1

Article

On Structures of Subpartitions Related toa Submodular Function Minimization '

Akira Nakayama

Abstract: For afmite set E.,let f :2・ii →R be a function with f (φ)= 0 satisfying at least oneof the two following inequalities such that VX,Y (: E f (X)十f (Y)> f (X し1Y)十f(Xn Y)and f (X)十f(Y)> f (X-Y)十f(Y-X).The former inequality iS Called SubmOdular,while some examples of the latter one are seen in a literature We consider a problem of minimizing the function f on the set SPAkof subpart1t1ons of A ⊂E;where k is some positive integer and the cardina」itv of each subpartition in SPAkis at mostk.We show some structures of subpartitions related to the submodula1・ function minimization_ Final]y we present two related topics One is with respect to Cont「aPolymat「old,the other is about a mnim1zatlon problem of symmetric submodu1ar function_

1. Introduction

Let E be a finite set and R be the set of reals. A function f :2E_,R is called a

submoduiar f unction on 2E if

(1) f (X)十f (Y)> f (X U Y)十f (X n Y) (VX,Y (:E).

A function9:2E →R is called supem od,ujaron2E if _9js submodular Let h :2g→R

be a function sat.isfying

(2) lt(.X)十h(Y)> h(_X - Y)十h(Y - X) (?X;Y (:E).

1The author ls grateful tea referee for his useful commerts otl theoriglnal versjon of ti e present paper.

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OnStructuresof SubpartltjonsRelatedtoaSubmodularFunf t1onMinimizatior1lAKfRANakayama1

Examples of the functio11-h such that ti satisfies(2)are seen ln l21.For A(:E d11d

some positive integer k let SPA k(resp.SP?)be the set of subpartitionsof A such that

for each subpartition li ef A,wehavelnl くk(resp.:Ii iく kl(VB ? n)).If k> Al,then

both SPAkand SP?are ejulvalent to the family SPA consisting of every subpartition

of A. We ol1iy consider the case when k くt A j,because we have SPA k = SP- 1

and s p? _ Si:「I ter any k 、>tA j. Moreover,for AC E and a pair (k,りof positive

integers with k d"l A l and l d"l A 1_ We regard {φ}as the unique subpartitio''of

empty set φ,and {φ}is called in it subpartiti on 010. Hence we ha,ve iφ}∈SPAk・

Throughout this paper we assume A? φ. For f :2E→R and n E SPAk f(if)is

defmed by f (n)= ∑B∈nf (B). Let fmln(SPA k)= minnespAkf (n)_ Simila「iy..We also

defmenotation fm1n(SP?)for SP?.

2 A few results

Letting mt= tEf l (t = 1,2),mt,2= l Ei'1U E? 1,and mt'2= lEi'm E2 1,eu「 main 「eSultS

are the following theorems.We will prove them later.

Theorem 2.f Lot El,E2 (二E be nonemptlf, and f :2E → R be a function unth

f (φ)= 0 satisfjlin9 at least one of th.e inequalities (1)and (2)for any X,Y (:E For

ttuo gtt1en positiue into9ers k (i = 1,2) such, that k d" mi,.there erist posi加1e mte9e「S ki

anti kいatisfyin9k,十k? < kt十k2十1,kl d" m1,2,k?d" max{1,m1'2}f and

(3 ) f mln (S P Et k1)十f ruit,(S P E. k ) > f min (S P EIUE k ) 十f min (S P E rli ;'.k ), 口

c orollary 2.2: Under tんe same condition asm theorem (2,1f ule hatle

(4 ) f min (S P E k ) 十f ml・(S P n j k ) ) f min(S P E Ui 2 k ') 十 f-min(S P Ell ni 2k? )'

