Ntchito yofufuza mwina ndi gawo losangalatsa kwambiri la maphunziro athu. Lingaliro ndikudziyesa nokha munjira yomwe mwasankha mukadali ku yunivesite. Mwachitsanzo, ophunzira ochokera kumadera a Software Engineering ndi Machine Learning nthawi zambiri amapita kukafufuza m'makampani (makamaka JetBrains kapena Yandex, koma osati).
Mu positi iyi ndilankhula za projekiti yanga mu Computer Science. Monga gawo la ntchito yanga, ndinaphunzira ndikugwiritsa ntchito njira zothetsera vuto limodzi lodziwika bwino la NP: vuto lakuphimba vertex.
Masiku ano, njira yosangalatsa yamavuto a NP-hard ikukula mwachangu - ma aligorivimu a parameterized. Ndiyesetsa kukupangitsani kuti mufulumire, ndikuuzeni ma algorithms osavuta a parameterized ndikufotokozera njira imodzi yamphamvu yomwe idandithandiza kwambiri. Ndinapereka zotsatira zanga pa mpikisano wa PACE Challenge: malinga ndi zotsatira za mayesero otseguka, yankho langa limatenga malo achitatu, ndipo zotsatira zomaliza zidzadziwika pa July 1.
Payekha
Dzina langa ndi Vasily Alferov, tsopano ndikumaliza chaka changa chachitatu ku National Research University Higher School of Economics - St. Ndakhala ndi chidwi ndi ma algorithms kuyambira masiku anga akusukulu, pamene ndinaphunzira ku Moscow sukulu No. 179 ndipo ndinachita nawo bwino pa Olympiads ya kompyuta.
Chiwerengero chochepa cha akatswiri mu ma aligorivimu a parameterized amalowa mu bar...
Chitsanzo chotengedwa m'buku
Tayerekezani kuti ndinu mlonda wapa bala m’tauni ina yaing’ono. Lachisanu lililonse, theka la mzinda umabwera ku bar yanu kuti mupumule, zomwe zimakupatsirani mavuto ambiri: muyenera kutaya makasitomala amphesa kunja kwa bala kuti mupewe ndewu. Pamapeto pake, mumatopa ndikusankha kuchita zodzitetezera.
Popeza mzinda wanu ndi wawung'ono, mumadziwa kuti ndi magulu ati a anthu omwe angamenyane nawo ngati athera limodzi mu bar. Kodi muli ndi mndandanda wa n anthu omwe abwera ku bar usikuuno. Mwaganiza zoletsa anthu ena a m’taunimo kuti asalowe m’balamo popanda wina kuchita ndewu. Panthawi imodzimodziyo, mabwana anu sakufuna kutaya phindu ndipo sadzakhala osangalala ngati simukulola zambiri. k anthu.
Mwatsoka, vuto pamaso panu ndi tingachipeze powerenga NP-zovuta vuto. Inu mukhoza kumudziwa iye , kapena ngati vuto lakuphimba vertex. Pazovuta zotere, nthawi zambiri, palibe ma aligorivimu omwe amagwira ntchito munthawi yovomerezeka. Kunena zowona, lingaliro losatsimikizirika komanso lolimba kwambiri la ETH (Exponential Time Hypothesis) limanena kuti vutoli silingathe kuthetsedwa munthawi yake. , ndiko kuti, simungaganize chilichonse chowoneka bwino kuposa kusaka kwathunthu. Mwachitsanzo, tiyerekeze kuti wina abwera kumowa wanu n = 1000 Munthu. Ndiye kufufuza kwathunthu kudzakhala zosankha zomwe zilipo pafupifupi - ndalama zopenga. Mwamwayi, oyang'anira anu akupatsani malire k = 10 ndi, kotero kuchuluka kwa zophatikizika zomwe muyenera kubwereza ndizocheperako: kuchuluka kwa magawo khumi ndi . Izi ndizabwino, koma sizimawerengedwa tsiku limodzi ngakhale pagulu lamphamvu.
Kuchotsa kuthekera kwa ndewu mu kasinthidwe kaubwenzi wovuta pakati pa alendo a bar, muyenera kusiya Bob, Daniel ndi Fedor. Palibe njira yomwe awiri okha adzasiyidwe.
