Polynomial solvability of cost-based abduction

14 Pages·2003·0.87 MB·English

laicifitrA
ecnegilletnI
REIVESLE laicifitrA ecnegilletnI 86 ( 1996) 157- 170
hcraeseR etoN
laimonyloP ytilibavlos fo desab-tsoc noitcudba *
eneguE sotnaS ,**arJ eneguE .S ’qbsotnaS
a
Department of Electrical and Computer Engineering, Air Force Instifute of Technology, Wright-Patterson
AFB, OH 45433-7765, USA
h Department of Computer and Information Sciences, Youngstown State University, Youngstown, OH 44555,
USA
devieceR yaM 1995; desiver enuJ 1996
Abstract
nI tnecer laciripme seiduts ew evah nwohs taht ynam gnitseretni desab-tsoc noitcudba smelborp
nac eb devlos yltneiciffe yb gniredisnoc eht raenil margorp noitaxaler fo rieht regetni margorp
.noitalumrof eW eit siht ot eht tpecnoc fo latot ytiraludominu morf krowten wolf ,sisylana a -nuf
latnemad tluser ni laimonylop .ytilibavlos morF ,siht ew nac enimreted eht laimonylop ytilibavlos
fo noitcudba smelborp ,dna ni ,noitidda tneserp a wen citsirueh for hcnarb dna dnuob ni the
laimonylop-non cases.
1. Introduction
nI desab-tsoc noitcudba 3,9], [ sesehtopyh evah detaicossa costs, dna the cost of
a proof si ylpmis the mus of the costs of the sesehtopyh deriuqer ot etelpmoc that
proof. selpmaxE of hcus proofs nac eb dnuof ni [ 1,3,9]. lartneC ot siht approach si
the esu of detcerid cilcyca graphs dellac SGADOAW (or, dethgiew RO/DNA detcerid
cilcyca graphs) ot tneserper spihsnoitaler neewteb sesehtopyh dna the ecnedive ot eb
.denialpxe Each edon stneserper emos ,noitisoporp dna the snoitcennoc ylticilpxe liated
the spihsnoitaler neewteb tnereffid .snoitisoporp tI has neeb nwohs ni [3] that feileb
noisiver ni naiseyaB skrowten [7] nac eb yletarucca deledom yb desab-tsoc .noitcudba
ehT costs are deterpretni as evitagen gol .seitilibaborp ,yletanutrofnU gnitupmoc the
laminim cost noitanalpxe has osla neeb nwohs ot eb drah-PN [ 31.
* laicepS sknaht ot eht suomynona sreweiver ohw depleh yltnacifingis evorpmi siht .repap
* sihT hcraeser saw detroppus ni trap yb RSOFA tcejorP #940006. :liam-E .lim.fa.tifa@sotnase
’ :liam-E .ude.usy.sic@sotnas
0004-3702/96/$15.00 thgirypoC @ 1996 reiveslE ecneicS .V.B llA sthgir .devreser
PIISOOO4-3702(96)00016-l
851 E. Sums JI; ES. Suntos/Artijicd Intelligence 86 (1996) 157-I 70
tneceR laciripme [ 2.91 seiduts evah nwohs that ynam gnitseretni desab-tsoc -cudba
noit smelborp nac eb devlos yltneiciffe gnimrofrep retteb naht gnitsixe tsrif-tseb search
seuqinhcet yb gniredisnoc the raenil margorp noitaxaler of rieht regetni margorp -rof
.noitalum nI ,ralucitrap a tnacifingis rebmun of test cases [ 2,9] were devlos hguorht
raenil gnimmargorp enola tuohtiw gnitroser ot hcnarb dna .dnuob raeniL gnimmargorp
si nwonk ot eb elbavlos ni laimonylop emit [ 6, IO].
