Download pdf Fundamentals of computation theory 23rd international symposium fct 2021 athens greece
Fundamentals of Computation Theory
23rd International Symposium FCT
2021 Athens Greece September 12 15
2021 Proceedings Lecture Notes in Computer Science 12867 Evripidis Bampis
(Editor)
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Computer Algebra in Scientific Computing 23rd
International Workshop CASC 2021 Sochi Russia September 13 17 2021 Proceedings Lecture Notes in Computer Science 12865 François Boulier (Editor)
Static Analysis 28th International Symposium SAS 2021 Chicago IL USA October 17 19 2021 Proceedings Lecture Notes in Computer Science Cezara Dr■goi (Editor)
Matchings. Let M = {m1 ,...,mM } beasetof M men, W = {w1 ,...,wW } be asetof W women,and N =min(M,W ).Inamatching,eachpersoniseither single,ormatchedwithsomeoneoftheoppositesex.Formally,weseeamatching asafunction μ : M∪W→M∪W ,whichisself-inverse(μ2 =Id),where eachman m ispairedeitherwithawomanorhimself(μ(m) ∈W∪{m}),and symmetrically,eachwoman w ispairedwithamanorherself(μ(w ) ∈M∪{w }). PreferenceLists. Eachpersondeclareswhichmembersoftheoppositesexthey findacceptable,thengivesastrictlyorderedpreferencelistofthosemembers.
Preferencelistsare complete whennooneisdeclaredunacceptable.Formally,we representthepreferencelistofaman m asatotalorder m over W∪{m},where w m m meansthatman m findswoman w acceptable,and w m w means thatman m preferswoman w towoman w .Similarlywedefinethepreference list w ofwoman w .
Stability. Aman-womanpair(m,w )isblockingamatching μ when m w μ(w ) and w m μ(m).Abusingnotations,observethat μ matchesaperson p with anunacceptablepartnerwhen p wouldprefertoremainsingle,thatiswhenthe pair(p,p)isblocking.Amatchingwithnoblockingpairisstable.Astablepair isapairwhichbelongstoatleastonestablematching.
Definition2(Popularitypreferences). Whenawoman w has popularity preferences,shegivesapositivepopularity Dw (m) toeachacceptablepartner m. Wesee Dw asadistributionoverheracceptablepartners,scaledsothatitsums to1.Sheusesthisdistributiontodrawherfavouritepartner,thenhersecond favourite,andsoonuntilherleastfavouritepartner.
Theorem1. Assumethateachwomanindependentlydrawsherpreferencelist fromaregulardistribution.Themen’spreferencelistsarearbitrary.Let uk be anupperboundontheoddsthatman mi+k isrankedbeforeman mi : ∀k ≥ 1,uk =max w,i P[mi+k w mi ] P[mi w mi+k ] w findsboth mi and mi+k acceptable
Thenforeachwomanwithatleastonestablepartner,inexpectationallof herstablepartnersarerankedwithin (1+2exp( k ≥1 kuk )) k ≥1 k 2 uk ofone anotherinherpreferencelist.
6H.Gimbertetal.
Theorem 1 ismostrelevantwhenthewomen’spreferencelistsarestrongly correlated,thatis,wheneverywoman’spreferencelistis“close”toasingle ranking m1 m2 ... mM .Thisclosenessismeasuredbytheoddsthatin someranking,somemanisrankedaheadofamanwho,intheranking m1 m2 ... mM ,wouldbe k slotsaheadofhim.
Wedetailbelowthreeexamplesofapplications,wheretheexpecteddifference ofranksbetweeneachwoman’sbestandworstpartnersis O (1),andthusher incentivestomisreportherpreferencesarelimited.
– Identicalpreferences. Ifallwomenranktheiracceptablepartnersusingamasterlist m1 m2 ··· mM ,thenall uk ’sareequalto0.ThenTheorem 1 statesthateachwomanhasauniquestablehusband,awell-knownresultfor thistypeofinstances.
– Preferencesfromidenticalpopularities. Assumethatwomenhavepopularity preferences(Definition 2)andthateachwomangivesman mi popularity2 i . Then uk =2 k andtheexpectedrankdifferenceisatmost O (1).
– Preferencesfromcorrelatedutilities. Assumethatwomenhavesimilarpreferences:eachwoman w givesman mi ascorethatisthesumofacommonvalue i andanidiosyncraticvalue η w i whichisnormallydistributed withmean0andvariance σ 2 ;shethensortsmenbyincreasingscores.Then uk ≤ maxw,i {2 P[η w i η w i+k >k ]}≤ 2e (k/2σ )2 andtheexpectedrank difference,byashortcalculation,isatmost4√πσ 3 (1+2e4σ 2 )= O (1).
Theorem2. Assumethateachwomanindependentlydrawsherpreferencelist fromaregulardistribution.Let uk beanupperboundontheoddsthatman mi+k isrankedbeforeman mi :
∀k ≥ 1,uk =max w,i
P[mi+k w mi ]
P
Furtherassumethatallpreferencesarecomplete,that uk =exp( Ω (k )),and thatmenhaveuniformlyrandompreferences.Then,inexpectationthefraction ofpersonswhohavemultiplestablepartnersconvergesto0.
NoticethatinthethreeexamplesofTheorem 1,thesequence(uk )k ≥1 is exponentiallydecreasing.TheassumptionsofTheorem 2 areminimalinthe sensethatremovingonewouldbringusbacktoacasewhereaconstantfraction ofwomanhavemultiplestablepartners.