Kodi izi zikutanthauza kuti ndi nthawi yololera kuti aliyense alowe? Tiyeni tione zina zimene mungachite. Chabwino, mwachitsanzo, simungalole kuti alowemo okhawo omwe angathe kumenyana ndi anthu ambiri. Ngati wina akhoza kumenyana osachepera ndi k +1 ndi munthu wina, ndiye simungamulole kuti alowe - apo ayi muyenera kutulutsa aliyense k +1 ndi anthu akutawuni, omwe angamenyane nawo, zomwe zingakhumudwitse utsogoleri.
Lolani kuti mutulutse aliyense yemwe mungathe molingana ndi mfundo iyi. Ndiye wina aliyense akhoza kumenyana ndi zosaposa k anthu. Kuwataya kunja k munthu, inu simungakhoze kuletsa china kuposa mikangano. Izi zikutanthauza kuti ngati pali zambiri kuposa Ngati munthu akhudzidwa ndi mkangano umodzi, ndiye kuti simungathe kuwaletsa onse. Popeza, ndithudi, mudzalowetsamo anthu omwe alibe mikangano, muyenera kudutsa magawo onse khumi mwa anthu mazana awiri. Pali pafupifupi , ndipo kuchuluka kwa magwiridwe antchitowa kumatha kusanjidwa kale pagulu.
Ngati mutha kutenga anthu omwe alibe vuto lililonse, nanga bwanji za omwe akuchita nawo mkangano umodzi wokha? Ndipotu, angathenso kuloledwa mwa kutseka chitseko kwa mdani wawo. Zowonadi, ngati Alice akusemphana ndi Bob yekha, ndiye kuti tikamulola Alice kuti atuluke pa awiriwa, sitidzataya: Bob akhoza kukhala ndi mikangano ina, koma Alice alibe. Komanso, n’zopanda nzeru kuti tisalole tonsefe kulowa. Pambuyo pa ntchito zoterezi palibenso alendo omwe ali ndi tsoka losathetsedwa: tili nawo okha mikangano, aliyense ali ndi anthu awiri ndipo aliyense amakhudzidwa osachepera awiri. Ndiye chomwe chatsala ndikukonza zosankha, zomwe zitha kuganiziridwa mosavuta theka la tsiku pa laputopu.
M'malo mwake, ndi kulingalira kosavuta mungathe kukwaniritsa mikhalidwe yabwino kwambiri. Zindikirani kuti tiyenera kuthetsa mikangano yonse, ndiye kuti, kuchokera pagulu lililonse lomwe likutsutsana, sankhani munthu m'modzi yemwe sitingalole. Tiyeni tilingalire ma aligorivimu otsatirawa: tengani mkangano uliwonse, womwe timachotsapo wina aliyense ndikuyamba mobwerezabwereza kuchokera pazotsalira, kenako chotsani winayo ndikuyambanso mobwerezabwereza. Popeza timaponyera wina pa sitepe iliyonse, mtengo wobwereza wa algorithm yotere ndi mtengo wa binary wakuya k, kotero kuti algorithm yonse imagwira ntchito kumene n ndi chiwerengero cha vertices, ndi m - chiwerengero cha nthiti. Mu chitsanzo chathu, izi ndi pafupifupi mamiliyoni khumi, omwe angawerengedwe mugawikana yachiwiri osati pa laputopu, komanso pa foni yam'manja.
Chitsanzo pamwambapa ndi chitsanzo parameterized algorithm. Ma algorithms a parameterized ndi ma algorithms omwe amayenda munthawi yake f (k) poly(n)kumene p - polynomial, f ndi ntchito computable mosasinthasintha, ndi k - magawo ena, omwe, mwina, adzakhala ochepa kwambiri kuposa kukula kwa vuto.
Malingaliro onse pamaso pa algorithm iyi amapereka chitsanzo kernelization ndi imodzi mwa njira zopangira ma algorithms a parameterized. Kernelization ndikuchepetsa kukula kwa vuto kukhala pamtengo wocheperako ndi ntchito ya parameter. Zotsatira zake nthawi zambiri zimatchedwa kernel. Chifukwa chake, mwa kulingalira kosavuta za madigiri a vertices, tinapeza quadratic kernel ya Vuto la Vertex Cover, lokhazikitsidwa ndi kukula kwa yankho. Palinso zokonda zina zomwe mungasankhe pa ntchitoyi (monga Vertex Cover Above LP), koma izi ndizomwe tikambirana.