eW yduts siht nonemonehp dna enimreted the snoitidnoc for gnivlos noitcudba -borp
smel ni laimonylop .emit ruO approach seit the tpecnoc of latot ytiraludominu morf
krowten wolf sisylana ot ruo desab-tsoc .melborp latoT ytiraludominu si a latnemadnuf
tluser ni noitazimitpo no laimonylop ytilibavlos [6, .]ol ,eromrehtruF for those -ba
noitcud smelborp gniriuqer hcnarb dna ,dnuob we tneserp a wen citsirueh desab no
ruo stluser hcihw yllaitnetop secuder the llarevo rebmun of sehcnarb that deen ot eb
.derolpxe
2. Solvability
desab-tsoC noitcudba smelborp are denifed as detcerid graphs, dellac WAODAGS (or,
dethgiew OR / AND detcerid cilcyca graphs), where the sedon tneserper ,snoitisoporp
the snoitcennoc tneserper lacigol ,spihsnoitaler dna the sthgiew at each edon tneserper
the costs [3,9]. ,yllacitnameS these costs nac eb treated as the evitagen smhtiragol of
lanoitidnoc seitilibaborp as nwohs ni [ 31.
Notation. R+ setoned the evitisop .slaer
Definition 2.1. A WAODAG si a elput-4 .G( c. r, S), :erehw
( 1) G si a detcerid cilcyca graph, G = (VE).
(2) c : V + +RI si dellac the rsoc .noitcnuf
(3) r : V ---t {AND, OR} si dellac the ./e&l hcuS sedon are dellac AND-nodes dna
OR-nodes .ylevitcepser
(4) S C V si dellac the ecnedive nodes.
Notation. roF each edon 4 E V, D, = p{ / (p. y) E ,}E the stnerap of q.
Notation. 1 . 1 setoned the ytilanidrac of a set.
nA noitanalpxe of the ecnedive si the emas as a proof. More ,yllamrof we enifed siht
as :swollof
Definition 2.2. A hturt assignment for a WAODAG W = (G, c, r, S) where G = (YE)
si a noitcnuf e : V ----t {true,false}. eW yas that hcus a noitcnuf si na explanation for
W fi dna ylno fi the gniwollof snoitidnoc :dloh
(1) t7’q E V s.t. r(q) = AND,e(q) = true + bfp E V s.t. (p,q) E E,e(p) = true.
(2) qit E V s.t. r(q) = OR, e(q) = true + 3p E V s.t. (p, q) E E dna e(p) = true.
(3) EqY S, e(q) =true.
E. Santos JK E.S. SantodArtijcial Intelligence 86 (1996) 157-170 159
no-one-home-7
.giF 1. A relpmis WAODAG. The edon-DNA house-quiet si the observation. The nodes no-shows,
no-one-home dna sgnos-dab era the sesehtopyh with detaicossa stsoc ,6 7 dna ,3 respectively.
oT order the elbissop ,snoitanalpxe we enifed the cost of na noitanalpxe as :swollof
Definition 2.3. ehT cost C of na noitanalpxe e for W = (G, c, r, S) where G = ( y E)
si
c
C(e) = c(q). (1)
}eurt=)q(+e{
nA noitanalpxe e hcihw seziminim C si dellac a best explanation for .W
Note that snoitidnoc ( 1) dna (2) ni noitinifeD 2.2 evoba are dexaler morf the lanigiro
noitinifed dnuof ni [9]. gnidroccA ot smeroehT 2.14 dna 2.15 morf [9], the tseb
noitanalpxe denifed here nac eb demrofsnart otni a tseb noitanalpxe for the lanigiro
.noitinifed hcuS a noitamrofsnart nac eb deveihca ni 0( )IEI steps.
nI .giF 1, emussa that teiuq-esuoh = true si the noitavresbo ot eb .denialpxe enO
elbissop noitanalpxe dluow eb the gniwollof tnemngissa of hturt :seulav {housequiet,
radio-off, bad-songs, ,ffo-vt }swohs-on era dengissa true dna }emoh-eno-on{ si dengissa
false. A dnoces ytilibissop si ot ngissa lla of meht ot true. dnA as a driht ,ytilibissop
we nac ngissa ,teiuq-esuoh{ ,ffo-oidar W-off, }emoh-eno-on ot true dna ,sgnos-dab{
}swohs-on ot false. eW dnif that ruo three snoitanalpxe evoba evah costs 9, 16 dna 7,
.ylevitcepser Of the three, ruo tseb noitanalpxe si the driht eno htiw the cost of 7.
