a note on m-convexity in polyhedral split decomposition of distances
TRANSCRIPT
Japan J. Indust. Appl. Math. (2012) 29:187–204DOI 10.1007/s13160-011-0052-y
ORIGINAL PAPER Area 2
A note on M-convexity in polyhedral splitdecomposition of distances
Shungo Koichi
Received: 19 October 2009 / Revised: 26 October 2011 / Published online: 17 December 2011© The JJIAM Publishing Committee and Springer 2011
Abstract Buneman’s decomposition of a metric can be obtained in a geometric wayas the polyhedral split decomposition of a polyhedral convex function associated witha metric. In discrete convex analysis, such a polyhedral convex function is consideredas an important example of M-convex functions. Motivated by this, we shed light onthe decomposition from the viewpoint of discrete convex analysis. We firstly showthat the decomposition is a sum of M-convex functions, where a sum of M-convexfunctions is not necessarily M-convex in general. We explain why the M-convexity ispreserved in the decomposition. We next show that a quadratic M-convex function issplit-decomposable at every point. This indicates simplicity of the geometric structureof a quadratic M-convex function. We finally give a characterization of M-convexpolyhedra in terms of their facets. This can be used to describe the M-convexity of apolyhedron associated with a function on a cross-free family.
Keywords M-convex function · Polyhedral split decomposition ·Split-decomposability · Tree metric · Cross-free family
Mathematics Subject Classification (2000) 52B05 · 52B40
1 Introduction
Let X be a finite set. An X -split is a pair {S, T } of S and T such that ∅ �= S ⊆ X,∅ �=T ⊆ X, S∩ T = ∅ and S∪ T = X. In this paper, a distance on X is defined as a func-tion d : X× X → R such that d(i, i) = 0 for all i ∈ X and d(i, j) ≤ d(i, k)+d(k, j)
S. Koichi (B)Department of Systems Design and Engineering, Nanzan University,Seireicho 27, Seto 489-0863, Japane-mail: [email protected]
123
188 S. Koichi
for all i, j, k ∈ X. In addition, a metric means a symmetric and nonnegative distance,that is, a distance d is a metric if d(i, j) = d( j, i) ≥ 0 for all i, j ∈ X. In particular,the split metric δ{S,T } : X × X → {0, 1} associated with an X -split {S, T } is definedby
δ{S,T }(i, j) :={
0 if i, j ∈ S or i, j ∈ T,
1 otherwise,
for all i, j ∈ X.
In phylogenetics, a method for constructing a phylogenetic tree from dissimilaritiesamong biological sequences is a basic tool to study the evolution of biological species;see [10]. To propose such a method, Buneman introduced the Buneman index in [2].The method can be mathematically interpreted as a decomposition of a metric d intoa sum of split metrics multiplied by the Buneman indices and a residual metric d ′.Moreover, the residual metric d ′ vanishes if and only if the metric d is a tree metric, thatis, there exists a tree with nonnegative edge lengths such that, for all i, j ∈ X, d(i, j)is equal to the length of the (unique) path in the tree between the vertices indexed byi and j. This is an important property of Buneman’s decomposition.
As another decomposition of a metric, Bandelt and Dress’ split decomposition [1]is known; see also [3]. This decomposition uses the isolation index, and, as is similar toBuneman’s decomposition, a metric is represented as a sum of split metrics multipliedby the isolation indices and a residual metric. However, until recently, the relationshipbetween the two decompositions was not clear.
In [5], Hirai introduced the polyhedral split decomposition of polyhedral convexfunctions to extend Bandelt and Dress’ split decomposition. Furthermore, recently,it was shown in [7] that the approach of Hirai is applicable to Buneman’s decom-position, that is, the two decompositions can be understood as special cases of thepolyhedral split decomposition. Since the approach of Hirai is purely geometric, thetwo decompositions now can be described in a geometric framework, and one can seethat their geometric structures provide their property. Hence, it is important to studythe geometric structures.
The aim of this paper is therefore to study the geometric structure of, in particu-lar, Buneman’s decomposition of a metric, which is easily extendable to a distance.A distance d on X can be regarded as a discrete function on � = {χi −χ j | i, j ∈ X}by setting d(χi−χ j ) = d(i, j), where χi is the characteristic vector of {i}. Taking the(homogeneous) convex extension of d on �, one obtains a polyhedral convex functiond. The polyhedral split decomposition is applied to this convex extension d. In thepresent paper, we regard the polyhedral split decomposition of d as a decompositionof an M-convex function, which is a function playing a central role in discrete convexanalysis advocated by Murota [8]. In discrete convex analysis, it is known that theconvex extension d of a distance d is a positively homogeneous M-convex function.Motivated by this fact, we shed light on the decomposition from the viewpoint ofdiscrete convex analysis.
We firstly show that the polyhedral split decomposition of a distance gives a decom-position of a polyhedral M-convex function into a sum of polyhedral M-convex func-tions. This is a remarkable phenomenon because, in general, a sum of M-convex
123
M-convexity in polyhedral split decomposition of distances 189
functions is not necessarily M-convex. As the polyhedral split decomposition isa geometric notion, we will give a geometric explanation to the reason why theM-convexity is preserved in the decomposition. In particular, we describe it in termsof polyhedral subdivisions induced by M-convex functions. Those polyhedral subdi-visions must consist of M-convex polyhedra.
This paper next deals with quadratic M-convex functions. It is known that thereis a one-to-one correspondence between tree metrics and quadratic M-convex func-tions [6]. For a quadratic M-convex function, a positively homogeneous M-convexfunction can be defined at each point of its domain. We show that such a positivelyhomogeneous M-convex function is representable as the sum of a tree metric anda linear function. A tree metric and its sum with a linear function were shown tobe split-decomposable in [7]. Hence, a quadratic M-convex function turns out to besplit-decomposable at every point of its domain. This split-decomposability indicatessimplicity of the geometric structure of a quadratic M-convex function.
We finally characterize the M-convex polyhedra, i.e., the base polyhedra of sub-modular systems, using their facets. It is shown that a polyhedron is M-convex ifand only if the normal vectors of two facets which share a ridge, i.e., a maximalproper face of a facet, is the characteristic vectors of two subsets Y, Z of X such thatY ⊆ Z , Y ⊇ Z , Y ∩ Z = ∅, or Y ∪ Z = X. This characterization immediately impliesthat a polyhedron associated with a function on a cross-free family is M-convex, whichis a well-known result.