Pace Challenge
Mpikisano (The Parameterized Algorithms and Computational Experiments Challenge) adabadwa mu 2015 kuti akhazikitse kulumikizana pakati pa ma aligorivimu a parameterized ndi njira zomwe zimagwiritsidwa ntchito pothana ndi mavuto owerengera. Mipikisano itatu yoyambirira idaperekedwa kuti mupeze kukula kwa mtengo wa graph (), kufunafuna mtengo wa Steiner () ndikufufuza ma vertices omwe amadula kuzungulira (). Chaka chino, limodzi mwamavuto omwe mungayeserepo linali vuto lakuphimba vertex lomwe lafotokozedwa pamwambapa.
Mpikisanowu ukutchuka chaka chilichonse. Ngati mukukhulupirira zoyambira, chaka chino magulu 24 adatenga nawo gawo pampikisanowu kuti athetse vuto lophimba vertex lokha. Ndikoyenera kudziwa kuti mpikisano sutenga maola angapo kapena sabata, koma miyezi ingapo. Magulu ali ndi mwayi wophunzira mabuku, amabwera ndi lingaliro lawo loyambirira ndikuyesera kuligwiritsa ntchito. Kwenikweni, mpikisanowu ndi ntchito yofufuza. Malingaliro a mayankho ogwira mtima kwambiri komanso kupereka mphotho kwa opambana adzachitika limodzi ndi msonkhano. (International Symposium on Parameterized and Exact Computation) monga gawo la msonkhano waukulu wapachaka wa algorithmic ku Europe . Zambiri zokhudzana ndi mpikisano womwewo zitha kupezeka pa , ndipo zotulukapo za zaka za m’mbuyomo n’zabodza .
Yankho chithunzi
Kuti ndithane ndi vuto lakuphimba vertex, ndidayesa kugwiritsa ntchito ma algorithms a parameterized. Nthawi zambiri amakhala ndi magawo awiri: malamulo osavuta (omwe amatsogolera ku kernelization) ndi malamulo ogawa. Malamulo osavuta ndikuwongolera zolowetsa mu nthawi ya polynomial. Cholinga cha kugwiritsira ntchito malamulo oterowo ndicho kuchepetsa vutolo kukhala vuto laling’ono lofanana nalo. Malamulo osavuta ndi gawo lokwera mtengo kwambiri la algorithm, ndipo kugwiritsa ntchito gawoli kumabweretsa nthawi yonse yothamanga m'malo mwa nthawi ya polynomial yosavuta. Kwa ife, malamulo ogawanika amachokera ku mfundo yakuti pa vertex iliyonse muyenera kuitenga kapena yoyandikana nayo ngati yankho.
Chiwembu chambiri ndi ichi: timagwiritsa ntchito malamulo osavuta, ndiye timasankha vertex, ndikupanga maulendo awiri obwerezabwereza: poyamba timatenga poyankha, ndipo winayo timatenga oyandikana nawo onse. Izi ndi zomwe timatcha kugawa (nthambi) m'mphepete mwa vertex.
Ndendende kuwonjezera kumodzi kudzapangidwa ku dongosololi mu ndime yotsatira.
Malingaliro ogawa malamulo (brunching).
Tiyeni tikambirane momwe tingasankhire vertex yomwe imagawanika.
Lingaliro lalikulu ndi ladyera kwambiri pamalingaliro a algorithmic: tiyeni titenge vertex ya digiri yayikulu ndikugawanika motsatira. Chifukwa chiyani zikuwoneka bwino? Chifukwa mu nthambi yachiwiri ya kuyitana kobwerezabwereza tidzachotsa ma vertices ambiri motere. Mutha kudalira graph yaying'ono yotsala ndipo titha kuigwira mwachangu.
Njirayi, ndi njira zosavuta zomwe zafotokozedwa kale za kernelization, zimadziwonetsera bwino ndikuthetsa mayesero ena a vertices zikwi zingapo kukula kwake. Koma, mwachitsanzo, sizikuyenda bwino pazithunzi za cubic (ndiko kuti, ma graph omwe digiri ya vertex iliyonse ndi atatu).