neviG a GADOAW W = (G, c, I, S) where G = (YE), tcurtsnoc a l-O integer linear
program, ,)W(L sa :swollof ehT laer selbairav of )W(L si the set of selbairav dexedni
yb V, that ,si qx{ 1 q E .}V esehT selbairav are denrevog yb the stniartsnoc
roF each xq,
l
( 1) fi r(q) = ,DNA neht
xq < xp (2)
for each p E D,;
160 .E sotnaS ;IJ .SE laic’itrA/shnaS ecnegilletnI 68 )6991( I-751 07
(2) fi r(q) ,RO= neht
c X,’ > x c/ (3)
’I E D‘,
roF each q E S, xq = .I
l
roF each xq, xq si rehtie 0 or .I
l
ehT evitcejbo noitcnuf gnieb deziminim si
c
.)q(c,x
(“)L(\V) =
:etoN eW evah ylno nekat a tesbus of the stniartsnoc morf the lanigiro noitalumrof [9]
for ruo .noitamrofsnart nI ,ralucitrap the nwod-pot stniartsnoc ni [ 91 are ton .deredisnoc ’
,niagA gnidrocca ot smeroehT 2.14 dna 2.15 morf [9], siht metsys si tneiciffus ot dnif
a tseb noitanalpxe for .VI
Theorem 2.4. An optimal noitulos for )W(L si a best explanation rof .W
roF .giF ,I we evah the gniwollof metsys of raenil :seitilauqeni
>
Xhouse-quiet 6 Xtv-off
5
Xhouse-quiet < .xradic-off
Xhouse-quiet = .I
7
-xbad-songs f Xn+one-home 3 -xtv-off
-xno-shows f Xnwxx-home rN> radio-offs
dna the gniwollof evitcejbo noitcnuf
)W(L@ =
sgnos-dabX3 + emoh-em&~ + ~-%-shows.
htiW the noitamrofsnart ot regetni raenil ,gnimmargorp we won enimaxe the latot
unimodularz’ty [ 6, lo] of the tniartsnoc .metsys tuohtiW ssol of ,ytilareneg emussa that
W si a yranib graph, that ,si each edon has rehtie 0 or 2 stnerap .ylno ynA GADOAW
nac eb yltcerid demrofsnart otni a yranib graph htiw at tsom a raenil esaercni ni the
rebmun of sedon dna edges.
sA denoitnem ni [9], the lamitpo noitulos ot )W(L yam ton eb largetni (or erom
,yllacificeps .)l-O redisnoC the elpmis GADOAW ni .giF 2. morF siht ,GADOAW ew
evah the gniwollof metsys of raenil :seitilauqeni
1
Xtv-ofF $- ffo-oidarX 3 Xhouse-quiet
Xhouse-quiet = 1,
3
&v-off < x no-one-home
1
ffwcidarX 6 Xno-one-home
2 sihT sdnopserroc ot eht decudni-mes tniartsnoc metsys fo 91 1
E. Sanros Jr, E.S. laic@rA/sornaS Intelligence 86 (1996) 157-l 70 161
Fig. 2. A WAODAG with a non-integraolp timalf or L(W)
dna the gniwollof evitcejbo noitcnuf
)W(L@ = 7&c-one-home.
eW nac ylisae show that the noitulos hcihw seziminim the evitcejbo noitcnuf si as
:swollof teiuq_esuohX = 1, ffoeidarX = 0.5, ff+vtX = 0.5, dna emoh_eno-onX = 0.5 htiw
)W(L@ = 3.5. ehT yramirp esuac for the 0.5s sesira morf the ytilauqeni for the OR-
edon .teiuq-esuoh
redisnoC the gniwollof lacinonac mrof noitazimitpo :melborp
nim nc
X
s.t. Ax < b,
;x roOsi 1.