This paper is organized as follows. Section 2 briefly describes the polyhedral anddiscrete split decompositions on the basis of [5]. In Sect. 3, we collect the resultsobtained in [7] on the polyhedral split decomposition of a distance, which are used inSect. 4. The new results of this paper are presented in Sect. 4. We give in Sect. 4.2a geometric explanation for the M-convexity in the decomposition. In Sect. 4.3,we apply the polyhedral split decomposition to a quadratic M-convex function toreveal its geometric structure. Section 4.4 gives a characterization of M-convex poly-hedra using their facets.
2 Polyhedral and discrete split decomposition
2.1 Preliminaries
Let Z be the set of integers. Let R and R+ be the sets of real numbers, and nonnegativereal numbers, respectively. We denote by Rn the n-dimensional Euclidean space withthe standard inner product 〈·, ·〉. Let 0 and 1 be the all-zero and all-one vectors in Rn,
respectively.We refer to an (n − 1)-dimensional affine subspace of Rn as a hyperplane. In par-
ticular, for (a, r) ∈ Rn × R with a �= 0, we define a hyperplane Ha,r := {x ∈Rn | 〈a, x〉 = r}, and closed half spaces H−a,r := {x ∈ Rn | 〈a, x〉 ≤ r} andH+a,r := {x ∈ Rn | 〈a, x〉 ≥ r}. A hyperplane Ha,r is called linear if Ha,r containsthe origin 0, that is, r = 0.
For a set S ⊆ Rn, we denote by cone S and lin S the conical hull and linear hullof S, respectively, i.e.,
123
190 S. Koichi
cone S ={∑
t∈T
λt t | T ⊆ S : a finite set, λ = (λt ) ∈ RT+
},
lin S ={∑
t∈T
λt t | T ⊆ S : a finite set, λ = (λt ) ∈ RT
}.
We say that a cone C is generated by a set S, or that S generates C if C = cone S.
Let P ⊆ Rn be a polyhedron, and let x be a point x in P. The tangent cone TP (x)
and normal cone NP (x) at the point x are the cones defined, respectively, by
TP (x) := {αy | x + y ∈ P, α ∈ R+}, NP (x) := {p ∈ Rn | 〈p, y − x〉 ≤ 0, y ∈ P}.
Since P is a polyhedron, both TP (x) and NP (x) are polyhedral cones. It is well knownthat TP (x) is the dual cone of NP (x) and vice versa.
A polyhedral complex C is a finite collection of polyhedra such that
(1) if P ∈ C, all the faces of P are also in C, and(2) the nonempty intersection P ∩ Q of two polyhedra P, Q ∈ C is a face of
P and Q.
A polyhedral subdivision of a polyhedron P is a polyhedral complex C with⋃Q∈C Q = P. For two polyhedral subdivisions C1 and C2, their common refine-
ment C1 ∧ C2 is defined by C1 ∧ C2 := {F ∩ G | F ∈ C1, G ∈ C2, F ∩ G �= ∅}. Inparticular, for a finite set H of hyperplanes, the polyhedral subdivision A(H) of Rn
is defined by
A(H) :=∧
Ha,r∈H{Ha,r , H+a,r , H−a,r }.
For a function f : Rn → R∪{+∞}, the effective domain of f, denoted by dom f,is defined by dom f := {x ∈ Rn | f (x) < +∞}, and the epigraph of f, denoted byepi f, is given by epi f := {(x, β) ∈ Rn × R | β ≥ f (x)}.
A convex function f is said to be polyhedral if its epigraph epi f is a polyhedron.Let f be a polyhedral convex function. By defining, for each proper face F of epi f,the set F ′ as F ′ := {x ∈ dom f | (x, f (x)) ∈ F}, one obtains a collection of subsetsof dom f. One can easily see that each set F ′ in the collection is a polyhedron, andtherefore the collection is a polyhedral subdivision of dom f, which we denote byT ( f ).
The polyhedral subdivision obtained by the sum of two polyhedral convex functionscoincides with the common refinement of the associated polyhedral subdivisions.
Lemma 1 For two polyhedral convex functions f, g, we have T ( f + g) = T ( f ) ∧T (g).
123
M-convexity in polyhedral split decomposition of distances 191
2.2 Polyhedral split decomposition
For a hyperplane Ha,r , the split function lHa,r : Rn → R associated with Ha,r isdefined to be the function such that the value lHa,r (x) of each point x in Rn is ‖a‖times the distance between the point x and the hyperplane Ha,r . That is, lHa,r is givenby
lHa,r (x) := |〈a, x〉 − r | (x ∈ Rn).
For a polyhedral convex function f : Rn → R and a hyperplane Ha,r , we definethe quotient cHa,r ( f ) of f by lHa,r as
cHa,r ( f ) := sup{t ∈ R+ | f − tlHa,r is convex}.
The quotient cHa,r ( f ) depends on the equation 〈a, x〉 = r representing a hyper-plane Ha,r , but the function cHa,r ( f )lHa,r is independent of the equation. From nowon, unless otherwise stated, a hyperplane H is assumed to be represented as H = Ha,r
for a normal vector a with ‖a‖ = 1, and so lH (x) is equal to the distance betweenx and H.
We define the set of hyperplanes H( f ) as
H( f ) := {H : hyperplane | 0 < cH ( f ) < +∞}.
In [5], it is shown that H belongs to H( f ) if and only if H ∩dom f is contained in theset of all points x ∈ dom f with the property that f is not linear in any neighborhood(in dom f ) of x . If dom f is full-dimensional, each hyperplane H ∈ H( f ) is uniquein its intersection with dom f, and hence the polyhedral split decomposition of f isunique. For this, and only for this, dom f needs to be assumed to be full-dimensional.
Theorem 1 ([5, Theorem 2.2]) A polyhedral convex function f : Rn → R ∪ {+∞}with dim dom f = n is uniquely decomposable as
f =∑
H∈H( f )
cH ( f )lH + f ′, (1)
where f ′ : Rn → R∪{+∞} is a polyhedral convex function with cH ′( f ′) ∈ {0,+∞}for any hyperplane H ′.