,aherekj = min{kt十k2,mt,2}and k2= min{kt十k? max{1,mt'2}}・

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3

行政社会論集 第12巻 第1号

Theorem 2.3: Under the same conditi on as in theor、em .12,1) for tuo gmen posi.ti,?e

i nt eger s Lt (i i d" m t) and t2 f 2 d" m 2) ule ha1,e

(5)fm,r,(SPii )+ fm,n(Sp??)> fm,n(sp?,?'?,j21)+ f mn(sp船 ''1),

uhe「e L12= max{11,12},n1(i i,i2)= min{mt2 L? 」十3 2m',:'-L','(L12- l)-l-

1}andn2(11、12)= min{mt'2,n1(ii ,i2)-1}. 口

First?we make some preparations for ft proof of theorem (2.1).Given twosubpartitions

nt ? SPE k, with kt d" mt for t = 1,2.colslder a family nl U II2 and a funct1on

f :2f1→R satisfying f (0)= 0 and at,least one of (1}and (2)for any X,Y (:E where

E,e E ls nonempty,k, is some positive integer and function h in (2)should be replaced

by f .Let A be the set of such functions.Note that nj U n21s not always a subpartition

of SPE,uE,,kt where k「 -= min{kl 十k2,mt,2}. For nl U n2 and f ? A,consider the

following algorithm 1.

Algorithm 1

Input:Il l U Ii2,t E A

Output:a subpartition nf of SPE- 12k1

Let nf = nlUI12.While we have two distinct sets B and Cot Il f satisfying B「1(.? (・

and (2),we repeat the following(6.a)~(6.b):

(6.a)Put n :- 「ff _ {B,C},If C(_B (resp.B(::C) then n' is

replaced by n'U{11'- C},(resp.n'し'{C_ B}).Otherwise,put

Il f :- Ilf U{B- C、C- B}.

(6.b)If f (fi - C)> 0(resp.f (C- B)> 0)fol B- ( ? φ(resp.B_ C? φ) then

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onslructures of subpartjtjmsRela1edto a Submodt11ar Functiotl Minimization lAKIRA Nakayama1

「If is replaced by nf_ {ii -C}(resj1.n - {C- B}). 口

After ajgorithm l isover,we have the fo11owi lg lemma:

Lemma 2.4: Ii; be the family I1 obtamed after j -tfi loop (6,a) ~~ ) of aL90- m i ,

ulhere nいs the input 111 U n 2. Th,en ue hat' (7)~112)

(7) For anti X ? n;,,fie have Y ? Ii ;_1 uch that X 〔:Y .

(8) IIi:ld"l n;_11.

(9) ∑xen.f (X)d" ∑x?n,_,f (X).

(10) Σx∈n l X ld" ∑xerl , l X 1- 2 and maxxen l X ld" maXxEn _,い'( l

(11) 1111「1 n 21d" l E'1「l E21.

(12) ln1U Ii 21d" l I I U12l.

(proof)It js easy tosee(7)~(10)by using inductionon j. Weonly prove(11)by

lnductjonon n =いl i n t 21.(12)fOllOws f「Om (11),l Il lU Il 'i i= 1n1- n21十l n1「11121

十1n2_ 1111,and nt _ n3_t ? SPE,_E,_, (t = 1,2). Assume that We have (Ii)to「

any E,,E2( E satisfyi11g n < k Consider the case when n = k 十1_ Choose a block

B ∈11m n2 We can write nt (t -_ 1,2)as li lt U{f1'},who「e nt = Il t- {Ii}・Then We

have n:∈SPE_B (t = 1,2).Ftom l nt 1,2(li t- B)lく k We haVe l n l n n2 1= 1[11n i fi l

十1d" l nt=12(Et- B)1十1d" l nt_1,2Et l- 口

If we consider output li ef algorithm 1,then from lemma (2.4)we have? .x?n~X)>

∑x n f (X)and l f1 1d"lnold" min{m12,k1十k・1}.Note that the number of tepetit1onS

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行政社会論集 第12巻 第1号

of (6.a)~(6.b)of algorithm l is at most li t.