Palinso lingaliro lina lozikidwa pa lingaliro losavuta: ngati graph ichotsedwa, vuto pazigawo zake zolumikizidwa lingathe kuthetsedwa paokha, kuphatikiza mayankho kumapeto. Izi, mwa njira, ndikusintha kwakung'ono kolonjezedwa mu chiwembu, chomwe chidzafulumizitsa kwambiri yankho: m'mbuyomu, pamenepa, tinkagwira ntchito yopangira nthawi kuti tiwerenge mayankho a zigawozo, koma tsopano tikugwira ntchito. kuchuluka. Ndipo kuti muthamangitse nthambi, muyenera kusintha graph yolumikizidwa kukhala yolumikizidwa.
Kodi kuchita izo? Ngati pali mfundo yofotokozera mu graph, muyenera kulimbana nayo. Malo ofotokozera ndi vertex kotero kuti ikachotsedwa, graph imataya kugwirizanitsa kwake. Magawo onse olumikizirana mu graph atha kupezeka pogwiritsa ntchito algorithm yachikale mu nthawi ya mzere. Njira iyi imafulumizitsa kwambiri nthambi.
Ma vertices aliwonse osankhidwa akachotsedwa, graph imagawanika kukhala zigawo zolumikizidwa.
Tidzachita izi, koma tikufuna zambiri. Mwachitsanzo, yang'anani macheka ang'onoang'ono a vertex mu graph ndikugawaniza ma vertices kuchokera pamenepo. Njira yabwino kwambiri yomwe ndikudziwira kuti ndipeze kudulidwa kochepa kwambiri padziko lonse lapansi ndikugwiritsira ntchito mtengo wa Gomori-Hu, womwe umamangidwa mu nthawi ya cubic. Mu PACE Challenge, kukula kwa graph ndi ma vertices masauzande angapo. Zikatere, mabiliyoni a maopareshoni ayenera kuchitidwa pamtundu uliwonse wa mtengo wobwereza. Zikuoneka kuti n’zosatheka kuthetsa vutoli mu nthawi imene wapatsidwa.
Tiyeni tiyese kukonzetsa yankho. Ma vertex ochepa omwe amadulidwa pakati pa ma vertices amatha kupezeka ndi algorithm iliyonse yomwe imapanga kuthamanga kwambiri. Mutha kuyilowetsa pa netiweki yotere , pochita zimagwira ntchito mofulumira kwambiri. Ndili ndi kukayikira kuti ndizotheka kutsimikizira kuyerekezera kwa nthawi yogwira ntchito , zomwe ziri zovomerezeka kale.
Ndinayesa kangapo kuti ndiyang'ane mabala pakati pa ma vertices mwachisawawa ndikutenga yokhazikika kwambiri. Tsoka ilo, izi zidabweretsa zotsatira zoyipa pakuyesa kotseguka kwa PACE Challenge. Ndinaziyerekeza ndi algorithm yomwe imagawaniza ma vertices a digiri yayikulu, ndikuyiyendetsa ndikuchepetsa kuya kwa kutsika. A aligorivimu kuyesera kupeza odulidwa motere anasiya ma graph akuluakulu. Izi ndichifukwa choti mabala adakhala osagwirizana kwambiri: atachotsa ma vertices 5-10, zinali zotheka kugawanika 15-20 okha.
Ndizofunikira kudziwa kuti zolemba za ma algorithms othamanga kwambiri amagwiritsa ntchito njira zapamwamba kwambiri posankha ma vertices ogawanika. Njira zotere zimakhala ndi kukhazikitsa zovuta kwambiri ndipo nthawi zambiri zimakhala zosagwira bwino nthawi komanso kukumbukira. Sindinathe kuzindikira omwe ali ovomerezeka kuchita.
Mmene Mungagwiritsire Ntchito Malamulo Osavuta
Tili ndi malingaliro okhudza kernelization. Ndiroleni ndikukumbutseni:
- Ngati pali vertex yokhayokha, chotsani.
- Ngati pali vertex ya digiri 1, chotsani ndikutenga woyandikana naye poyankha.
- Ngati pali vertex ya digiri osachepera k +1 ndi, bweza.