Obviously, we have T (αlH ) = {H, H+, H−} for any α > 0. Hence, by Lemma 1,the decomposition (1) produces a subdivision as in (2) below.
Lemma 2 Suppose that a polyhedral convex function f : Rn → R∪{+∞} is decom-posed as (1). Then, the polyhedral subdivision T ( f ) is represented as
T ( f ) = A(H( f )) ∧ T ( f ′). (2)
123
192 S. Koichi
2.3 Discrete split decomposition
In this paper, a discrete function means a function defined on a finite set of points inRn . Let K be a finite set of points in Rn . If K contains the origin 0, we assume thatf (0) = 0 for any discrete function f : K → R.
For a discrete function f : K → R, the homogeneous convex closure of f on Kis defined by
f (x) := inf
⎧⎨⎩
∑y∈K
λy f (y)
∣∣∣ ∑y∈K
λy y = x, λy ≥ 0 (y ∈ K )
⎫⎬⎭ (x ∈ Rn). (3)
Since K is a finite set, f is a polyhedral convex function with dom f = cone K .
Furthermore, by definition, f is positively homogeneous, i.e., f (αx) = α f (x) holdsfor any α ≥ 0 and x ∈ Rn .
For a function g : Rn → R, we denote the restriction of g to K by gK . A discrete
function f : K → R is said to be convex-extensible if it satisfies fK = f. If f is
convex-extensible, we call f the homogeneous convex extension of f, or the extensionof f, for short.
Applying the polyhedral split decomposition to the extension of a discrete function,we obtain the discrete split decomposition.
Theorem 2 ([5, Theorem 3.2]) Let f : K → R be a discrete function on K withdim cone K = n. Then, f is uniquely decomposable as
f =∑
H∈H( f )
cH ( f )l KH + f ′,
where f ′ : K→R satisfies cH ′( f ′) ∈ {0,+∞} for any linear hyperplane H ′. Fur-thermore, we have
f =∑
H∈H( f )
cH ( f )lH + f ′.
If, in addition, f is convex-extensible, then f ′ is also convex-extensible.
What is worth discussing is a relation between K and H( f ). Since f (0) = 0,
each hyperplane H ∈ H( f ) is linear. By the definition of f , every cone in T ( f ) isgenerated by a subset of K . Since T ( f ) = A(H( f )) ∧ T ( f ′) as in Lemma 2, thisfact imposes the K -admissibility on H( f ): A set H of linear hyperplanes is calledK -admissible if H satisfies the following conditions (A1) and (A2).
(A1) Each hyperplane in H intersects with the relative interior of cone K .
(A2) Each cone in A(H) ∧ cone K is generated by a subset of K .
123
M-convexity in polyhedral split decomposition of distances 193
For simplicity, a linear hyperplane H is called K -admissible if the set {H} isK -admissible. We denote by HK the set of all K -admissible linear hyperplanes. SinceH( f ) is K -admissible, one has H( f ) ⊆ HK for any discrete function f : K → R.
The K -admissibility also relates to the split-decomposability. A function f on afinite set K with dim cone K = n is said to be split-decomposable if the residue f ′given by f ′ := f −∑
H∈H( f ) cH ( f )l KH is equal to the restriction of a linear function
to K .
In fact, an addition of the restriction of a linear function to a discrete function fdoes not change the quotient of f and hence the split-decomposability is unaffected,which follows from the next lemma. Note that the quotient cH (〈q, ·〉) equals zero forany hyperplane H.
Lemma 3 Let f be a discrete function on K with dim cone K = n. For any α > 0and any vector q ∈ Rn, we have α f + (〈q, ·〉)K = α f + 〈q, ·〉.
Moreover, according to the following proposition, every split-decomposable func-tion is constructed essentially from a K -admissible set of hyperplanes.
Proposition 1 ([5, Proposition 3.5]) Let K ⊆ Rn be a finite set with dim cone K = n.
For a subset H of HK and positive weights αH > 0 (H ∈ H), let f :=∑H∈H αH l K
H .
Then the following conditions (a), (b) and (c) are equivalent.
(a) f (x) ={∑
H∈H αH lH (x) (x ∈ cone K ),
+∞ (x �∈ cone K ).
(b) H = H( f ) and αH = cH ( f ) for every H ∈ H.
(c) H is K -admissible.
3 Split decomposition of a distance
In this section, we give a brief description of results in [7] obtained by applying thepolyhedral and discrete split decompositions to a distance.
Let X := {1, 2, . . . , n}, and identify RX with Rn naturally. For a subset Y of X,
the characteristic vector χY is defined by χY (i) := 1 if i ∈ Y and χY (i) := 0 if i /∈ Y.
For simplicity, we write χi instead of χ{i} for a singleton set {i}.
3.1 Homogeneous convex extension of a distance
Let � be the finite set defined by
� := {χi − χ j | i, j ∈ X}.
A distance d on X can be regarded as a discrete function on the set � by setting
d(χi − χ j ) = d(i, j) (i, j ∈ X).
It is important that d is a distance since the convex-extensibility of d on � is guaranteedby the triangle inequality.
123
194 S. Koichi
Lemma 4 A discrete function f : �→ R with f (0) = 0 is convex-extensible if andonly if f satisfies f (χi − χ j ) ≤ f (χi − χk)+ f (χk − χ j ) for all i, j, k ∈ X.
We deal with the polyhedral split decomposition of the extension of d on �. How-ever, since cone � = H1,0 and hence dim cone � = n − 1, we cannot directly applythe polyhedral split decomposition. To cope with this difficulty, we extend � to �1 :=�∪ {1,−1}, and define d(1) = d(−1) := 0. Then, cone �1 = RX . The extension ofd on �1 is given, for each x in RX , by
d(x) =
inf
⎧⎨⎩
∑i, j∈X
λi j d(χi − χ j )
∣∣∣∣∣∑
i, j∈X λi j (χi − χ j )+ λ11+ λ−1(−1) = x,
λi j ≥ 0 (i, j ∈ X), λ1 ≥ 0, λ−1 ≥ 0
⎫⎬⎭ .