Next,given n,U n2 and f E A the following algorithm devides the initial family

111 U n2 into a modified family n f and a family △of sets of El n E2,

Algorithm2

Input: n l UIl2,t EA andΔ:-_ φ

Output: n ,△

Stope:put n = n,u n2.

Step 1:Carryout Algorithm l with inputs n and f .

Step2:Let Il f be output n ot step 1.If we have two distinct sets B and Cot nff

Satisiying B「、 C:ラ1:φ,then clo the following and return to step 1:

Put 「I := (nff - {fi e })U f fi U C}.

If f (11'U C)> 0,then put l ff := ll f1- {BU C}.If f (Bn C)< 0,then

put△:= ΔU{BnC}. 口

Note that B et∈ii satisfy (1)at each step 2 of algorithm 2,because algorithm i is

used as a subroutine at step 1.For t= 1,2,let

(13) n t ≡ {Bt.11Bt,21- 1Bt r lt1}

where r(t)(r(t)d" k,)is some positive integer. After aigorithm2isover,we have the fol1owinglemmas:

Lemma 2.5 Let n (resp.△,)be the famii11 n f (resp.△)obtained after j_th 1olyp of

algori.thm 2,ulhereng = n lUn2 and△o= φare the mputs. Then u,e have (14)~(18)

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OnStrllctm・esol Stlbpart通onsRelatedtoa Submoduiar Function Minimlzatior1(AKIRANakayama1

(14)l n ld"l Il ,1-1.

(15)0 d"lΔ,1- l △i_l ld" 1,f (n「)d" f (n f ,),f (Δi)d" f (Δ,_1)d" 0。

(16)∑xen lX ld" ∑x?nil_,lX l- 1.

(17)maxxal ,、X ld" maxxEn.1X ld" min{2maxx∈n_ l X l -1,

1 x?nr_,、X l - l「I l l + 1,m,,2}.

(18)maxxe?,_,l.X ld" maxxc△,1X ld" max{maxx?:?__,lX 1,max)(?n ,い(1-1}・

u,h.ere f (△,)一∑xEA,f(X),

(Proof)Use induction on i and lemma (2.4). 口

Lemma 2.6 Let l,≡{(t,j):l d" j d" r(t)}for t= 1,2_Th,en ule have(19)~(2t).

(19)Each X ∈「f is ufri tten as X -_ U- (Ul,_ ,x,B,?pi)f

ulh,ore JtXj is some subset of It and Bt? i,ulhich is nonmcreasm9 uith respect te l (i .e.,

Bt?pi ( Bt?pi-1) is a subset of Btp ∈lit. We mail ha?e X = B.?pi f or some(s,p)? I IUI2.

I n this casef me re9ard J3X sj as empt11 set,

(20)For anti dAstjnct pair X Y ? n「 (Ut_12JtX)n (U, 12Jty )= φ,

ulhereX 三Ut 1,2(U(t- ,x,Bt? i)and Y 二一Ut-1,2(U(t- ,?lBt?f ).

(Proof)Use induction on i and lemma (2.4). 口

Let 11,三Uxal Ul,,- 1.B,?'' for t= 1,2.Then lemma (2.6)shows that n1 is a subpar_

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tition ol E, and that each element of n is expressed as a union of all the elements In

some subfamily of Ill U nj.

Le m m a 2 ・7 : Su pp os e that X ∈ n j i S tiln tten as X 三 Ut= j .1j (U (t_ txj B t? Ii ). I f tlj er e

elnsts Z ∈nj - {.Xl such that 時 i (:Z for (t,p)∈J X,then- laue

(21) l {Zf ∈I1 - {X'}:jill ( Zf}l= 1,

(22) There e:flist JaZ,.C 13_, and a f amli11 {B?_i,,∈nj_t ;(3 - t ,q,)∈J?_,;.}such,

that Bt??i ( U(3_tf- : jB?二itq.