Ndi ziwiri zoyambirira zonse zimamveka bwino, ndi chachitatu pali chinyengo chimodzi. Ngati mu vuto lazithunzithunzi za bar tinapatsidwa malire apamwamba k, ndiye mu PACE Challenge mumangofunika kupeza chophimba cha vertex cha kukula kochepa. Uku ndikusintha kwanthawi zonse kwa Mavuto Osaka kukhala Mavuto Osankha; nthawi zambiri palibe kusiyana pakati pa mitundu iwiri yamavuto. M'zochita, ngati tikulemba solver pavuto lakuphimba vertex, pangakhale kusiyana. Mwachitsanzo, monga mu mfundo yachitatu.
Kuchokera pamawonedwe okhazikitsa, pali njira ziwiri zopitira. Njira yoyamba imatchedwa Iterative Deepening. Zili motere: titha kuyamba ndi zopinga zomveka kuchokera pansipa payankho, kenako ndikuyendetsa ma algorithm athu pogwiritsa ntchito cholepheretsa ichi ngati chopinga payankho lochokera pamwamba, osapita m'munsi mobwereza kuposa chopingachi. Ngati tapeza yankho lina, ndilotsimikizika kuti ndiloyenera, apo ayi tikhoza kuonjezera malire awa ndikuyambanso.
Njira ina ndikusunga yankho labwino lomwe lilipo ndikuyang'ana yankho laling'ono, kusintha gawoli likapezeka k kudulidwa kwakukulu kwa nthambi zosafunikira pakufufuza.
Pambuyo poyesa maulendo angapo ausiku, ndidakhazikika pazophatikizira njira ziwiri izi: choyamba, ndimayendetsa ma aligorivimu anga ndi mtundu wina wa malire pakuzama kwakusaka (kusankha kuti zitenge nthawi yocheperako poyerekeza ndi yankho lalikulu) ndikugwiritsa ntchito bwino kwambiri. yankho lopezeka ngati malire apamwamba ku yankho - ndiko kuti, ku chinthu chomwecho k.
Zotsatira za digiri 2
Tathana ndi ma vertices a degree 0 ndi 1. Zikuwoneka kuti izi zitha kuchitika ndi ma vertices a digiri 2, koma izi zidzafuna ntchito zovuta kwambiri kuchokera pa graph.
Kuti tifotokoze izi, tiyenera kusankha ma vertices mwanjira ina. Tiyeni tiitane vertex ya digiri 2 vertex v, ndi oyandikana nawo - vertices x и y. Kenako tidzakhala ndi milandu iwiri.
- pamene x и y - oyandikana nawo. Ndiye mukhoza kuyankha x и yndi v kufufuta. Zowonadi, kuchokera pamakona atatuwa ndikofunikira kubweza ma vertices osachepera awiri, ndipo sitidzataya ngati titenga. x и y: mwina ali ndi anansi ena, ndi v Sanabwere.
- pamene x и y - osati anansi. Kenako zimanenedwa kuti ma vertices onse atatu amatha kumamatidwa kukhala amodzi. Lingaliro ndiloti mu nkhani iyi pali yankho mulingo woyenera kwambiri, mmene ife kutenga mwina v, kapena ma vertices onse awiri x и y. Komanso, mu nkhani yoyamba tidzayenera kutenga oyandikana nawo onse poyankha x и y, koma chachiwiri sikofunikira. Izi zimafanana ndendende ndi milandu pamene sititenga vertex yomatira poyankha komanso tikamatero. Kungotsala pang'ono kuzindikira kuti muzochitika zonsezi kuyankha kuchokera ku ntchito yotereyi kumachepa ndi chimodzi.
Ndikoyenera kudziwa kuti njira iyi ndi yovuta kuigwiritsa ntchito moyenera munthawi yoyenera. Gluing vertices ndi ntchito yovuta; muyenera kukopera mndandanda wa oyandikana nawo. Izi zikachitika mosasamala, mutha kukhala ndi nthawi yothamanga mopanda asymptotically (mwachitsanzo, ngati mumakopera m'mbali zambiri mukamatira). Ndinakhazikika pakupeza njira zonse kuchokera ku ma vertices a digiri ya 2 ndikusanthula mulu wa milandu yapadera, monga mizunguliro yochokera ku ma vertices kapena kuchokera ku vertices yonse kupatula imodzi.