For any point x ∈ RX , x is represented as x = x ′ + λ1 for some x ′ ∈ cone � andλ ∈ R. Since d(1) = d(−1) = 0, we have d(x) = d(x ′), which allows us to identifythe extension of d on � with that on �1.
Figure 1c illustrates the extension on � of a metric d on X = {i, j, k}. Since a pointin � is on the linear hyperplane H1,0 as in Fig. 1a, we can project {(χi −χ j , d(i, j)) |i, j ∈ X} to a 3-dimensional space as shown in Fig. 1b.
3.2 Discrete split decomposition of a distance
To apply the polyhedral split decomposition, we need the set H�1 of �1-admissiblehyperplanes. Let {S, T } be an X -split, and let (S, T ) denote the ordered pair of S and T .
Then, we define the vector a(S,T ) by
a(S,T ) = |S| |T ||X |(
χS
|S| −χT
|T |)
. (4)
Since Ha(S,T ),0 = Ha(T,S),0, we write H{S,T } for Ha(S,T ),0. Although ‖a(S,T )‖ �= 1, weuse a(S,T ) to make the quotient cH{S,T }(d) simple.
(a) (b) (c)
Fig. 1 The homogeneous convex extension on � of a metric d on X = {i, j, k}
123
M-convexity in polyhedral split decomposition of distances 195
Proposition 2 ([7, Proposition 9.3]) The set H�1 of �1-admissible hyperplanes isgiven by
H�1 = {H{S,T } | {S, T } : an X-split} ∪ {H1,0}.
Furthermore, for each H{S,T }, the quotient cH{S,T }(d) by a split function lH{S,T } isgiven by cH{S,T }(d) = max{0, cH{S,T }(d)}, where cH{S,T }(d) is represented as
cH{S,T }(d) = 1
2min
i, j∈S, k,l∈T
{d(i, k)+ d(l, j)− d(i, j)− d(l, k),
d(i, l)+ d(k, j)− d(i, j)− d(k, l)
}.
If, in particular, d is a metric, we have cH{S,T }(d) = bd{S,T }, where bd
{S,T } is the Bunemanindex defined by
bd{S,T } :=
1
2min
i, j∈S, k,l∈T{d(i, k)+ d( j, l)− d(i, j)− d(k, l)} .
Theorem 3 ([7, Theorem 9.6]) Let d : X × X → R be a distance. Then d can bedecomposed as
d =∑
{S,T }∈�(d)
cH{S,T }(d)l �1H{S,T } + d ′, (5)
where �(d) = {{S, T } | {S, T } : an X-split with cH{S,T }(d) > 0 } and d ′ : X × X →R is a distance with cH{S,T }(d
′) = 0 for any X-split {S, T }. Furthermore, we have
d =∑
{S,T }∈�(d)
cH{S,T }(d) lH{S,T } + d ′. (6)
An interesting case of the discrete split decomposition is the case of split-decom-posable discrete functions. According to Proposition 1, a split-decomposable functionon �1 is given by a sum of split functions for an �1-admissible set of hyperplanes.
The �1-admissibility of a set of hyperplanes can be translated to the pairwise com-patibility of X -splits. Two X -splits {S, T } and {U, V } are called compatible if at leastone of the sets S ∩U, S ∩ V, T ∩U and T ∩ V is empty. For a subset H of H�1 , wedenote
�H := {{S, T } | {S, T } : an X -split with H{S,T } ∈ H}.
Proposition 3 ([7, Proposition 9.8]) A set of hyperplanes H ⊆ H�1 is �1-admissibleif and only if �H is pairwise compatible.
123
196 S. Koichi
Fig. 2 The polyhedral split decomposition of a metric on X = {i, j, k}
Furthermore, the decomposition (5) can be interpreted as a decomposition of dinto a sum of split metrics and a distance d ′. It is easily seen that, for a split functionlH{S,T } , we have lH{S,T }(χi − χ j ) = 1 if |{i, j} ∩ S| = 1 and lH{S,T }(χi − χ j ) = 0
otherwise. Hence, l �1H{S,T } can be identified with the split metric δ{S,T }. Since H{S,T } is
�1-admissible, the extension of l �1H{S,T } coincides with lH{S,T } by Proposition 1.
Proposition 4 For an X-split {S, T }, the split function lH{S,T } is the homogeneousconvex extension of the split metric δ{S,T }.
It is known that a metric d is a tree metric if and only if d is represented as a sumof split metrics for pairwise compatible X -splits.
Proposition 5 ([7, Proposition 9.7]) A metric d on X is a tree metric if and only if theextension d of d on �1 is decomposed as
d =∑
{S,T }∈�(d)
cH{S,T }(d)lH{S,T } .
By Proposition 5, a split-decomposable function f on �1 (with f (1) = f (−1) = 0)corresponds to a sum of a tree metric and a linear function.
Figure 2 illustrates the polyhedral split decomposition of a metric on X with|X | = 3. It is known that every 3-point metric can be represented as a sum of splitmetrics, i.e., d ′ = 0 in the decomposition (5).
4 M-convexity and split decomposition
M-convexity is a key concept in discrete convex analysis introduced by Murota [8].In discrete convex analysis, the following is known.
Theorem 4 ([8, Theorem 6.59]) For a distance d on X, the homogeneous convexextension d of d on � is a positively homogeneous M-convex function on cone �.
Conversely, for any positively homogeneous M-convex function f on cone �, there isa distance d f on X such that f is equal to the homogeneous convex extension d f ofd f on �.
By this theorem, the polyhedral split decomposition as in (6) can be considered asthat of a positively homogeneous M-convex function. In this section, from this pointof view, we shed light on the M-convexity and split decomposition.
123
M-convexity in polyhedral split decomposition of distances 197
4.1 M-convex functions
Instead of giving the original definitions of the M-convex polyhedron and function,we list their characterizations, which we use later.
A function f : 2X → R ∪ {+∞} is called submodular if for all Y, Z ⊆ X, fsatisfies
f (Y )+ f (Z) ≥ f (Y ∩ Z)+ f (Y ∪ Z).
A submodular function defines a polyhedron, called an M-convex polyhedron, or abase polyhedron, as is described in [8, p. 118].