ehT ix dnopserroc ot ruo xq where each q si a edon .evoba x si dellac the noitulos
vector. Each ic ni c sdnopserroc ot the cost of each edon detneserper yb iX dna c si
dellac the cost vector. ,yllaniF we nac etirwer ruo evoba WAODAG raenil seitilauqeni
as a xirtam dna rotcev noitacilpitlum where A si dellac the constraints matrix dna b the
constraints vector.
evresbO that each nmuloc ni the xirtam A sdnopserroc ot a euqinu edon ni W dna
eciv .asrev roF .giF 1, we evah the gniwollof :metsys
Xhouse-quiet
&v-off
=x Xradio-off c=
sgnos-dabX
Xno-one-home
&c-shows _
162 E. Suntos JI; ES. Santos/Artijicial InIelligence 86 (I 996) 157-I 70
1 -I 0 0 0 0 0
1 0 - 1 0 0 0 0
I 0 0 0 0 0 1
=A * =b
1- 0 0 0 0 0 1-
0 1 0 1- I- 0 0
0 0 I 0 -I -1 0
Next, we enifed the tpecnoc of ytirap as :swollof
noitinifeD 2.5. neviG na detceridnu elpmis ,elcyc
~={~p1~p2~~~p2.p3~~...~~pn-l~pI~}.
ni ,G the parity of r si
p{i E V 1 r(p) = OR, (p,p) E 7, (p,q) lt 7, dna ,D = l}}q,p{ eludom 2. (5)
,eromrehtruF fi the ytirap slauqe ,I neht r si na odd-cycle. ,esiwrehtO ti si na even-cycle.
,ylevitiutnI ytirap si a erusaem no the rebmun of sedon-RO dna rieht stnerap that are
desrevart yb the .elcyc eW won dnetxe siht revo .W
noitinifeD 2.6. W si parity-balanced fi for yna detceridnu elpmis elcyc r ni ,G r si na
.elcyc-neve
.noitatoN roF ,yticilpmis we nac osla tneserper r yb the ecneuqes of sedon tp{ , .,2~ . . ,
)np erehw
pn = PI.
noitinifeD 2.7. A si dias ot eb totally unimodular fi lla of sti erauqs secirtambus evah
tnanimreted lauqe ot rehtie 0, + ,I or - .I
redisnoC the gniwollof meroeht no latot unimodulurity morf [ 10, meroehT 19.3( )vi ] :
ammeL 2.8. A si totally unimodular ffi each collection fo columns fo A can be tilps
otni owt strap so taht the sum fo the columns ni one part minus the sum fo the columns
ni the other part si a vector htiw entries only 0, ,I+ and - 1.
Based no ruo noitavired of A morf ,W the seirtne ni A are rehtie 0, ,I+ or .1-
,eromrehtruF for yna xirtambus A’ demrof yb ylirartibra gnitceles sets of snmuloc morf
A, each row ni A’ lliw evah eno of the gniwollof noitarugifnoc of :sorez-non
( 1) llA zeros.
(2) enO .lf
(3) enO lf dna eno .1-
(4) owT .s1-
(5) enO 1+ dna two .s1-
E. Santos JK E.S. Santos/Artijicial Intelligence 86 (1996) 157-170 163
htiW regards ot ammeL 2.8 no snmuloc ,gnittilps the tsrif two snoitarugifnoc lliw evah on
effect. ehT driht noitarugifnoc seriuqer that the two snmuloc devlovni tsum gnoleb ni the
emas ,gnipuorg ,esiwrehto the stluser for that row lliw eb f2. ehT htruof noitarugifnoc
seriuqer that the two snmuloc eb separated. ehT tsal noitarugifnoc stibihorp gnicalp the
l+ nmuloc ni eno puorg dna htob the - 1s ni the other .puorg
,ylevitiutnI ammeL 2.8 stneserper a ycnedneped graph neewteb the snmuloc ni the
.xirtam sA gnol as siht graph si ,etitrapib we nac noititrap pu the snmuloc ot yfsitas the
.ammel ruO laog si ot show that ecnalab-ytirap seetnaraug a etitrapib graph dna eciv
.asrev eW won evorp the gniwollof meroeht no latot :ytiraludominu
meroehT 2.9. A si totally unimodular ffi W si parity-balanced.