Kuphatikiza apo, ndikofunikira kuti ntchitoyi ikhale yosinthika, kotero kuti pobwerera kuchokera ku recursion timabwezeretsa graph ku mawonekedwe ake oyamba. Kuti nditsimikizire izi, sindinachotse mindandanda yam'mphepete mwa ma vertices ophatikizidwa, kenako ndidangodziwa m'mbali zomwe zikuyenera kupita komwe. Kukhazikitsa kwa ma graph uku kumafunanso kulondola, koma kumapereka nthawi yofananira. Ndipo kwa ma graph a makumi angapo a m'mphepete mwake, imalowa mu cache ya purosesa, yomwe imapereka zabwino zambiri pa liwiro.
Mzere wa kernel
Pomaliza, gawo losangalatsa kwambiri la kernel.
Poyamba, kumbukirani kuti mu ma graph a bipartite chivundikiro chocheperako cha vertex chikhoza kupezeka pogwiritsa ntchito . Kuti muchite izi, muyenera kugwiritsa ntchito algorithm kuti mupeze kufanana kwakukulu komweko, ndiyeno gwiritsani ntchito theorem .
Lingaliro la kernel ya mzere ndi ili: choyamba timagawanitsa graph, ndiye kuti, m'malo mwa vertex iliyonse. v tiyeni tiwonjezere nsonga ziwiri и , ndipo m’malo mwa m’mphepete uliwonse inu v tiyeni tiwonjezere nthiti ziwiri и . Chithunzi chotsatira chidzakhala bipartite. Tiyeni tipeze chivundikiro chocheperako cha vertex mmenemo. Ma vertices ena a graph yoyambirira amafika pamenepo kawiri, ena kamodzi kokha, ndipo ena sanatero. Nthanthi ya Nemhauser-Trotter imanena kuti pamenepa munthu amatha kuchotsa ma vertices omwe sanagunde ngakhale kamodzi n’kubweza amene anagunda kawiri. Kuphatikiza apo, akuti mwa ma vertices otsala (omwe amagunda kamodzi) muyenera kutenga theka ngati yankho.
Tangophunzira kumene kusiya kuposa 2k nsonga Zowonadi, ngati yankho lotsalalo lili pafupifupi theka la ma vertices onse, ndiye kuti palibenso ma vertices onse okwana 2k.
Apa ndinatha kupita patsogolo pang’ono. Zikuwonekeratu kuti kernel yopangidwa motere imatengera mtundu wocheperako wa vertex wophimba womwe tidatenga pa graph ya bipartite. Ndikufuna kutenga imodzi kuti chiwerengero cha vertices chotsalira chikhale chochepa. Poyamba ankatha kuchita zimenezi m’kupita kwa nthawi . Ndidabwera ndikukhazikitsa algorithm iyi munthawi yake , motero, pachimake ichi chikhoza kufufuzidwa mu ma graph a mazana masauzande a vertices pa gawo lililonse la nthambi.
chifukwa
Zoyeserera zikuwonetsa kuti yankho langa limagwira ntchito bwino pamayesero a ma vertices mazana angapo ndi m'mphepete zikwi zingapo. M'mayesero oterowo ndizotheka kuyembekezera kuti yankho lidzapezeka mu theka la ola. Kuthekera kopeza yankho mu nthawi yovomerezeka, makamaka, kumawonjezeka ngati graph ili ndi chiwerengero chokwanira cha vertices ya digiri yapamwamba, mwachitsanzo, digiri 10 ndi apamwamba.
Kuti achite nawo mpikisano, mayankho adayenera kutumizidwa . Kutengera zomwe zaperekedwa pamenepo , yankho langa pamayesero otseguka limakhala lachitatu mwa makumi awiri, ndi kusiyana kwakukulu kuchokera kwachiwiri. Kunena zowona, sizikudziwikiratu momwe mayankho adzawunikiridwa pa mpikisano wokha: mwachitsanzo, yankho langa limadutsa mayeso ocheperako kuposa yankho lachinayi, koma kwa omwe adutsa, limagwira ntchito mwachangu.
Zotsatira za mayeso otsekedwa zidzadziwika pa July XNUMXst.
Source: www.habr.com