
paramavadhi vihitam

nyayamaaya inam vihithathinte oru maanadandamaanu maximine share (mms). vyathyastha moolyangalulla oru koottam enangal nalkiyaal, enangale formula _ 1 bhagangalaayi vibhajikkunnathiloodeyum ettavum kuranja moolyathode bhagam edukkunnathiloodeyum nedaan kazhiyunna paramavadhi moolyamaanu "1-outtu-oof-enn maximine-share".
oro ejantinum avante/avalude 1-outtu-oof-"enn" maximine-share pole mikacha oru bundle labhikkukayaanenkil vyathyastha moolyanirnnayangalulla formula _ 1 ejantumaarkkidayil enangalude vihithathe mms-fair ennu vilikkunnu. aanupaathikathayude maanadandatthinte elavu enna nilayil eric budish aanu mms fairness kandupidichathu-oro ejantinum thulya vibhajanam (oro vibhavathinteyum formula _ 3) pole mikacha oru bundle labikkum. enangal vibhajikkappedumbol aanupaathikatha urappunalkaan kazhiyum, pakshe avaa avibhaajyamaakumbol allah, ellah ejantumaarkkum ore moolyanirnnayamundengilum. ithinu vipareethamaayi, ore ejantumaarkku ellaayppozhum mms nyaayabodham urappunalkaan kazhiyum, athinaal ejantumaar vyathyasthamaanenkilum ithu aanupaathikathaykk swaabhaavika badalaanu.
prachodanavum udaaharanangalum.
samana enangal. formula 4 samana enangal formula 1 aalukalkkidayil nyaayamaayi anuvadikkendathundennu aadyam karuthuka. oro vyakthikkum formula 6 enangal labhikkanam, ennaal formula 4 formula 1 kondu harikkaanaavaathathinaal ithu asaadyamaanu. swaabhaavikamaaya randaamathe mikacha nyaayabodha maanadandam formula _ 6 aduthulla poornnasamkhyayilekku raundu cheyyuka, oro vyakthikkum kuranjathu formula _ 10 enangalenkilum nalkuka ennathaanu. formula 10-ll thaazhe enangal sweekarikkunnathu "valare anyaayamaan"-ithu enangalude avibhajyathayaal nyaayeekarikkappedaatha oru aneethiyaanu.
vyathyastha enangal. ippol enangal vyathyasthamaanennum oro inathinum vyathyastha moolyamundennum karuthuka. udaaharanathinu, formula _ 12 umm formula _ 13 umm, enangalude moolyangal formula _ 14 aanennu karuthuka, formula _ 15 lake cherkkunnu. enangal vibhajichuvenkil, nammal oro vyakthikkum formula _ 16 nte moolyam nalkum (allengil, avaa munbathe khandikayilennapole poornnasankhya moolyangalilekku maathram vibhajichuvenkil, kuranjathu formula _ 17), pakshe ithu saadhyamalla. partition formula _ 18 prakaram moonnu ejantumaarkkum urappunalkaan kazhiyunna ettavum valia moolyam 7 aanu. anoupachaarikamaayi, formula _ 19 ennathu formula _ 1 kondu harikkappedunna motham moolyamaanu.
eee maximine moolyam nedunna settu formula _ 21 neey "1-outtu-oof-3 maximine-share" ennu vilikkunnu-yadhaartha settine formula _ 22 bhagangalaayi vibhajichu ettavum kuranja moolyamulla bhagam eduthu nirmmikkaan kazhiyunna enangalude ettavum mikacha upavibhaagamaanith. athinaal, eee udaaharanathil, oro ejantinum kuranjathu formula _ 19 nte moolyamengilum nalkunnuvenkil oru alokkeshan mms-fair aanu.
vyathyastha moolyanirnayangal. oro ejantum oro inathinum vyathyasthamaaya moolyam nalkunnuvennu karuthuka, udaaharanathina
ippol, oro ejantinu vyathyastha mms undu
ivide, oru alokkeshan mms-fair aanu, athu alissinu kuranjathu formula _ 19 nte moolyavum jorj kuranjathu formula _ 28 nte moolyavum dinaykku kuranjathu formula _ 22 nte moolyavum nalkunnu. udaaharanathinu, jorjinu aadyathe rund enangalude formula _ 35 nalkunnu, adutha rund enangalude formula _ 36, dhina avasana inaa formula _ 37, mms-fair aanu.
vyaakhyaanam. oru ejantinte 1-outtu-oof-formula _ 1 emm. emm. ezz. oru ejantinu oru alokkeshanil ninnu labhikkumenn prathekshikkunna paramavadhi upayogamaayi vyaakhyaanikkaam, matellaa ejantumaarkkum "ore" mungananakalundengil, avanu ellaayppozhum ettavum mosham vihitam labhikkumbol. thaazhepparayunna vaadathe adisthaanamaakki oru ejantinu arhatayullah ettavum kuranja upayogamaanitha matellaa ejantumaarkkum enneppole thanne mungananakalundengil, enikku eee upayogapradatha nalkunna oru vihithamenkilum undu, athu matellaa ejantumaareyum (durbalamaayi) mechappeduthunnu; athinaal enikku kurachu nalkaan oru kaaranavumilla.
oru badal vyaakhyaanam ithaanu athiraalikalkkethire vibhajikkanum thiranjedukkaanum ejantinu urappunalkaan kazhiyunna ettavum ishtappetta bandil ejantu thante mikacha vihitam nirddeshikkukayum backie edukkunnathinu munbu oru ohari thiranjedukkaan matellaavareyum anuvadikkukayum cheiyunnu.
