on structures of subpartitions related to a submodular ... · 1 article on structures of...
Post on 14-Jul-2020
2 Views
Preview:
TRANSCRIPT
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.
- 42-
2
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}}・
- 43 -
口
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
- 44 --
4
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
- 45 -
5
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)
一46 -
6
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_
- 47 -
7
行政社会論集 第12巻 第1号
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
- 48-
8
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
- 49 -
9
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
- 50 -
10
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).
- 51-
11
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).
- 52 -
12
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.
一 53 -
13
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 ))
-,54 -
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.
- 55 -
15
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.
- 56 -
top related