Theorem 5 A nonempty polyhedron B ⊆ RX is an M-convex polyhedron if and onlyif B is given by
B = {x ∈ RX | 〈χY , x〉 ≤ f (Y ) (Y ∈ 2X ), 〈χX , x〉 = f (X)}
for a submodular function f : 2X → R ∪ {+∞} with f (∅) = 0 and f (X) < +∞.
The following is a characterization of an M-convex polyhedron using a tangentcone.
Theorem 6 ([4, Theorem 17.1]) A nonempty polyhedron B ⊆ RX is an M-convexpolyhedron if and only if, for every point x ∈ B, the tangent cone TB(x) at point x isgenerated by a subset of �.
The following theorem is essentially shown as Theorem 5.2 in [9]. M-convexity ofa polyhedral convex function f depends on that of polyhedra in the subdivision T ( f ).
Theorem 7 A polyhedral convex function f : RX → R ∪ {+∞} with dom f �= ∅ isM-convex if and only if each polyhedron in T ( f ) is M-convex.
4.2 M-convexity of split functions
In general, a sum of M-convex functions is not necessarily M-convex. Then, we recog-nize that the polyhedral split decomposition of a positively homogeneous M-convexfunction as in (6) is a remarkable phenomenon for M-convex functions.
Theorem 8 The polyhedral split decomposition of a positively homogeneous M-con-vex function as in (6) is a decomposition of a polyhedral M-convex function into a sumof polyhedral M-convex functions.
Proof As is mentioned in Proposition 4, a split function lH{S,T } is the extension of the
split metric δ{S,T } on X. By Theorem 3, d ′ is the extension of a distance d ′ on X . There-fore, by Theorem 4, lH{S,T } for all {S, T } ∈ �(d) and d ′ are positively homogeneousM-convex functions (if they are restricted to cone �). ��
123
198 S. Koichi
The rest of this subsection is devoted to unraveling why M-convexity is preservedin (6). Since the polyhedral split decomposition is a geometric notion, we explain itespecially in terms of geometry. To this end, we focus on the polyhedral subdivisionT (d) since the M-convexity of a function d is equivalent to the M-convexity of allcones in T (d) (precisely, T (d) ∧ H1,0) as in Theorem 7.
As is mentioned in Theorem 3, the following holds:
∑{S,T }∈�(d)
cH{S,T }(d) lH{S,T } + d ′ =∑
{S,T }∈�(d)
cH{S,T }(d)l �1H{S,T } + d ′. (7)
Since lH{S,T } is the extension of l �1H{S,T } , this equation means that the sum of the exten-
sions of l �1H{S,T } and d ′ is equal to the extension of the sum of l �1
H{S,T } and d ′. This kindof commutativity is the underlying reason for the preservation of M-convexity. Bydefinition, the right-hand side of (7) is obviously M-convex, and hence its inducedsubdivision consists of M-convex cones. Therefore, the addition on the left-hand sideof (7) must produce a refinement of the subdivision composed of M-convex cones.
4.3 Quadratic M-convex functions and their split-decomposability
We start with the property of M-convex functions as described in the next theorem;see also [8, Theorem 6.61].
Theorem 9 ([9, Theorem 4.15]) For an M-convex function f : ZX → R ∪ {+∞}and x ∈ dom f, define γ f,x (u, v) := f (x + χu − χv)− f (x) (u, v ∈ X). Then γ f,x
is a distance on X.
For each x ∈ dom f, we regard γ f,x as a discrete function on �1 by setting
γ f,x (χi − χ j ) = γ f,x (i, j) (i, j ∈ X), γ f,x (1) = γ f,x (−1) = 0.
Then, by Lemma 4 and Theorem 9, γ f,x is convex-extensible on �1. In addition, thediscrete split decomposition is applicable to γ f,x .
We are particularly interested in the case that f is a quadratic M-convex func-tion on ZX ∩ H1,0 represented as f (x) = 1
2 x�Ax for x ∈ ZX ∩ H1,0 with somecoefficient matrix A. Recall that a quadratic M-convex function can be constructedfrom a tree metric, and a tree metric is a split-decomposable function on �1. For adistance d : X × X → R, the distance matrix of d is a matrix D = (di j ) defined bydi j := d(i, j) for all i, j ∈ X.
Theorem 10 ([6, Theorem 3.1]) A quadratic form f (x) defined on ZX ∩ H1,0 isM-convex if and only if there exists a tree metric d : X × X → R+ such that
f (x) = −1
2x�Dx (x ∈ ZX ∩ H1,0), (8)
where D is the distance matrix of d.
123
M-convexity in polyhedral split decomposition of distances 199
By applying the discrete split decomposition to γ f,x , we find the following theorem,which indicates a simple geometric structure of a quadratic M-convex function.
Theorem 11 Let f be a quadratic M-convex function on ZX ∩H1,0 represented as (8)for the distance matrix D of a tree metric d on X. Then, for any point x ∈ ZX ∩ H1,0,
we have T (γ f,x ) = T (d) and furthermore the function γ f,x (·) is split-decomposable.
Proof For f and x ∈ ZX ∩ H1,0, γ f,x is given by
γ f,x (u, v) = f (x + χu − χv)− f (x)
= −1
2(x + χu − χv)
�D(x + χu − χv)+ 1
2x�Dx
= −x�Dχu + x�Dχv − 1
2χ�u Dχu + χ�v Dχu − 1
2χ�v Dχv
= 〈−x�D, χu − χv〉 + d(u, v) (u, v ∈ X).
Therefore, γ f,x can be regarded as a discrete function on �1 as follows:
γ f,x (·) = d(·)+ (〈−x�D, ·〉)�1 .
By Lemma 3, we have γ f,x = d + 〈−x�D, ·〉, and hence T (γ f,x ) = T (d). More-over, since the quotient cH{S,T }(γ f,x ) depends only on d, cH{S,T }(γ f,x ) coincides withcH{S,T }(d) = max{0, bd
{S,T }}. Since d is a tree metric, γ f,x is split-decomposable byProposition 5. ��
4.4 Facets of an M-convex polyhedron
In this subsection, we apply a result of the polyhedral split decomposition to charac-terize M-convex polyhedra using their facets.