mms-fairnessine inipparayunna charchaa prakriyayude falamaayum visheshippikkaam. oru nischitha vihitam nirddeshikkappedunnu. oro ejantinum enangalude oru badal partition nirddeshichukondu athine ethirkkaan kazhiyum. ennirunnaalum, angane cheiyunnathinu munbu matellaa ejantumaareyum avarude punk thiranjedukkaan addeham anuvadikkanam. athinaal, oru ejantu thante nilavile bandilinekkaal "ellah" bandilukalum mikacha oru partition nirddeshikkaan kazhiyumenkil maathrame oru alokkeshane ethirkkoo. oru ejantum athirkkunnillenkil oru alokkeshan mms-fair aanu, athaayath.iii.oro ejantinum, oro paartteeshanilum, thante nilavile vihithathekkaal durbalamaaya oru bundle undu.
aupachaarika nirvachanam.
formula _ 39 anuvadikkenda vibhavathe prathinidheekarikkunna oru koottamaayirikkatte. formula 39-nte upavibhaagangalile eathenkilum yadhaartha function formula 40 ayirikkatte, athu avayude "moolyam" prathinidheekarikkunnu. formula 39 ll ninnulla formula 40-nte 1-outtu-oof-"n" maximine-share ingane nirvachikkappedunnu formula 44 ivide, formula 39-nte ellah partitionukalkkum mukalilulla formula 1 disjoyint subsettukalilekkum ettavum kuranjathu paartteeshanile ellah formula 1 subsettukalkkum mukalilumaanu. melpparanja udaaharanangalil, formula 39 oru koottam poornnasamkhyakalaayirunnu, athaayath "v" ennathu koottal functionaayirunnu, athaayath formula 49 ennathu formula 50 leeii poornnasamkhyakalude thukayayi nirvachikkapettu. udaaharanathinu, nammal formula 51 kaanichu, avide paramavadhi partition formula 52 aanu. oru saadhaarana nyayamaaya alokkeshan prashnatthil, ore resource formula 39 nuu mukalilulla formula 54 vyathyastha moolyangalulla formula 1 vyathyastha ejantumaaraanu. ejantu formula 57 nte 1-outtu-formula 1 mms moolyam formula 58 kondu soochippikkunnu. oru "alokkeshan" ennathu "n" jody thirichulla-disjoyint subsettukalude oru vektaraanu formula 39 "-- oru ejantinu oru upavibhagam. oro ejantu formulakkum _ 57, formula_62.an alokkeshane "ejantinte oru mms partition" formula _ 57 ennu vilikkukayaanenkil, ellah formulakalkkum aaa formula _ 64 undengil, oru alokkeshan formula _ 60 neey "mms-fair" allengil lalithamaayi "mms alokkeshan" ennu vilikkunnu.iii.formula _ 66 nte mms-nulla formula paramavadhi vardhippikkunna paartteeshanukalilonnaan alokkeshan.
emm. emm. ezz. vihithangalude nilanilpp.
mms vihitam nilanilkkumennu urappilla. procasia vaangum kurokkaavayum formula _ 12 ejantukalum formula _ 68 enangalum upayogichu oru udaaharanam nirmmichu, athil oro ejantinum 1-outtu-oof-3 millimeter vihitam urappunalkunnilla. avarude udaaharanathil, formula _ 70, formula _ 71 ennivayaal soochikayilaakkiya formula _ 69 objectukal undu. oro ejantu formula _ 72 umm oro object formula _ 73 neey moolyanirnnayam cheiyunnu formula _ 74 ivide formula _ 75 formula _ 76 nekkaal cheria moolyangalulla prathyeka 3-bye-4 maatriksukalaan. oro ejantinum vasthukkale formula _ 78 objectukalude formula _ 22 upavibhagangalaayi vibhajikkaan kazhiyumennu avaa theliyikkunnu, athaayath oro upavibhagathileyum moolyangalude koottam 4,055,000 aanu, athinaal ellah ejantumaarudeyum mms aanu. oro mms allocation oro ejantinu krithyamaayi 4 prathyeka vasthukkal nalkanamennu avaa theliyikkunnu, pakshe atharamoru alokkeshan nilavililla. athinaal, oro allocation kuranjathu oru ejantinu paramavadhi 4,054,999 enna moolyam nalkunnu. avar eee udaaharanam saamanyavalkkarikkukayum oro formulakkum _ 79 formula _ 80 inangalumaayi atharamoru udaaharanam undennu kaanikkukayum cheythu.
feej, sapir, tabber enniva nilavilillaattha phalam mechappeduthi, formula _ 81 ejantukalum formula _ 82 enangalum upayogichu oru udaaharanam nirmmichu, athil mms alokkeshan illa. eee saahacharyathil oro ejantinum 40 emm. emm. ezz undu, ennaal ettavum mosham oof ejantu enangalkku 39 enna samyojitha moolyathil maathrame urappu nalkaan kazhiyoo. eathu formulakkum _ 79, formula _ 84 enangalkku oru emm. emm. ezz alokkeshan nilavilillenn avaa kaanikkunnu. formula _ 1 thulyamaanengil, avaa formula _ 86 enangalilekkulla boundu mechappeduthunnu. eee sandarbhangalil, ettavum mosham ejantinu avarude mmsinte oru formula _ 87 vihitam labikkum.