(Proof)For (t,p)E J,X,let Z1and Z2be distinct.elementsof n - {Xl such that 1;ill '⊂

ZJ (j = 1,2).From lemma(2_6),we have (t,p)? J,?13(j = 1,2).Hence,B,?,f is eeyore(i

with each family{B3t ',j?:(3_t,q)? J??,,j}(j = 1,2)_Choose aft element e∈B,??i.The11

there exist (3- t,1b)E Jj? t.j (j = 1,2)such that ii 3.jq ? nj_t and e ? B,?f 「l fia t,n

113Z2j 9.From B?3f?1( B3_t , (3 = 1,2),we must haveq1= q2_From lemma (2,6),we

have a contradiction_ 口

Lem ma 2.8 1f Y ∈△, is determined at step 2 of i -th.toop,then ule hat,e

(2.3) Y i s ulr itt en as Y = 「l t= l.2(U1t plf:K,,B tl pi ),

u,here K,, is some subset of j, and B,?pi ∈目,((t p)? K,)for t = 1 2 Note that y

mail beurritten as Y = B?f? f or some B,??i E1l j.

(24) Suppose that Y is .written as Y = 「.1,__l.2(U1tp1EK,.、Bt?t ). Then tot af 11 Pat「

((1,p ) (2,q)) E K t,j x K2,, and anti int 9er u1 (u > i), ule ha ll,e no Z ? △,,f such

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onstrucluresof subpartjtjons Rejatedtoa SubmodujarFunetion Minimization(AKIRANakayama1

that B「業出n il境'⊂Z.

(proof)Assume that we have VW∈111_1 satisfying VnWji;φjust after(i- l)-th loop・

we only consider the case when we have nov W∈jil l satisfying V 「、l W≠φand (2)

at the beginningof step l in i_th1oop_The proof of the case when we have V:W?::ni l

satisfying v n w ? φand (2)is similar. At the beginning of Step 2 of i-th loop in

algorithm2,let c andDbe distinct elements of nff satisfying e nD≠φ・Note he「e that

we have 1l f = Ilf = nj l ln this case.From lemma (2.6).,C andDa「e exp「eSSed as C=

Ut_12(Ult_ ,?, B常一1)and ,D= Ut=12(Ult- ,o,_,Bt?pi-i).F「Om Y = C n D We haVe

Y = nt=l 2((U(tplt Jc_,Ej常一1)U(Ul3_t- 30_ Ii?)j p1))・ Let Kt ,= JtC_1U J3Dt,i_l fO「

t = 1,2 Then we have(23).For any X∈I1 -{CUD},we ha:ve J,X_1= J,X,Whole t=

1,2 Let ((1,p),(2,(1))be any element in K1.j x Kt.,then we have(1)p)? J1?iU Jj, and

(2,q)? J1(jul 2?j From JtCuo = JtC_1UJtf)_l (t = 1,2),we have{(1;P)(2,q)}( Jf uDUCUD 口

J 2 yi '

Lemm a 2.9; Δ, is a subpartiti on of E1「、E2.

(proof)Assume that we have X,Y ∈△, satisfying X 「1 Y ≠φand X ≠Y-Let u (「eSP・

u)be the number such that X (resp.Y)is taken in△at Step2of u-th(「eSP-11-th)loop

Assume u <11 From lemma (2.8) X and Y are written as X = n,_1,2(Ul',pi?K,,uBi 「)

and Y = nt_l2(U(tp)EK B?f ). For 、.? ∈X n Y we have ((1,P・:),(2,q・))? Kt,,・ X K2・''

and ((1?p,),(2,q:,))∈Kt,u>く K2,.such that at ∈(B?p?n B2?j?)n (B?'p? n B2?;??),Since

11? and n,are subpartjtjonsof E, for t = 1,2,we have P.,・= P,,and q'・,= q,・Heiite,We

have ◆,「p' n gj三c y,This contradicts lemma(2.8). 口

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行政社会論集 第12巻 第1号

Lemma 2・10 Let 9(Il ,△i)= f (Ii )十f (△j), Then ulehaue

(25)、Il ld" l n1U n21- i,

(26)l△j ld" i,

(27)9(Il ,△:)d" g(Il 11△j_l)