Proof. (+) teL A eb yllatot raludominu dna emussa that W si ton .decnalab-ytirap sihT
seilpmi that there stsixe na elcyc-ddo r ni .G Let r = ,1~( ,2~ . . . , },,p where p,, = .lp
nA edon-RO p si dias ot eb detelpmoc ni r fi ti setubirtnoc ot the ytirap of r, ,.e.i
r(p) = OR, >p,p( E ,T (p,q) E 7, dna D, = .}q,p{ The rebmun of detelpmoc sedon
ni r si odd.
etirweR 7 as :swollof
where (41,. . . , }kq are lla the detelpmoc sedon-RO ni 7. Let j,ia eb the nmuloc ni
A
detaicossa htiw j,ip for j = 1,. . . , inr dna i = 1,. . . , k. mroF a noitcelloc of snmuloc
htiw siht set dna llac the xirtam A’. morF ammeL 2.8, fi the snmuloc ni A’ nac eb
,tilps neht the snmuloc of yna xirtambus of A’ demrof yb gnivomer rows morf A’ nac
osla eb .tilps erehT yam eb sedon-RO ni r whose stnerap are htob osla ni r tub si ton
.detelpmoc etanimilE the rows gnitaler hcus sedon-RO ot rieht ,stnerap ,.e.i those morf
ytilauqeni (3). ,woN the gniniamer rows ni A’ nac eb ni yna noitarugifnoc except the
htfif eno as debircsed .reilrae
,...,l=ihcaeroF.mialC ..,l=jrofj,iplla,k . , mi must belong ni the same group.
redisnoC yna two evitucesnoc sedon j,ip dna t+j,ip ni r. enO si ylirassecen the tnerap
of the other. tuohtiW ssol of ,ytilareneg yas j,ip si the .tnerap fI t+i,,ip si na ,edon-DNA
neht there si a row ni A’ that seifsitas noitarugifnoc (3) htiw the l+ ni t+j,ia dna 1-
ni .j,iU ,ylralimiS siht sdloh neve fi t+j,ip si na ,edon-RO ecnis ti tonnac eb a detelpmoc
.edon sihT seilpmi that the two sedon tsum gnoleb ot the emas puorg ecnis the snmuloc
tsum gnoleb ni the emas part ni order ot yfsitas ammeL 2.8.
etoneD these spuorg ylpmis yb .iP
.mialC iP can never be grouped htiw 1+iP rof i = 1,. . . , k - 1 and kP cannot be grouped
htiw .IP
roF each detelpmoc edon ,iq l_iln,t_ip dna 1,ip are the stnerap of iq for i = 2,. . . , k,
dna for 91, the stnerap are Lm,kP dna p1 ,I. ,oslA there tsum tsixe a row of noitarugifnoc
(4) evoba hcus that the two s1- dnopserroc ot the .stnerap sihT seilpmi that these two
sedon tsum syawla eb ni separate spuorg ot yfsitas ammeL 2.8. ,suhT iP nac reven eb
depuorg htiw ,+iP for i = 1, . . . , k - 1 dna kP tonnac eb depuorg htiw .IP
164 E. Santos JI; ES. Sontos/Artijicial Intelligence 86 (1996) 157-I 70
roF k = 1, ew evah na ycnetsisnocni where 1.ip dna ~~~,1~ tsum htob eb ni the emas
puorg as llew as htob eb separated.
roF k > 1, siht sevael su htiw the melborp of gnigrem the iP litnu we ylno evah
two spuorg elihw ylsuoenatlumis gniyfsitas the gnipuorg dna noitarapes .stniartsnoc eW
nac tneserper siht yb na detceridnu graph A whose sedon dnopserroc ot the iP dna arcs
neewteb the sedon dnopserroc ot a noitarapes .noitidnoc ruO melborp won secuder ot
whether d si .etitrapib
,ylraelC A si a elpmis elcyc of odd htgnel ecnis the rebmun of detelpmoc sedon si
odd. ,suhT A tonnac eb etitrapib [ .15 .noitcidartnoC
)-e( Let W eb decnalab-ytirap dna tel ,ta{ ,2~ . , },a eb yna noitcelloc of snmuloc
ni A. Let A’ eb the xirtambus of A demrof morf the a,.