mms alokkeshanukal nilanilkkumennu urappillengilum, kramarahithamaya sandarbhangalil, mms alokkeshanukal uyarnna saadhyathayode nilanilkkunnuvennu theliyikkappettittund. rund kesukalil ithu sariyaanennu kurokawa, prokaashya, wang enniva kaanichu
amanaattidikal, maarkkakkikikal, nixad, saberi ennivayum kramarahithamayi srishtikkappetta sandarbhangalil mms-fair alokkeshanukal uyarnna saadhyathayode nilanilkkunnuvennu theliyikkunnu.
mms alokkeshanukal ellaayppozhum nilanilkkunnuvennu pala klaasukalilum theliyikkappettittund. ellah "enn" ejantumaarkkum ore moolyanirnayangal ullappol, nirvachanamanusarichu oru mms alokkeshan ellaayppozhum nilanilkkunnu (ellah ejantumaarkkum ore mms paartteeshanukal undu). oru mms alokkeshan nilanilkkunna oru saamaanya cases chila formula _ 92 ejantumaarkku samanamaaya moolyanirnayangal undaakumbozhaanu. oru mms alokkeshan pinneedu vibhajikkukayum thiranjedukkukayum cheyyaam formula _ 92 samanamaaya ejantumaar enangale formula _ 94 bandilukalaayi vibhajikkunnu, avaa oronnum avayude mms pole mikachathaanu; formula _ 1-aam ejantu ettavum uyarnna moolyamulla bundle thiranjedukkunnu; athe ejantumaar backie formula _ 92 bundleukal edukkunnu. prathyekichum, rund ejantumaar ullappol, oru mms alokkeshan ellaayppozhum nilanilkkunnu.
inipparayunna kesukalil mms alokkeshanukal nilavilundennu buvarttum lemitre theliyichu
pinneedulla phalam pinneedu kurokawa, prokaashya, wang enniva formula 100 aayum feej, sapir, tabber enniva formula 101 aayum mechappeduthi. moonnu ejantukaludeyum onpathu enangaludeyum negatteevu udaaharanam kaaranam, formula 94 ejantukaludeyum formula 104 enangaludeyum ellah udaaharanagalkkum ellaayppozhum mms alokkeshanukal undennathupole nilavilulla ettavum valia sthiramaaya formula 102 ithaanu. formula 94 nte moolyam enthuthanneyaayaalum. inipparayunna sandarbhangalil mms alokkeshanukal nilavilundennu hammal kaanichu
oro inathinum 0,1 allengil 2 enna moolyam varunna "trirashtra moolyanirnnayangalude" kaaryatthil mms alokkeshanukal nilavilundennum polynomial samayat kandethaamennum amanthedikal, markkakkakkikal, nixad, saberi enniva kaanichu.
ekadesha kanakk.
budish 1-oof-"n" mms-nuu oru ekadesha kanakk avatharippichu-"1-oof-(formula _ 112") mms-oro ejantinum "n" + 1 bandilukalaayi vibhajikkunnathiloodeyum ettavum moshamaayath nedunnathiloodeyum kuranjathu labhikkunnu. pothuve, eathenkilum "d"> "n" nuu, oraalkk "1-oof-d" mms-neey "-n" mms-nte 1-nte ekadeshamaayi kanakkakkaam, koodaathe oro ejantinum "i": formula 113 nottu "1-oof-d" mms-nte moolyam "d"-nte durbalamaayi kurayunna functionaanennu oru alokkeshan thirayuka. ithine "oordinal ekadesha kanakk" ennu vilikkunnu, kaaranam ithu bandilukalude rankingine maathram aasrayichirikkunnu, avayude krithyamaaya moolyangaleyalla.
procasia wang vyathyastha tharatthilulla ekadesha kanakk avatharippichu-mmsnulla "gunana ekadesha kanakk": oro ejantinteyum moolyam avante/avalude mmsinte moolyathinte kuranjathu oru bhinnasankhya "aarr" aanenkil, athaayath oro ejantinu "aii": formula 114 rund algorithangalkkidayil oraalkk thiranjedukkaan kazhiyumennu karuthuka aadyathethu oru gunana ekadesha kanakk urappunalkunnu (iii.jii. 3/4-fraction emm. emm. ezz), randaamathethu oru oordinal ekadesha kanakk (udha.jii. 1-outtu-oof-(3 "n"/2) mms). eee rund gyaarandikalil eathaanu kooduthal? utharam moolyangale aasrayichirikkunnu.
pothuve, eathenkilum poornnasankhyayaaya "k"-nuu, "n"-nte 1-nte "mms" kuranjathu "k"-nte 1-nte "nk" mms-nte madangu opttimal 1-nte "nk" mms partition edukkuka, koodaathe bundleukal "n" suuppar bandilukalaayi grouppucheyyuka, avayil oronnilum "k" yadhaartha bundleukal adangiyirikkunnu. eee suuppar bandilukal oronninum ettavum cheria yadhaartha bandilinte "keey" eratti moolyamundu. athinaal, oru 1/"k"-multiplicative ekadesha kanakk kuranjathu 1-oof-"nk" oordinal ekadesha kanakk pole valuthaanu, ennaal mukalil paranja udaaharanathilennapole 1-oof-("nk-" 1) oordinal ekadesha kanakk ennathinekkaal cheruthayirikkaam. prathyekichum, r ≤1/2-nte eathoru "r"-multiplicative ekadeshavum kuranjathu 1-oof-(2 "n") oordinal ekadesha kanakk pole mikachathaanu, pakshe 1-oof-(2 "n"-1) oordinal ekadesharekhayekkaal moshamayirikkaam.