(Proof)Use induction on i.Note that△o= φinitially and that nlUn2U{BUC}? nlU

Ii2at step2of algorithm2. 口

Let 「l and Δ..be the families obtained after algorithm21s over. Then we see that

n (resp. △_)is a subpartition of E,U E2 (resp.El n E2). Now,we give a proof of

theorem (2.1). Let f (11't)= minn?spg,kt(Il)for t = 1,2,and αi =j il l.,13 = l △i i,

γ= ∑xen .X l,5,= maxxen lX l,and,,= maxx,?,lX l.Apply inputs n?U n2

and f to algorithm2.

Proof of Theorem (2.1),From lemmas(2.4)and(2_10),we have

ai 十βd" lnfj U n 21d" min{k1十k2,mt2} (0d" i d" ln11U n2 l -1).

Let k1=1n l and k2:=:l △.。1,where we define h2= I ter 1△.。l_ 0.Note here that

hi d" m1,2andk2d" max{1,m1,2}. Then ffomlemmas (2.4)~(2.10)we have theore1n (2.1). 口

Nextf we only give a proof of theorem(2.3).

Proof of Theorem (2.3): Let I°= {i :1 d" i d"l n,U n? l -1,5,_1 < 6,}. We

oniy consider the case when I' :≠φ. Let I'= {?l,j2,- ,is}. From max{ii,t2}十

Σ1d"md" (ij -(5jm_l)d" ml2 we have s d" m12- max{il i2}From lemna(2.5),we haye

5j d" 2 (11o- l)十1(0d" i d"11lj U Ilj 1-1).Fromδo= max{ii,12}and i < mt,2- 1,we

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OnStnjcturesof SubpartjtjonsRelatedtoa Sabmodular Fmction MinimizationlAKIRA Nakayama1

have

6j d" 2m',2-''a'{1'- (max{i1,i2}- l)十1 (0d" i d"l n?U nj l - 1).

0n theother hand.,froln? d" max{t1,12}and

51n un _1d" γIn un 2- am un _2十1 < -Io- In 'l U u 1l 十3 = ?o- a a十3,

we have

_ , 1 ・ , .

0, く lm _l_m il l _ ー 、一 一ハ max{ii,i2}J - ' - - '- ' - ' ' '

Let j'= max{i :1d" l d"l n?U 11i11_1,f.j_1< E,}. If there exists noi',then we have

0= ?1= _ = ,lnu11_1d" 61n,un1_l- 1.0therwise,from (18)we have t,・_1< 6,,_1- i

and t1= tj.for any j (f 'd" J d"l n?Unj 1-1)_In both cases we have e.d" 111n- t1_1- 1fO「

any i (0d" i d"l nj U n:2l -1),Hence,we have theorem (2_3). 口

3 Two related topics

In this section,we present two related topics.0ne is with respect to cont「aPolymat「Old,

the other is about a minimization problem of symmetric submodular function.Fi「St,We

show that the above theorem (2.1)leads a result on contrapolymat「OidaSa CO「olla「y.

second we describe some properties of a specific(i.e.,symmetric)function SafiSf1'lg at

least one of the inequalities(i)and (2).

Before descrlb1ljg the following corollary of theorem (2、i),we deflue Cont「aPolyma-

trojd The corollary go'neralizes theorem 7.f in 12]a little. Let p : 2E - R b( a

supem odular function sudl that p(φ)= 0.p is called monotone-nondecreasing if

(28) p(X)d"p(y) (X(Y(_E).

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行政社会論集 第12巻 第1号

For a supermodular,monotone-nondecreasing,(integer-valued)function with1J(φ)= 0,

consider a polyhedron

(29) C(p)≡{a;∈RE:::?(B)> p(.11) VB (:E}.

