>A2>···>Am.
Consider the submodularity inequalities (5) for∆( ,≤). Then, for any ∈(∆ˆ( ,≤))∗, there exists A∗∈suppλ such that A∗>A holds for any A∈ . Hence we define a function f : →R as
f(Ai) =−εi (Ai∈ ={A1,A2, . . . ,Am}),
whereε ∈R+ is a sufficiently small positive real. Then f satisfies strict submodularity inequalities. This implies that submodular cone ∆(
,≤∗) has interior points. By Corollary 4,∆( ,≤)is regular. 2
For two posets 1= ( ,≤1)and 2= ( ,≤2)with a common ground set , we define meet 1∧ 2= ( ,≤1∧2) as
A≤1∧2B⇐⇒def A≤1B and A≤2B (A,B∈ ). (11) The next theorem shows that if two greedy posets define the same greedy fan, then their meet also defines the same one.
Theorem 13 Let 1= ( ,≤1)and 2= ( ,≤2)be greedy posets on . If∆ 1 =∆ 2, 1∧ 2is also greedy and satisfies∆ 1∧ 2=∆ 1=∆ 2.
PROOF: From the definitions ofΦ and the meet 1∧ 2, we see
Φ 1∧ 2(X)⊆Φ 1(X)∪Φ 2(X) (X⊆V).
Hence, it suffices to show thatΦ 1∧ 2 satisfies (M2). Suppose that there exist A,B∈Φ 1∧ 2(X)such that A∩B is nonempty. Then we have A,B∈Φ 1∧ 2(A∪B)by (M2). Then A or B is not contained byΦ 1(A∪B)∪Φ 2(A∪B). We assume A6∈Φ 1(A∪B)∪Φ 2(A∪B). Then there exists C∈Φ 1(A∪B)such that C≥1A holds. We claim that A and C are disjoint. Suppose that A and C intersect. Then we have C∈Φ 1(A∪C). We show A6∈Φ 2(A∪C)and C6≥2A.
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If A∈Φ 2(A∪C), then A,C∈Φ 2(A∪C)must be disjoint. This contradicts the assumption. If C≥2A, then we have C≥1∧2A. This contradicts A∈Φ 1∧ 2(A∪B). Hence there exists D∈Φ 2(A∪C)such that D≥2A and D6=C. Then D∩C=/0 and hence D⊂A (strict inclusion). This implies D<2A, which contradicts D∈Φ 2(A∪C). Hence we have A∩C=/0 and C⊂B (strict inclusion). By Lemma 9, we have C<1B. This contradicts C∈Φ 1(A∪B). 2
For a poset , we denote by the set of all linear extensions of . The above theorem implies that for any family
⊆2V\ {/0}which contains every singleton, there uniquely exists a set of greedy posets ( )on the ground set such that
(H1) the family of linear extensions{ | ∈ ( )}forms a partition of (
,⊆). (H2) for two linear extensions 1∗, 2∗∈ ( ,⊆), we have
∆ ∗
1 =∆ ∗
2 ⇔ ∃ ∈ ( ), 1∗, 2∗∈ (12)
A set of posets on a common ground set which satisfies the condition (H1) above is called holometry, which was in-troduced by Tomizawa [14], [15], [16] in 1983 as a combinatorial abstraction of normal fans of base polyhedra. For poset ( ,≤)we define an order cone ( ,≤)⊆R as
( ,≤)={x∈R |x(A)≤x(B) (A,B∈ , A≤B)}. (13) The set of polyhedral cones consisting of order cones{ | ∈ ( )}and their faces is denoted byΣg( ). In fact, Σg( )forms a polyhedral fan as follows, where detailed proof will be given in [8].
Theorem 14 For a family which contains every singleton,Σg( )forms a polyhedral cone subdivision of order cone
( ,⊆).
A holometry is called a hypergeometry if its associated set of order cones given above forms a polyhedral cone subdi-vision [14], [15], [16]. The above theorem states that ( )is a hypergeometry. Analogously to the secondary fan [7], [12], we call this polyhedral fanΣg( )the secondary greedy fan of . So it is natural to ask whether there exists some polyhedron P⊆R whose normal fan coincides withΣg( ). If such a polyhedron P exists, each edge vector is parallel to χ{A}−χ{B}for some A,B∈ . A well-known characterization of base polyhedra by edge directions [13] implies that P is a base polyhedron associated with some (ordinary) submodular function defined on the set of lower ideals of the poset( ,⊆) (see [5]).
Problem 15 Does there exist a base polyhedron whose normal fan coincides with a secondary greedy fan ?
2.4 Greedy Fans by Set Systems
Here, we discuss the case when is a greedy poset ordered by inclusion(⊆). In this case, the associated holometry ( ) is a singleton, i.e., ( ) ={( ,⊆)}. We observe the following.
Proposition 16 ( ,⊆)is greedy if and only if it satisfies (S0) for any e∈V , we have{e} ∈ .
(S1) for any intersecting pair A,B∈ , we have A∪B∈ .
The condition (S1) implies thatΦ( ,⊆)(X)forms the unique maximal partition of X , where ”unique maximal” means that any partitionΠ⊆2 of X is a refinement ofΦ( ,⊆)(X), that is, for any C∈Πthere exists C0∈Φ( ,⊆)(X)such that C⊆C0. The submodularity inequalities for a greedy fan∆( ,⊆)are explicitly given as follows.
Theorem 17 Let( ,⊆)be a greedy poset ordered by inclusion. The submodularity inequalities for∆( ,⊆)are given by f(A) +f(B)≥f(A∪B) +
∑
C∈Φ( ,⊆)(A∩B)
f(C) (A,B∈ : intersecting), (14)
A
∑
∈f(A)≥f([
A∈
A)
( ⊆2 : pairwise disjoint withSA∈ A∈ and∀ 0⊂ ⇒SA∈ 0A6∈ ). (15) In the case of =2V\ {/0}, equations (14) and (15) coincide with ordinary submodularity inequalities.
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