Let V’ = ,p{ ,p2,. . eb the sedon detaicossa htiw the snmuloc .ia tcurtsnoC the
,y ,,}
hpargbus ’G = (V’, E’) morf G as :swollof roF lla ,i j morf 1 ot ,II fi )i.p3ip( E E, neht
)iP>,P( E E’.
eW yas na edon-RO si ylkaew detelpmoc htiw respect ot G’ fi lla sti stnerap are
osla ni .’G eW evresbo the :gniwollof neviG )ip,;p( E E’, )i( )jp(r = DNA or )ii(
)ip(r = OR dna i,p si ton ylkaew detelpmoc fi dna ylno fi there stsixe yltcaxe eno row
ni ’A of noitarugifnoc (3) hcus that the sorez-non are ylno ni ia dna .jU
enifeD a noitaler R no V’ hcus that piRp, ffi i = j or there stsixe na detceridnu path
r morf ip ot jp where r does ton niatnoc yna ylkaew detelpmoc .edon-RO ,ylraelC R si
na ecnelaviuqe noitaler dna tel u = ,,TC{ .,2~( . , }~TC eb the noititrap no V’ decudni yb
R. ,ylevitiutnI these klf tneserper the yrassecen gnipuorg of snmuloc/sedon ot eetnaraug
latot .ytiraludominu
Next, tcurtsnoc na detceridnu graph A whose sedon dnopserroc ot each .kT( roF each
row of noitarugifnoc (4) no A’, fi the s1- rucco ni dna neht tcennoc (rk, ot ffk,
Ui U,i,
where ip E ,klf dna i.p E .,kT(
.mialC kff si never directly connected ot itself.
fI htob ip dna p,, gnoleb ot ,kT( neht there si na detceridnu path morf ip ot jp
gniniatnoc on ylkaew detelpmoc .sedon-RO yB noitcurtsnoc of A morf ,W ip dna i,p are
the stnerap of emos edon-RO q edistuo of V’. eW nac tcurtsnoc a elcyc yb gnidulcni
q. yB noitinifeD 2.5, siht si na .elcyc-ddo sihT dluow tcidartnoc ruo noitpmussa of
ecnalab-ytirap for .W
,yllaniF we tsum redisnoc rows of noitarugifnoc 5 ni A’. hcuS rows etacidni ylkaew
detelpmoc sedon-RO ni .’G emussA ip si hcus a edon dna ,ip dna zjp are the stnerap of
Pi.
.mialC Both p.i, and p,? cannot belong ni the sume noititrap pk.