Let B be a polyhedron of dimension n − 1 in a hyperplane H1,c for c ∈ R. Then,each facet F of B, which is of dimension n − 2, is included in the intersection ofH1,c and the hyperplane HaF ,rF for some (aF , rF ) ∈ RX ×R such that ‖aF‖ = 1, aF
is normal to 1 and B ⊆ H−aF ,rF. We denote by NB the set of the normal vectors aF
corresponding to all the facets F of B. Note that NB is finite. Two normal vectors inNB are called adjacent if their corresponding facets of B share a ridge, i.e., a face ofdimension n − 3. A characterization of M-convex polyhedra using their facets is asfollows.
Theorem 12 Let B be a polyhedron of dimension n − 1 in a hyperplane H1,c. Thepolyhedron B is an M-convex polyhedron if and only if
(1) any normal vector in NB is a positive multiple of the vector a(S,T ) for someX-split {S, T }, and
(2) the two X-splits corresponding to adjacent normal vectors in NB are compatible.
123
200 S. Koichi
The proof of Theorem 12 is given in Sect. 4.4.1. We here present an applicationof Theorem 12. Consider a polyhedron B of dimension n − 1 that is expressed asB = {x ∈ RX | 〈χY , x〉 ≤ f (Y ) (Y ∈ F), 〈χX , x〉 = f (X)} for a functionf : F → R on a family F of subsets of X with ∅, X ∈ F . Since B is in a hyper-plane H1, f (X), the normal cone NB(x) for a point x in B contains the linear spaceL := {α1 | α ∈ R}. Hence, one can reset the vector χY defining a facet of B to bethe vector a(Y,X\Y ) as in (4) for the X -split {Y, X \Y }without changing B. Therefore,B satisfies (1) of Theorem 12. It is known that, if F is a cross-free family, then B isM-convex; see [4, p. 43]. This result is an immediate corollary of Theorem 12. Indeed,two X -splits {Y, X \ Y } and {Z , X \ Z} are compatible if and only if Y and Z are notcrossing, that is, Y ⊆ Z , Y ⊇ Z , Y ∩ Z = ∅, or Y ∪ Z = X. Hence, if there is nocrossing pair in F , then B satisfies (2) of Theorem 12, and therefore B is M-convexby Theorem 12.
4.4.1 Proof of Theorem 12
Let x be a point in B. Recall that TB(x) and NB(x) are the tangent and normal conesat x, respectively.
The only-if part. We firstly give a proof of the only-if part. The proof is based onTheorem 6 and the fact that
(∗) a proper face F of TB(x) of dimension n−k (k ≥ 2) belongs to the intersectionof the hyperplane H1,0 and the hyperplanes whose normal vectors in NB togetherwith the vectors ±1 generate a face of NB(x) of dimension k, and furthermorethe linear hull of F coincides with the intersection.
Proof (the only-if part) Suppose that F is a facet of TB(x). According to (∗), lin Fcoincides with the intersection Hai ,0 ∩ H1,0 for some ai ∈ NB . By Theorem 6, F isgenerated by a subset of �, and hence Hai ,0∩ H1,0 is also generated by a subset of �.Therefore, Hai ,0 is �1-admissible. By Proposition 2, we have (1).
Suppose that G is a ridge of TB(x). Since every face of NB(x) of dimension 3 isgenerated by the vectors ±1 and some two adjacent normal vectors in NB, we havelin G = Hai ,0 ∩ Ha j ,0 ∩ H1,0 for some adjacent normal vectors ai , a j ∈ NB . ByTheorem 6 again, G is generated by a subset of �, and hence Hai ,0 ∩ Ha j ,0 ∩ H1,0 isalso generated by a subset of �. Since H−ai ,0
∩ Ha j ,0 ∩ H1,0 and Hai ,0 ∩ H−a j ,0∩ H1,0
contain facets of TB(x), they are generated by a subset of �. Consequently, each conein A({Hai ,0, Ha j ,0}) ∧ H1,0 is generated by a subset of �; namely, {Hai ,0, Ha j ,0} is�1-admissible. By Proposition 3, the two X -splits corresponding to ai and a j arecompatible. Thus, we have (2). ��
The if part. We next prove the if part. We denote by Dx the family of the sets Ysuch that χY ∈ NB(x). In particular, Dx contains both ∅ and X since both 0 and 1are in NB(x). The proof of the if part uses the following lemma, whose proof is givenlater.
Lemma 5 For any point x in B, the family Dx is a distributive lattice, i.e., for allY, Z ∈ Dx , both Y ∩ Z and Y ∪ Z are in Dx .
123
M-convexity in polyhedral split decomposition of distances 201
Proof (the if part) This proof is a direct adaption of the proof of Theorem 17.1 in [4].Since B is in a hyperplane H1,c and satisfies (1), B is expressed as
B = {y ∈ RX | 〈χY , y〉 ≤ f (Y ) (Y ∈ 2X ), 〈χX , y〉 = f (X)} (9)
for some function f : 2X → R ∪ {+∞} with f (∅) = 0 and f (X) = c. According toTheorem 5, it is sufficient to show the submodularity of f. Let Y, Z be subsets of X.
Suppose that M := max{〈χY + χZ , y〉 | y ∈ B} < +∞. Since B is expressed as (9),we have f (Y ) + f (Z) ≥ M. Let y∗ ∈ B attain the maximum M. Then, χY + χZ iscontained in NB(y∗) and thus in NB(y∗)∩RX+. We show that both χY∪Z and χY∩Z arein NB(y∗). Since cone {χW | W ∈ 2X } = RX+ and B is expressed as (9), NB(y∗)∩RX+is generated by the set {χW | W ∈ Dy∗}. Then, we may assume that χY + χZ is rep-resented as χY + χZ = ∑m
k=1 λkχWk for some subset {Wk | k = 1, . . . , m} of Dy∗and positive numbers λk for k = 1, . . . , m. Since, by Lemma 5, Dy∗ is a distributivelattice and
⋃mk=1 Wk = Y ∪ Z , the set Y ∪ Z is a member of Dy∗ and therefore
χY∪Z ∈ NB(y∗). For each i ∈ Y ∩ Z , let Ji := ⋂{Wk | i ∈ Wk (k = 1, . . . , m)}.Then, Ji ⊆ Y ∩ Z for each i ∈ Y ∩ Z ; otherwise, there is j ∈ (Y ∪ Z) \ (Y ∩ Z) with(χY + χZ )( j) > 1, which is impossible. Since Dy∗ is a distributive lattice, we haveJi ∈ Dy∗ for all i ∈ Y∩Z , and furthermore
⋃{Ji | i ∈ Y∩Z}, which is equal to Y∩Z ,
is also in Dy∗ . Thus, we obtain χY∩Z ∈ NB(y∗). Since χY + χZ = χY∩Z + χY∪Z ,
we have
f (Y )+ f (Z) ≥ M = 〈χY + χZ , y∗〉= 〈χY∩Z , y∗〉 + 〈χY∪Z , y∗〉 = f (Y ∩ Z)+ f (Y ∪ Z).