charakkukalude emm. emm. ezz-fair alokkeshan.
gunita ekadesha kanakk.
procasia wang ellaayppozhum oru "rn"-fraction mms kandethunna oru algoritam avatharippichu,
soothravaakyam _ 118
ivide ottathavana ("n") "n" nuu cherutho thulyamo aaya ettavum valia ottathavana poornnasamkhyayaanu. prathyekichum, "r3" = "r4" = 3/4, "n" vardhikkumbol athu kurayunnu, athu ellaayppozhum 2/3 nekkaal valuthaanu. "n" sthiramaakumbol avayude algoritam "m" ll time polynomialil pravartthikkunnu, pakshe athinte rantaim "n" ll exponential aayirikkam.
anathedes, maarkkikis, nixad, saberi enniva niravadhi mechappetta algorithangal avatharippichu
barman krishnamoorthiyum avatharippichu
ghodsi, hajiyaaghaayi, seddigin, seddigin, yaami ennivar avatharippichu
garg, mcgloflin, tackie ennivar randara-moonnu fraction mms-fairnessinaayi oru lalithamaaya algoritam avatharippichu, athinte vishakalanavum lalithamaanu.
garg takiyum avatharippichu
akrami, gurg, sharmma, taaki enniva gargh tucky avatharippicha algoritam vishakalanam mechappeduthukayum vishakalanam lalithamaakkukayum formula _ 121 lake nilanilpp urappu mechappeduthukayum cheiyunnu.
innuvare, oru "r"-fraction mms alokkeshan ellaayppozhum nilanilkkunna ettavum valia "r" eathaanennu ariyilla. ithu formula _ 122 num formula _ 123 num edayilulla eathu samkhyayum aakam.
kramamaya ekadesha kanakk.
thulya varumaanathil ninnulla ekadesha malsara santhulithaavastha ellaayppozhum "1-oof-(formula _ 112") emm. emm. ezz. urappunalkunnuvennu budish kaanichu, ennirunnaalum, eee vihitathinu adhika vitharanavum athilum pradhanamaayi-adhika aavasyavum undayirikkaam ellah ejantumaarkkum anuvadicha bandilukalude aakethuka ellah enangaludeyum koottathekkaal alpam valuthayirikkaam. cheria ennam seettukal cherthu oru cheria adhika vitharanam sariyaakkaan kazhiyumennathinaal atharam oru pishak kozhs alokkeshanil nyaayayukthamaan. ennaal classic fair division prashnam enangal cherkkillenn anumaanikkunnu.
adhika vitharanavum aavasyavum illaathe, inipparayunna ekadesha kanakkukal ariyappedunnu
innuvare, 1-outtu-oof-dii mms alokkeshan ellaayppozhum nilanilkkunna ettavum cheria "dii" eathaanennu ariyilla. ithu "n" + 1 num 3 "n/" 2 num edayilulla eathu samkhyayum aakam. ettavum cheria ooppan cases "n" = 4 aanu.
adhika niyanthranangal.
uthppannam paramavadhi vardhippikkal karajiani, kurokawa, moulin, prokaashya, shhaa, wang enniva kaanikkunnath "maiks-naash-welfare alokkeshan" (yoottilittikalude uthppannathe paramavadhi anuvadikkunna alokkeshan) ellaayppozhum formula _ 125-fraction mms nyaayamaayathaanennum athu irukiyathaanennum.
sathyasandhatha amanaattidikal, birmbakal, markkakkakkikal enniva ekadesha mms-fair alokkeshanukalkkaayi sathyasandhamaya samvidaanangal avatharippichu (thanthraparamaaya fair division kaanuka):
cardinality parimithikal enangale vibhagangalaayi thirichirikkunnu, oro ejantinu oro vibhaagathil ninnum paramavadhi "kha" enangal labikkum. mattoru vidhathil paranjaal, bundleukal oru partition maatroyidinte swatantra settukalaayirikkanam.
samgharsha graph hammal hetlaandum enangal thammil samgharsha graph ulla mattoru krameekaranam padikkunnu (udaaharanathina enangal sambhavangale prathinidheekarikkunnu, oru ejantinu oresamayam rund evantukalil pankedukkaan kazhiyilla). vairudhya graphile alavu "dii" aanenkil athu (2 ", enn") aanenkil, oru 1/"dii"-fraction mms alokkeshan polynomial samayat kandethaamennum 1/3-fraction mms alokkeshan ellaayppozhum nilanilkkunnuvennum avaa kaanikkunnu.
connectivity enangal oru graafilaanu sthithicheyyunnathu, oro bhaagavum bandhippicha oru subgraph aayirikkanam.
emm. emm. ezz.-fair jolikalude vihitam.
assis, rauchekkar, shreyn, walsh ennivar mms ashayam "jolikal" (negatteevu yoottilittikalulla enangal) aayi vyapippichu. jolikale sambandhichidatholam, gunana ekadesha gadakangal 1 nekkaal valuthaanennum (kurachu jolikalkku uyarnna upayogakshamatha ullathinaal) oordinal ekadesha gadakangal "n" nekkaal cheruthaanennum shradhikkuka. avar avatharippichu
barman krishnamoorthiyum 4/3 fraction mms (krithyamaayi, formula _ 126) nedunna oru algoritam avatharippichu. samana yanthrangalude shedyulinginulla ell. pii. tii algorithathinte saamaanyavalkkaranamaayi algorithathe kaanaan kazhiyum.