,esiwrehtO there si na detceridnu path morf p,,, ot 1iP gniniatnoc on ylkaew detelpmoc
.sedon-RO eW nac tcurtsnoc a elcyc ni the path yb gnidulcni .ip yB noitinifeD 2.5, siht
si na .elcyc-ddo sihT dluow tcidartnoc ruo noitpmussa of ecnalab-ytirap for .VI
ecniS the stnerap of a ylkaew detelpmoc edon-RO tonnac eb ni the emas ,noititrap
ecalp na arc ni A neewteb the gnidnopserroc snoititrap gniniatnoc the .stnerap
sA ,erofeb we tsum won evorp that A si .etitrapib emussA that A has na htgnel-ddo
.elcyc Each arc ni A sdnopserroc ot na edon-RO ni .G eW nac tcurtsnoc na detceridnu
.E sotnaS ;IJ .S.E laicjitrA/sotnaS ecnegilletnI 68 )6991( 071-751 561
elcyc ni G yb tsrif gnitcurtsnoc path stnemges hcihw are free of ylkaew detelpmoc
sedon-RO ni the sedon of A gnieb desrevart dna nioj meht aiv the sedon-RO detneserper
yb the arcs. ,ecneH we evah na elcyc-ddo ni .G .noitcidartnoC ,eroferehT A si etitrapib
dna ’A seifsitas the snmuloc gnittilps of ammeL 2.8. 0
ehT xelpmiS Method for gnivlos raenil smargorp searches gnola the secitrev of the
nordehylop denifed yb the tniartsnoc .metsys denibmoC htiw the gniwollof meroeht morf
[ 10, meroehT :])ii(3.91
Theorem 2.10. A si totally unimodular rofffi each integral vector b, the polyhedron
x{ ( x > 0, xA < }b has only integral vertices.
sihT seetnaraug that the lamitpo noitulos dnuof yb xelpmiS lliw eb .largetni ,ecneH
xelpmiS enola lliw evlos GADOAW W fi W si decnalab-ytirap tuohtiw gnitroser ot
hcnarb dna .dnuob
,revewoH xelpmiS has na laitnenopxe esac-tsrow .emit-nur raeniL gnimmargorp -borp
smel nac eb devlos ni laimonylop emit hguorht sdohtem hcus as s’nayihcahK ]OI[
tub these sdohtem do ton ylirassecen dnif na lamitpo noitulos hcihw si .largetni tI si
elbissop for there ot tsixe na largetni dna a largetni-non noitulos htob of hcihw are
lamitpo emas( evitcejbo .)eulav xelpmiS searches for na lamitpo noitulos yb gnirolpxe
the emertxe stniop of the tniartsnoc space [4,6]. latoT ytiraludominu seetnaraug that
lla emertxe stniop are .largetni ,ecneH xelpmiS lliw dnif na largetni .lamitpo sA ti snrut
,tuo gnikat the noitulos detareneg htiw s’nayihcahK ,dohtem the lamitpo largetni noitulos
nac eb dnuof ni laimonylop emit nehw W si .decnalab-ytirap sihT swollof morf [ 10,
meroehT 16.21.
s’teL redisnoc two elpmis SGADOAW ni .sgiF 3 dna 4 where the tsrif si of neve ytirap
dna the other si of odd .ytirap Let
=X
ehT gnidnopserroc tniartsnoc secirtam are
gnidroccA ot ammeL 2.8, we dnif that gnikat the tsal three snmuloc of ,ddoA there si
on yaw ot tilps meht otni two sets ot yfsitas the .ammel nO the other ,dnah A,,, does
deedni yfsitas the snoitidnoc for latot .ytiraludominu
166 .E sotnaS Jr; .SE luicijitrA/mnaS ecnegilletnI 68 I( )699 I-751 07
: 7-emoh-eno-on
;(----,‘
I’ \
.giF 3. A elpmis neve ytirap WAODAG
i 7-emoh-eno-on ’
teiuq-esuoh ,
.giF 4. A elpmis ddo ytirap WAODAG.
Proposition 2.11. Cost-based abduction htiw parity-balanced SGADOAW can be solved
ni polynomial time.
ehT latot ytiraludominu of a xirtam nac eb denimreted ni laimonylop emit gnidrocca
ot [ 10, meroehT 20.31. ,ecneH we nac enimreted ecnalab-ytirap osla ni laimonylop
.emit nA evitanretla ot siht approach si ot check for ecnalab-ytirap yltcerid for each
detceridnu elcyc ni the graph. ehT melborp of gnitareneg the selcyc nac eb ylisae enod
))l+n()lEI+IVI((Oni w h ere n si the rebmun of selcyc .detareneg ,yllacisaB a -htped
tsrif search si desu yberehw hguorht na tnagele approach ot gnidda edges lliw etareneg
each .elcyc sliateD dna sisylana of the mhtirogla nac eb dnuof ni [ 81.