C(p)is called a contrapohJmatf、old_

c orollary 3.1:Let g :2E →R be a function such, tliat -9 ∈A andQ= {z? RE:

_,> 0,~(A)> g(A) (vA c E)1. Then Q is a contrapoi11matroid C(p),uh,ore p is the

u,u ,tue supermo,iuiar f unction dejiried blf

(30) p(A)= maxn?spA-9(n) (A⊂E)_

(proof)First we show C(p)= Q. Let ?;∈C(p)and A (: E. From subpartition

{A}of A,we have:r(A) > p(A)> -g({A})= g(A). From {φ}∈St:lei,we have

a-(e)= 111({e})> p({e})> 0 for e ? E He1ice,we have ? > 0_This means C(pi ( Q.

On the other hand,let.9(nA、==;maxn∈spAg(]l )and n,4= {At,A2,_ ,At}.For z? Q,

We haVe -(A) = Σ1くくt:,(A,)十一(A - Ui<iくtAj) > :i:1くjくt9(Ai)= P(A). Note lie「e

that ・(A_ U1<d"tAj)> 0.Therefore we proved C(p)= Q.As p is clearly monotoue-

nondecreasing.,a11we have toshow is that p is supemodular.For 、dny X.Y(::E fie'''

theorem(2.1)with kt= lX la11dk2=lY l,we have

(31) p(_X)十p(Y)= -(milln?sp (-9)(11)十11111m?sp、(-9)(n))

d" _(n,1n1,?s, u,,(_g)(n)+ u,i,1l1?sp、,_,、,( g)(n))- p(X u y )+ p(x r1 y ).

Hence we have this corollary. 口

we describe theorem7,f in :21 in detail. Cotlsider a gtaph or digraph G= (t/,E).

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onstruct11resof stlbpartjtjonsRelatedtoaSubmodularFunctjonMinimizatiot1(AKIRANakayama1

For AB⊂v',d(A,B)denotes themmber of edges between A- Band B- A in any

direction. We denote d(A)≡d(A,l/- A)for A ⊂V_ DefineR(A)三max{r(u,,,1):

u ∈A t,? v _ Al ter A(二V,where r(u,t1)(u,t1∈1/)is a l1omlegative intege「-valued

function on the pair 01 vertices that serves as the demand for edge-connectivity between

u and,, The theorem corresponds to corollary(3.1)with P(A)= R(A)-d(A)for any

A C V.

We describe some comments on another topic related te a submoduiar functiOnm111-

1m1zat1on problem Recently,M.Queyranne(同)proposed a fi「St PolynOmial a11d COn1-

blnator1al ajgorjthm for minimizing a specific submodular function though the「e a「e no

such algorithms for general submodular functions.The specific function f :E - R is

s11mmetrlc If f (s)= f (El _ S)for any subset S of E.Now?we consider a Symmet「io

submodular function satisfying at least oneof the inequalities (1)and (2)- eu「 P「Oh-

lorn of mlnlm12:ing the latter function generalizes a submodular function mi1liililZat,Ion

problem hi the fo11owilig we show a recursive property on a probiem of miliimlZing a

symmetric submodular fimct1on?atisfyii1g at least one of the inc(1ualities (i)and (2).

Le m m a 3 2 ; 1f 9 j s s11m m etri c, th,on to t cach e ? E ther e c a s ts a s u bset A ct E Su ch

that g(A)= minse2Eg(S)and e ∈A・

(proof)Let A be a subset of fJsatlsfying that 9(A)= minsa g9(S)・If e? A,th- by

9(A)=9(E_A)we have this lemma. 口

For e ∈E let Ee = E _ {e}. Consider function he :2Ee →R defmed as he(S )-

min{g(e十sf),9(s )}(S ∈2E)1 where e十S = S U{e},Then we have t・he fOnOWing

lemma.