jolikal" ennathinaayi oru 11/9-fraction mms-fair alokkeshan ellaayppozhum nilanilkkunnuvennum 5/4-fraction mms alokkeshan polynomial samayat kandethaamennum huwang, luu enniva theliyikkunnu. samana yanthrangalude shedyulinginulla multifit algorithathinte saamaanyavalkkaranamaayi avarude algorithathe kaanaan kazhiyum.
mixed vaaluyeshanukalulla" mms-fair alokkeshan, athaayath, kulkarni, mehta, tackie enniva padikkunnu.iii., saadhanangalum jolikalum ullappol. avar theliyikkunnu
saankethikavidyakalum algorithangalum.
pariharam maattathe yadhaartha prashnathinu vividha normalisationukal prayogikkaan kazhiyum. thaazhe, "o" ennathu ellah vasthukkaludeyum koottamaanu.
scaling.
oro ejantu i-num, ellah moolyanirnayangalum oru factor formula 127 (vyathyastha ejantumaarkku vyathyasthamaayirikkam) upayogichu skeyil cheiyukayaanenkil, oro ejantinte mms ore ghadakam upayogichu skeyil cheiyunnu; athinaal, yadhaartha udaaharanathile oro mms allocation skeyil cheytha udaaharanathile oru mms alokkeshanaanu. oro ejantinteyum mms krithyamaayi formula _ 97 ayirikkumenna tharatthil moolyanirnayangal skeyil cheiyunnathu saadhaaranamaanu. eee scalinginu sesham, mms ekadesha prasnangal ingane prasthaavikkam
melpparanja scaling oro ejantinteyum mms kanakkaakkendathund, ithu oru enpi-haard prasnamaanu (multiway nambar partitioning). vegathil cheyyaan kazhiyunna oru badal scaling ithaanu
oru vasthu anuvadikkuka.
130 enna soothravaakyathil ninnu oru object formula _ 149 neekkam cheythaal. oro ejantinum, formula _ 97-outtu-oof-(formula _ 92) mmsw.aarr.tii. bakkiyulla settu formula _ 153 kuranjathu addehathinte formula _ 97-outtu-oof-formula _ 1 mms dablyu aanu.aarr.tii. yadhaartha settu formula _ 130. kaaranam, yadhaartha mms paartteeshanil, formula _ 92 bhagangal kedukoodaathe thudarunnu. ippol, oro ejantinum formula 129-nte moolyam nalkaan nammal lakshyamidunnuvennu karuthuka. eathenkilum object formula 159 kuranjathu oru ejantinu formula 129-nuu moolyamullathaanenkil, atharam oru ejantinu namukk formula 159 ekapaksheeyamaayi nalkaam, sheshikkunna vasthukkal sheshikkunna ejantumaarkku anuvadikkaan munnott pookaam. athinaal, namukk w ennu anumanikkaam.ell.oo.jii. athha
eee normalisation phaast scalingle ekapaksheeyamaaya monodon moolyanirnayangalilum (noonn-additteev polum) pravartthikkunnu.
bagg niraykkunnu.
ellah ejantumaarum paramavadhi 167 enna soothravaakyathaal moolyanirnnayikkunna oru vasthuvine "167-cheria soothravaakyam" aayi soochippikkuka. ellah vasthukkalum formula _ 167-cheruthaanennu karuthuka. oru ozhinja bagg eduthu, baginu kuranjathu oru ejantinenkilum formula 129 nalkunnathuvare oro vasthuvinteyum moolyam kondu niraykkuka. thudarnnu, atharam oru ejantinu bagg ekapaksheeyamaayi nalkuka. ellah vasthukkalum formula _ 167-cheruthaanennullathinaal, bakkiyulla ejantumaar baginu paramavadhi formula _ 172 ll moolyam nalkunnu; eee moolyam mathiyaya cheruthaanenkil, bakkiyulla moolyam mathiyaya valuthaanu, athinaal namukk aavarthanaparamaayi munnott pookaan kazhiyum. prathyekichum, bagg poorippikkal inipparayunna pariharangal nalkunnu
eee bagg-filling algorithangal phaast scalingle polum pravartthikkunnu, athinaal avaa polynomial samayathaan pravartthikkunnathu-avaykku krithyamaaya mms moolyam ariyendathilla. vaasthavathil, rund algorithangalum mms paraamarshikkaathe prasthavikkaan kazhiyum
parishkkaricha bagg poorippikkal ellah vasthukkalum formula _ 167-cheruthakkaamenna vyavastha inipparayunna reethiyil elavu nalkaam. chila soothravaakyam edukkuka. 167-cheria (i.) enna soothravaakyamillaatha oru vasthuvine soochippikkuka.iii., kuranjathu oru ejantengilum formula _ 167 ennu moolyanirnnayam cheytha) oru "formula _ 167-valia vasthu" aayi. mikka formula objectukal formula 167-valuthaanennu karuthuka. oru formula _ 167-valia object formula _ 159 edukkuka, oru bagil vaykkuka, oru ejantu kuranjathu formula _ 129-nuu moolyamundennu soochippikkunnathuvare formula _ 167-cheria vasthukkal kondu niraykkuka. atharam oru ejantengilum undaayirikkanam, kaaranam chila ejantu formula _ 57 moolyangal formula _ 159 chila formula _ 199-ll. eee ejantinu, mikka formula _ 92-umm formula _ 167-valia vasthukkal avasheshikkunnu. munbathe normalisation prakaram, eee vasthukkal ippozhum formula 129-cheruthaanu, athinaal formula 57-nte motham moolyam paramavadhi formula 204 "aanu, athinaal sheshikkunna formula 167-cheru vasthukkalude moolyam kuranjathu formula 206 aanu."