Therefore, f is submodular. ��We finally give a proof of Lemma 5.
Proof (Lemma 5) If x is in the relative interior of B, we have Dx = {∅, X}, and henceDx is a distributive lattice. Then, suppose that x is not in the relative interior of B. Thetangent cone TB(x) is a polyhedron of dimension n−1 in H1,0, and so we may considerthe set NTB (x) of the normal vectors corresponding to all the facets of TB(x). Note thatNTB (x) is a subset of NB, and furthermore NTB (x) generates the cone NB(x) ∩ H1,0.
By (1) of Theorem 12, each normal vector in NTB (x) is represented as a(S,T )/‖a(S,T )‖for some X -split {S, T }. We denote by Fx and Fx the families of the sets S and T,
respectively, in all the ordered pairs (S, T ) with a(S,T )/‖a(S,T )‖ ∈ NTB (x). We con-sider the cones C(x) := cone {χS | S ∈ Fx } and C(x) := cone {−χT | T ∈ Fx }. Inaddition, we consider the family Kx of the sets Y such that χY + α1 ∈ C(x) for someα ≥ 0, and Kx of the sets Z such that −χZ − β1 ∈ C(x) for some β ≥ 0. We firstlynote that, since NB(x) contains the linear space L = {α1 | α ∈ R},
(∗1) for any y ∈ NB(x), there is z ∈ NB(x) ∩ H1,0 with y − z = α1 for someα ∈ R.
We next claim that
(∗2) Y ∈ Kx ⇐⇒ a(Y,Z) ∈ NB(x) ∩ H1,0 ⇐⇒ Z ∈ Kx , where {Y, Z} is anX -split.
123
202 S. Koichi
Indeed, Y ∈ Kx implies a(Y,Z) ∈ NB(x) ∩ H1,0, and so does Z ∈ Kx since χY =a(Y,Z) + (|Y |/|X |)1,−χZ = a(Y,Z) − (|Z |/|X |)1, and NB(x) includes both C(x)
and C(x). Conversely, suppose that a(Y,Z) ∈ NB(x) ∩ H1,0. Then, since NTB (x)
generates NB(x) ∩ H1,0, there are nonnegative numbers λS for S ∈ Fx such thata(Y,Z) =∑
S∈FxλSa(S,T ). Since χS = a(S,T )+ (|S|/|X |)1 for each S ∈ Fx , we have
a(Y,Z) +∑
S∈Fx
λS(|S|/|X |)1 =∑
S∈Fx
λSa(S,T ) +∑
S∈Fx
λS(|S|/|X |)1
=∑
S∈Fx
λS{a(S,T ) + (|S|/|X |)1}
=∑
S∈Fx
λSχS .
Note here that a(Y,Z) +∑S∈Fx
λS(|S|/|X |)1 = χY + α1, where α = −|Y |/|X | +∑S∈Fx
λS(|S|/|X |). Since∑
S∈FxλSχS ∈ C(x), we have Y ∈ Kx . Similarly, we
obtain Z ∈ Kx from a(Y,Z) ∈ NB(x)∩ H1,0, and (∗2) is proved. Then, it follows from(∗2) that Kx contains all the complements of the nonempty sets in Kx and vice versa.Furthermore, by (∗1) and (∗2), Kx ∪ {X} amounts to Dx .
Now, our aim is to establish that we have Y ∩ Z ∈ Kx for all Y, Z ∈ Kx , andY ∩ Z ∈ Kx for all Y, Z ∈ Kx . If it is established, we have Y ∪ Z ∈ Kx for allY, Z ∈ Kx with Y �= ∅ �= Z and Y ∪ Z �= X since (X \Y )∩ (X \ Z) ∈ Kx . Therefore,Dx turns out to be a distributive lattice.
In order to accomplish our aim, we consider the family Gx of the sets Y such thatχY ∈ C(x). Obviously, Fx ⊆ Gx ⊆ Kx . We firstly show that
(∗3) for all Y, Z ∈ Gx with Y ∪ Z �= X, we have Y ∩ Z ∈ Gx .
We define the partial order < on the pairs {Y, Z} of Y, Z ∈ Gx by {Y, Z} < {U, V } ifY ∪ Z � U ∪ V or, Y ∪ Z = U ∪ V and Y ∩ Z � U ∩ V . We prove (∗3) by inductionon this order < . Note that if Y ⊆ Z , Y ⊇ Z , or Y ∩ Z = ∅, obviously Y ∩ Z ∈ Gx .