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行政社会論集 第12巻 第1号

L e m m a 3・3 : i f 9 1S svm m etf、i i (r esp .subm odujar) then he i s slfmm et rtc (1- subm od

ular)

(P「cot) For S ∈2Ee.,we have h,e(Ee_ S ) = min{9(e十(Ee_ s )),g(Ee_ s )} =

min{、g(Ii -S ),9(E-(S 十e))}= min{9(S ),9(S 十e)}= he(S ).Hence,he is symmetric.

Next,suppose that 91s s1lbmodular.Then we show that for any pair (St,Tf)∈2Ele x 211

(32)h,e(S )十h.(Tf)> he(St U T)十he(St r1 T)

Weo11ly Consider t.he (、tse when g(1十S )< 11(1))and g(e十T )> 9(T').The remail・line

cases are similar.11i this case we have

(33) he(S )十he(T1> g(e十(S UT'))十9(S nT 、1> he(S'UT )十he(St∩Tf) 口

Lemma 3.4:If 9 1s a function satisfym9 (2) sols h,e.

(Proof)Suppose that g satisfies(2).Thou we show that for any pair (S,T')∈2iie x2a・

(34)he(S )十he(T )> he(St _T)十he(T _ sf)

We Only Consider the case when 9(e十S')< 9(S )and g(e十T')くg(「 ).The remaining

Cases are similar_In this case we have

(35) he(S )十h,e(T )> 9(S -T )十g(「 - S )> he(S'_Tf)十he(Tf _ sf) 口

We carefully observe the above two lemmas Consider the follwing inequalities: For (S ,.T')∈2E x 2E ,

(36) 、g(S )十9(T)> 9(St U Tf)十g(S n T ).

(37) 9(e十St)十g(e十T)> 9((e十S )U(e十T))十9((e十s )n (e十T ))

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14

off structures of stlbpart通onsRelatedtoa Submodular Function Minimization(AKIRANakayama1

(38) 9(e十St)十9(T'?> 9((e十S)U T)十g((e十‘S )n T')・

(39) g(S )十9(T'?> g(S -T)十9(T -T )-

(40) 9(e十S )十9(e十T)> g((e十S )- (e十T ))十9((e十T)- (e十S ))・

(41) g(e十S )十g(Tf)> g((e十S )-T)十g(T'- (S 十e)).

Then we have the fo通owing lemma.

Lemma 3.5: if g is a function satlsfjlin9 at least one of the tu1oine91iatitieS(1)and(21,

so ls he,

(Proof)Suppose that g satisfies at leastoneof (1)and(2).Then we need to show(32)tnd

(34)for any pair (S ,T)? 2gex2fi .Weofliy cousider the case when9(e十S )く9(S )and

9(e十T)>9(「 ).The remaining cases are similar.In this case we have the left-hand-side

of (32)is equal to9(e十S')十g(T),If (38)holds,then we have(32).Otherwise,f「Om(41)

we have(34).Note that the right_hand_side of (38)etluais9(e十(SfUTf))十9(S nTf)and

the right,_hand_side of (41)e(lualsg(e十(S -T))十9(T -St)・ 口

From the above lemmas we see that if 91s a symnetrle submodula「 function Satisfying

at least oneof the two inequalities(1)and (2)f so ls he_

4. References

n T Na1toh and A. Nakayama: Note on Stnlctures of SubpartitionS Related te a

submodu1ar Function Mjnjmjzatjon Instjtut,c ot Socio-EcOnOmiC Planning,U11iV・ of

Tsukuba,July,1993.

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行政社会論集 第12巻 第1号

[21 A.Fra11k: Augmenting graphs to meet edge-connectivity,SIAM Journal on Discrete

Mathemati cs,5(1992)25-53.

同M.Queyranne: Minimizi11g symmetric submodular functions,Mathematicat P1ogram-

mng Societ1182(1998)3-12.

Acknowledgment

The author wishes to thank Professor Takeshi Naitohof Faculty of Economics,Shiga University for valuable advices and suggestions on the original version ofthe present article.

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