oordar cheiyunnu.
ellah ejantumaarkkum objectukalil ore oordinal ranking undengil oru udaaharanam "oordar" cheiyunnu, athaayath.e, oro ejantu formulakkum formula 57, formula 209. avabodhajanyamaayi, oordar cheytha instansukal kadinamaanu, kaaranam ejantumaar thammilulla samgharsham ettavum valuthaanu. theerchayaayum, negatteevu udaaharanam kramappeduthiyirikkunnu-vasthukkalude kramam nirnnayikkunnath matricks formula _ 210 aanu, ithu ellah ejantumaarkkum thulyamaanu. ithu aupachaarikamaayum theliyikkaavunnathaan. oro oordar cheytha udaaharanathinum, oru formula _ 129-fraction mms alokkeshan kandethunna oru algoritam namukkundennu karuthuka. ippol, namukk oru pothu inam-alokkeshan instans formula _ 212 nalkiyittundu. nammal athu inipparayunna reethiyil pariharikkunnu.
athinaal formula _ 129-fraction mms alokkeshanukalkkaayi thirayumbol, namukk dablyu anumanikkaam.ell.oo.jii. athha
rund vasthukkal anuvadikkuka.
o" 1, "o" 2 ennee rund vasthukkal nammal kandethunnundenn karuthuka, oru ejantu "i" kuranjathu "r" neey vilamathikkunnu, mattu ejantumaar paramavadhi 1 neey vilamathikkunnu. appol eee rund vasthukkalum "i" lake anuvadikkaam. mattu ejantumaarkku, 1-outtu-oof-("n"-1) mmsw.aarr.tii. bakkiyulla settu kuranjathu addehathinte 1-outtu-oof-"enn" mms dablyu aanu.aarr.tii. yadhaartha settu "oo". kaaranam, yadhaartha mms paartteeshanil, kuranjathu "enn"-2 bhaagangalenkilum kedukoodaathe thudarunnu, athesamayam kedukoodaattha rund bhaagangalum samyojippichu kuranjathu 1 moolyamulla orotta bhagam roopappeduthaan kazhiyum. eee normalisation additteev moolyanirnnayangalil maathrame pravartthikkoo.
koodaathe, udaaharanam oordar cheythathaanennu karuthuka, koodaathe "ooo" ll ninnu "ooo", "oonn", "oonn" + 1 (i.iii.n"-th, ("n" + 1)-ettavum uyarnna moolyamulla enangal). thudarnnu oro ejantinum, 1-outtu-oof-("n"-1) mmsw.aarr.tii. bakkiyulla settu kuranjathu addehathinte 1-outtu-oof-"enn" mms dablyu aanu.aarr.tii. yadhaartha settu "oo". kaaranam, pravukkuzhal thathwamanusarichu, oro ejantinteyum oru emm. emm. ezz bhaagamenkilum {"o" 1, enna settil ninnu rando athiladhikamo vasthukkal adangiyirikkanam..., + 1} ll. eee enangal nalkiyirikkunna vasthukkal maattisthaapikkaan upayogikkaam, athinte falamaayi "n"-1 bhagangal kuranjathu 1 enna moolyathil varum. ithinartham, "oonn", "oonn" + 1 vasthukkalkk chila ejantumaarkku "i" ennathinu kuranjathu mms moolyamengilum undengil, namukk avaa "i" ennathinu nalkaam, sheshikkunna vasthukkal sheshikkunna ejantumaarkku anuvadikkaan munnott pookaam. athinaal, namukk w ennu anumanikkaam.ell.oo.jii. athha
eee normalisation phaast scalingle polum pravartthikkunnu. ithu parishkkaricha bagg-fillingumaayi samyojippikkunnath 2/3 fraction mmacnaayi inipparayunna lalithamaaya algorithathilekku nayikkunnu.
mms paraamarshikkaathe polum eee algoritam urappunalkaan kazhiyum
algoritam prasnangal.
mmsumaayi bandhapetta niravadhi adisthaana algorithangal ivayaanu
mattu neethi maanadandangalumaayi bandhappettathaan.
oru ejantinu "i" ennathinu "asooya-rahita-ozhivakkal-sii-enangal" (udah "sii") ennu vilikkunnu, matellaa ejantumaarkkum "jeey" ennathil ninnu neekkam cheythaal, "aii" bakkiyullavaye asooyappedunnillenkil, paramavadhi "sii" enangalude oru koottam nilavilundenkil. oru iii. eph. 0 alokkeshane "asooyayillaathath" ennu vilikkunnu. udaaharanathinu, raundu-robin inam alokkeshan vazhiyo allengil asooya-graph nadapadikramatthiloodeyo iii. eph. 1 alokkeshanukal kandethaam.
oru alokkeshane "i" enna ejantinu "proportional-eksept-sii-aitams" (prop * "sii") ennu vilikkunnu, "i" yude bundleck purathu paramavadhi "sii" enangalude oru koottam undengil, athu ellah enangaludeyum settil ninnu neekkam cheythaal, "i" bakkiyullathil kuranjathu 1/"n" engilum moolyanirnnayam cheiyunnu. oru prop * 0 alokkeshane "aanupathikam" ennu vilikkunnu.