Then, we may assume that {Y, Z} is a crossing pair, i.e., all of Y ∩ Z , Y \ Z , Z \ Yand X \ (Y ∪ Z) are nonempty. The induction therefore starts with the case that{Y, Z} is minimal with respect to < among crossing pairs. Hence, at every step of theinduction, we may assume that, for all pairs {U, V } with {U, V } < {Y, Z}, we haveU ∩ V ∈ Gx . Since {Y, Z} is a crossing pair, the X -splits {Y, X \ Y } and {Z , X \ Z}are incompatible. If cone {χY , χZ } forms a 2-dimensional face of C(x), then, by(∗2), cone{a(Y,X\Y ), a(Z ,X\Z)} also forms a 2-dimensional face of NB(x)∩ H1,0, andhence a(Y,X\Y )/‖a(Y,X\Y )‖ and a(Z ,X\Z)/‖a(Z ,X\Z)‖ are adjacent, which contradicts(2) of Theorem 12. Therefore, cone {χY , χZ } does not form a 2-dimensional face ofC(x). This means that χY + χZ can be represented as χY + χZ = ∑
W∈G λW χW
for some G with {Y, Z} �= G ⊆ Gx and positive numbers λW for W ∈ G. Note thatW ⊆ Y ∪ Z for every W ∈ G. For each i ∈ Y ∩ Z , there is Ui ∈ G such that(i) i ∈ Ui and (ii) Ui satisfies none of Y ⊆ Ui , Z ⊆ Ui , and Y Z ⊆ Ui , whereY Z denotes the symmetric difference of Y and Z . Indeed, if every W ∈ G withi ∈ W satisfies Y ⊆ W, Z ⊆ W, or Y Z ⊆ W, then there is j ∈ Y Z such
123
M-convexity in polyhedral split decomposition of distances 203
that (∑
W∈G λW χW )( j) > (χY + χZ )( j) = 1, which is impossible. In fact, for thoseUi , (ii) implies {Y, Ui } < {Y, Z} or {Ui , Z} < {Y, Z}. If {Y, Ui } < {Y, Z}, thenY ∩Ui ∈ Gx . Moreover, since {Y ∩Ui , Z} < {Y, Z}, we have Ui ∩Y ∩ Z ∈ Gx . In thecase of {Ui , Z} < {Y, Z}, we have similarly Ui ∩Y ∩ Z ∈ Gx . If Ui ∩Y ∩ Z = Y ∩ Zfor some i ∈ Y ∩ Z , then we have Y ∩ Z ∈ Gx , and hence we may assume thatUi ∩ Y ∩ Z � Y ∩ Z for all i ∈ Y ∩ Z . Since {Ui ∩ Y ∩ Z , W } < {Y, Z} forany W ∈ G, the set Ji := ⋂{W | i ∈ W ∈ G ∪ {Y, Z}} belongs to Gx for eachi ∈ Y ∩ Z . Furthermore, J := {Ji | i ∈ Y ∩ Z} is a partition of Y ∩ Z . Indeed, ifJi ∩ Jk �= ∅ for Ji , Jk ∈ J with Ji �= Jk, then, since (
∑W∈G λW χW )(l) = 2 for
l ∈ Ji Jk, we have (∑
W∈G λW χW )(m) > 2 for m ∈ Ji ∩ Jk, which contradicts(∑
W∈G λW χW )(m) = (χY + χZ )(m) = 2. Since χY∩Z =∑J∈J χJ is in C(x), we
have Y ∩ Z ∈ Gx . Thus, we have (∗3).We then show that, for all Y, Z ∈ Kx , we have Y ∩ Z ∈ Kx . Let Y, Z ∈ Kx , and
let χY + αY 1, χZ + αZ 1 ∈ C(x), where αY , αZ ≥ 0. Since C(x) is generated by thecharacteristic vectors of the sets in Fx , we may assume that χY +αY 1+χZ +αZ 1 =∑
W∈F λW χW for some F ⊆ Fx and positive numbers λW for W ∈ F . For eachi ∈ Y ∩ Z , let Ji := ⋂{W | i ∈ W ∈ F}. Then, we claim that Ji ∈ Kx . Indeed, letWi := {W | i ∈ W ∈ F}. If W1 ∪W2 �= X for W1, W2 ∈Wi , then since Gx satisfies(∗3), we have W1 ∩ W2 ∈ Gx and so reset Wi ← (Wi \ {W1, W2}) ∪ {W1 ∩ W2}.This process can be repeated until Wi = {Ji } or otherwise the union of any two setsin Wi is equal to X. In the latter case, the complements of sets in Wi are mutuallydisjoint. From this, one can see that
∑W∈Wi
χW = χJi + αJi 1 for some αJi ≥ 0.
Therefore, we have Ji ∈ Kx in the latter case as well as in the former case. By thesame argument above, J := {Ji | i ∈ Y ∩ Z} is a partition of Y ∩ Z . For each J ∈ J ,
there is αJ ≥ 0 with χJ + αJ 1 ∈ C(x) since J ∈ Kx . Since∑
J∈J (χJ + αJ 1) =χY∩Z + (
∑J∈J αJ )1 ∈ C(x), we have Y ∩ Z ∈ Kx .
Similarly, it can be shown that, for all Y, Z ∈ Kx , we have Y ∩ Z ∈ Kx . Thiscompletes the proof. ��Acknowledgments The author is grateful to Satoru Iwata for his useful comments. The author thanksKazuo Mutora for his helpful suggestions. The author thanks the referees for careful reading and helpfulcomments. The present work is supported by the 21st Century COE Program on Information Science andTechnology Strategic Core.
References
1. Bandelt, H.-J., Dress, A.W.M.: A canonical decomposition theory for metrics on a finite set. Adv.Math. 92, 47–105 (1992)
2. Buneman, P.: The recovery of trees from measures of dissimilarity. In: Hodson , F.R., Kendall, D.G.,Tautu, P. (eds.) Mathematics in the Archaeological and Historical Sciences, Edinburgh UniversityPress, Edinburgh (1971)
3. Dress, A., Huber, K.T., Koolen, J., Moulton, V., Spillner, A.: Basic Phylogenetic Combinatorics. Cam-bridge University Press (to appear)
4. Fujishige, S.: Submodular Functions and Optimization, 2nd edn. Annals of Discrete Mathematics,vol. 58. Elsevier (2005)
5. Hirai, H.: A geometric study of the split decomposition. Discrete Comput. Geom. 36, 331–361 (2006)6. Hirai, H., Murota, K.: M-convex functions and tree metrics. Japan J. Indust. Appl. Math. 21, 391–
403 (2004)
123
204 S. Koichi
7. Koichi, S.: The Buneman index via polyhedral split decomposition. METR 2006-57. University ofTokyo (2006)
8. Murota, K.: Discrete Convex Analysis. SIAM, Philadelphia (2003)9. Murota, K., Shioura, A.: Extension of M-convexity and L-convexity to polyhedral convex func-
tions. Adv. Appl. Math. 25, 352–427 (2000)10. Semple, C., Steel, M.: Phylogenetics. Oxford University Press, Oxford (2003)
123