ef0 ennathu prop * 0 ennum ef1 ennathu prop * ("n"-1) ennum soochippikkunnu. koodaathe, eathenkilum poornnasankhyayaaya "c" 0 nuu, ef "c" ennathu prop * (("n"-1) "c") soochippikkunnu. n" = 2 aayirikkumbol vipareethamaaya ardham sariyaanu, pakshe "n"> 2 aayirikkumbol allah.
inipparayunna maximine-share ekadesha kanakkukal prop * ("enn"-1) soochippikkunnu, athinaal ef 1 umm
melpparanja prathyaaghaathangal chuvade chithreekarichirikkunnu
oru ejantinu "i" ennathinu "i" ennathil ninnu "j" ennathil ninnu neekkam cheytha eathenkilum "i" ennathil ninnu "i" ennathil ninnu asooyappedunnillenkil "x" ennathil ninnu "asooyayillaatha-eathenkilum-inam" ennu vilikkunnu. efx ennathu ef1 nekkaal karsanamaayi shakthamaan. ithu inipparayunna mms ekadesharekhakale soochippikkunnu
grouppukalkku emm. emm. ezz.
oro rund ejantumaarkkum "i", "j" ennivaykku, ejantu "i" nuu "i", "j" ennivaykku labhikkuna enangalil parimithappedutthiyirikku 1-outtu-oof-2 maximine-sharengle labhikkukayaanenkil oru alokkeshane pervise-maximine-share-fair (pii. emm. emm. ezz-fair) ennu vilikkunnu. oru pii. emm. emm. ezz alokkeshan ellaayppozhum nilavilundo ennu ariyilla, pakshe oru <aii. dii. 1> ellaayppozhum nilavilundu.
keey" valippamulla ejantumaarude oro upavibhagathinum, eee upavibhaagathin labhikkuna enangalil parimithappedutthiyirikku "keey" maximine-share subgroupile oro amgathinum labhikkunnundenkil, oru alokkeshane group thirichulla-maximine-share-fair (gmms-fair) ennu vilikkunnu.
additteev moolyanirnnayangalkkop, vividha nyaayabodha ashayangal inipparayunna reethiyil bandhappettirikkunnu
ejantumaarude moolyanirnnayam bainari allengil samaanamaakumbol gmms alokkeshanukal nilanilkkumennu urappunalkunnu. pothuvaaya additteev moolyanirnnayangalkkop, randara gram alokkeshanukal nilavilundu, avaa polynomial samayathil kandethaam.
vyathyastha avakashangalulla ejantumaarkku emm. emm. ezz.
ejantumaarkku "vyathyastha avakashangal" ("asamamaya oharikal" allengil "asamamaya avakashangal" ennum vilikkunnu) ullappol, uyarnna avakashangalulla ejantumaarkku uyarnna vihitam urappunalkunnathin emm. emm. ezz. nyaayabodham sweekarikkanam. vividha tharatthilulla mattangal nirddeshikkappettittund. thaazhe, avakashangal nalkunnathu oru vekttar formula _ 242 aanennu nammal anumaanikkunnu, ivide formula _ 243 ejantu "formula _ 57" nte avakaashathe prathinidheekarikkunnu.
waited-mms neethi.
farhadi, ghodsi, hajiyaaghaayi, laahi, pennock, seddiggin, seddiggin enniva bharamullah maximine share (wms) avatharippikkunnu, ithu nirvachichirikkunnathu formula _ 245 avabodhapoorvvam, opttimal wms kaivarikkunnath oru paarteeshanaanu, athil bhagam "jeey" yude moolyam ejantu "jeey" yude avakaashathinu aanupaathikamaanu. udaaharanathinu, ellah moolya pravartthanangalum thukayaanennum avakasha vekttar "t" = (1/6,11/24,9/24) aanennum karuthuka. appol partition formula 246 ({1,3}, {5,6}, {9}); oro bhaagathinteyum moolyam formula _ 57 ennathinaal ithu opttimal aanu. formula _ 249, formula _ 250. ellah "n" avakashangalum thulyamaakumbol formula _ 251.
oro ejantinteyum moolyam kuranjathu formula _ 252 aanenkil "sii" yude vihithathe "wms-fair" ennu vilikkunnu. ellah "enn" ejantumaarkkum ore moolyanirnayangal ullappol, oru wms-fair alokkeshan ellaayppozhum nirvachanam anusarichu nilanilkkunnu. ennaal vyathyastha moolyanirnnayangalil, saadhyamaaya ettavum mikacha gunana ekadesha kanakk 1/"n" aanu. ε> 0 valare cheria sthiraankamaaya 2 "n"-1 charakkukalum "n" ejantukalum upayogichu mukalile paridhi theliyikkunnu
ellah ejantumaarkkum opttimal wms partition undu "cheria" ejantumaarkku (1,...n"-1) ithu partition ({1},) aanu..., {"n"-1}, {"n"}), "valia" ejantinu ("n") ithu ({"n" + 1}, aanu..., {2 "n"-1}, {1..n"). athinaal, ellah ejantumaarkkum "i" enna formula 253 (thaarathamyathinaayi, cheria ejantumaarkku formula 254, ennaal valia ejantinu formula 255).
dablyuemesinte eathoru gunana ekadesha kanakkilum, ellah ejantumaarum oru positteev moolyam nedanam. ithinartham cheria ejantumaar kuranjathu 1 inathil "n"-1 engilum edukkum ennaanu..n", athinaal atharam oru inam valia ejantinu avasheshikkunnu, athinte moolyam ekadesham 1 ennathinekkaal ekadesham 1/"n" aanu.
1/"n"-fraction WMMS-fair alokkeshan ellaayppozhum nilavilundu, ithu raundu-robin inam alokkeshan vazhi kandethaam. oru niyantritha saahacharyathil, oro ejantum "i" oro gunanavum paramavadhi formula _ 256 kondu vilamathikkunnu, oru 1/2 fraction WMMS-fair alokkeshan nilavilundu, bagg-fillingin samanamaaya oru algoritam upayogichu ithu kandethaam oro ejantinteyum moolyanirnayangal soothravaakyam _ 257 kondu gunikkunnu; oro aavarthanathilum, thripthiyillaatha oru ejantinu (formula _ 258 ll thaazheyulla moolyamulla oru ejantu) oru inam nalkunnu, avar athine ettavum kooduthal vilamathikkunnu. eee algoritam oro ejantinum "i" enna formulayengilum _ 258 aayi neekkivaykkunnu, mikkavarum formula _ 256. praayogikamaayi, oru WMMS-fair alokkeshan mikkavarum ellaayppozhum nilanilkkunnu.
oordinal-mms neethi.
babayof, nizan, talgam-kohen enniva vyathyastha avakashangalulla ejantumaarkku oordinal mms ekadeshathinte swaabhaavika vipuleekaranam avatharippikkunnu. eathenkilum rund poornnasamkhyakalude formula _ 261, "c", moolya function "v" ennivaykku, ivide formula _ 262 nirvachikkuka, "c" yude ellah paarteeshanukalilum formula _ 115 disjoyint subsettukalaayi paramavadhi, formula _ 264 bhaagangalude ellah "unionukalilum" ettavum kuranjathu. udaaharanathinu, partition ({1,6}, {3,5}, {9}) prakaram formula _ 265. ippol, oordinal maximine share (ooms) nirvachikkunnath formula _ 266 udaaharanathinu, ejantu "i" yude avakasham kuranjathu 2/3 vare valuthaya eathenkilum yadhaartha samkhyayaanenkil, avanu kuranjathu "c" yude 2-outtu-oof-3 millimeter engilum arhathayund. 268 enna soothravaakyathil samthrupthiyulla ananthamaaya niravadhi jody soothravaakyangal undengilum, avayil parimithamayavayil palathum maathrame anaavashyamalla (mattullavar soochippikkunnilla), athinaal parimithamaaya samayat oom kanakkukoottaan kazhiyum. oru alokkeshan "z" 1..oro ejantinteyum moolyam kuranjathu formula _ 269 aanenkil "zn" neey "ooms-fair phoor entaitlent-vector w" ennu vilikkunnu.
moolyangale aashrayich ooms dablyuemminekkaal uyarnnatho thaazhnnatho aakam
eathu vilayum pankidunna neethi.
babayof, isra, feej ennivar nyaayabodhathinaayi moonnaamathe maanadandam avatharippichu, athine avar eathenkilum vila vihitam (apps) ennu vilikkunnu. avar athine rund thulya reethikalil nirvachikkunnu; avayilonnu vyakthamayum maximine shareinte shakthippeduthalaan. enangale "dii" disjoyint bandilukalaayi vibhajikkunnathinupakaram, ovarlaappu cheyyaavunna "eathenkilum" bandilukalude shekharam thiranjedukkaan ejantinu anuvaadamundu. ennaal, ejantu oro bandilinum oru bharam nischayikkanam, athaayath thookkathinte aakethuka kuranjathu 1 aayirikkanam, koodaathe oro inavum motham bharam ejantinte avakashamaaya bandilukalil pedunnu. ettavum kuranja moolyamulla positteev-veytt bundle moolyamaanu apps. aupachaarikamaayi formula _ 277, ellah koottam bandilukalilum paramavadhi, bandilukalilekkulla chila bhaarangalkk, ellah bandilukaludeyum motham bharam kuranjathu 1 aanu, oro inathinteyum motham bharam paramavadhi "tii" aanu. oro ejantinum kuranjathu anchil moonnu bhaagamenkilum avarude apps undennu urappunalkunna oru polyomial-time algoritam undu.
apps ellaayppozhum ooms pole uyarnnathaanan opttimal "l"-outtu-oof-"d" partition nalkiyaal, l/d ≤ "ti" ulla oraalkk 1/"d" bharam bhaagangalude samyojanathinu nalkaam.., ell", bhagangal 2-nte union..l" + 1, angane (oru chaakrika reethiyil), oro bhaagavum krithyamaayi "l" unionukalil ulppeduthiyittundu. athinaal, oro inavum paramavadhi "l"/"d" enna bharamullah bandilukaludethaanu, athaayath paramavadhi "ti". ejantinu atharam ettavum kuranja moolyamulla bundle urappunalkunnu, athu kuranjathu "ell"-outtu-oof-dii "mms aanu.
chila sandarbhangalil, apps oomsinekkaal karsanamaayi kooduthalaanu. rund udaaharanagal ithaa
apps dablyuemminekkaal uyarnnatho thaazhnnatho aakam; udaaharanagal ooms vezhsas dablyuemminu upayogikkunnathinu thulyamaanu