– Preferencelistsofwomen. Ifweremovetheassumptionthat uk isexponentiallydecreasing,theconclusionnolongerholds:considerabalancedmarket balanced(M = W )inwhichbothmenandwomenhavecompleteuniformly randompreferences;thenmostwomenhave ∼ ln N stablehusbands[19, 25].
Preferencelistsofmen. Assumethatmenhaverandompreferencebuiltas follows:startingfromtheordering w1 ,w2 ,...,wM ,eachpair(w2i 1 ,w2i )is swappedwithprobability1/2,forall i.Asymmetricdefinitionforwomen’s preferencessatisfythehypothesisofTheorem 2,with u1 =1and uk =0for all k ≥ 2.Thenthereisa1/8probabilitythatmen m2i 1 and m2i areboth stablepartnersofwomen w2i 1 and w2i ,forall i,henceaconstantexpected fractionofpersonswithmultiplestablepartners. – Incompletepreferences. Consideramarketdividedintogroupsofsize4of theform {m2i 1 ,m2i ,w2i 1 ,w2i },whereamanandawomanaremutually acceptableiftheybelongtothesamegroup.Onceagain,withconstantprobability, m2i 1 and m2i arebothstablepartnersofwomen w2i 1 and w2i
1.2RelatedWork
Analyzinginstancesthatarelessfar-fetchedthanintheworstcaseisthemotivationunderlyingthemodelofstochasticallygeneratedpreferencelists.Aseries ofpapers[19, 22, 24–26]studythemodelwhere N menand N womenhavecompleteuniformlyrandompreferences.Asymptotically,andinexpectation,the totalnumberofstablematchingsis ∼ e 1 N ln N ,inwhichafixedwomanhas ∼ ln N stablehusbands,whereherbeststablehusbandhasrank ∼ ln N andher worststablehusbandhasrank ∼ N/ ln N .
Theorem1. Assumethateachwomanindependentlydrawsherpreferencelist fromaregulardistribution.Themen’spreferencelistsarearbitrary.Let uk be anupperboundontheoddsthatman mi+k isrankedbeforeman mi :
While aman m issingleandhasnotproposedtoeverywomanhefindsacceptable, do m proposestohisfavoritewoman w hehasnotproposedtoyet. If m is w ’sfavoriteacceptablemanamongallproposalsshereceived, w accepts m’sproposal,andrejectsherprevioushusbandifshewasmarried.
Definition3(separator). A separator isaset S ⊆M ofmensuchthatinthe men-optimalstablematching μM ,eachwomanmarriedtoamanin S prefers himtoallmenoutside S :
Lemma1. Givenaseparator S ⊆M,everystablematchingmatches S tothe samesetofwomen.
Proof. Let w ∈ μM (S )andlet m bethepartnerof w insomestablematching. Since μM isthewoman-pessimalstablematchingbyTheorem 1, w prefers m to μM (w ).Bydefinitionofseparators,thatimpliesthat m ∈ S .Hence,inevery stablematching μ,womenof μM (S )arematchedtomenin S .Byacardinality argument,menof S arematchedby μ to μM (S ).
Definition4(prefixseparator,block). A prefixseparator isaseparator S suchthat S = {m1 ,m2 ,...,mt } forsome 0 ≤ t ≤ N .Givenacollectionof b +1 prefixseparators Si = {m1 ,...,mti } with 0= t0 <t1 < <tb = N ,the i-th block istheset Bi = Sti \ Sti 1 with 1 ≤ i ≤ b
Abusingnotations,wewilldenote S astheprefixseparator t and B asthe block (ti 1 ,ti ]
Lemma2. Givenablock B ⊆M,everystablematchingmatches B tothesame setofwomen.
Proof. B equals Sti \ Sti 1 forsome i.ApplyingLemma 1 to Sti andto Sti 1 provestheLemma.
Lemma3. Considerawoman wn whoismatchedby μM andlet B =(l,r ] denoteherblock.Let x denotethenumberofmenfromabetterblockthatare rankedby wn betweenamanof B and mn :
x = |{i ≤ l |∃j>l,mj wn mi wn mn }|
10H.Gimbertetal.
Thenin wn ’spreferencelist,thedifferenceofranksbetween wn ’sworstandbest stablepartnersisatmost x + r l 1.
Proof. Since μM iswoman-pessimalbyTheorem 1, mn isthelaststablehusband in wn ’spreferencelist.Let mj denoteherbeststablehusband. In wn ’spreferencelist,theintervalfrom mj to mn containsmenfromherown block,pluspossiblysomeadditionalmen.Suchaman mi comesfromoutside herblock(l,r ]andsheprefershimto mn :since r isaprefixseparator,wemust have i ≤ l .Thus x countsthenumberofmenwhodonotbelongtoherblock butwhoinherpreferencelistarerankedbetween mj and mn
Ontheotherhand,thenumberofmenwhobelongtoherblockandwhoin herpreferencelistarerankedbetween mj and mn (inclusive)isatmost r l
Together,thedifferenceofranksbetween wn ’sworstandbeststablepartners isatmost x +(r l ) 1.SeeFig. 1 foranillustration.
l =2, r =8and x =1
mi with i ≤ l mi with l<i ≤ r
mi with r<i
x = |{|∃ ,
Fig.1. Preferencelistof wn ,with n =6.Theblockof wn isdefinedbyaleftseparator at l =2andarightseparatorat r =8.Colorswhite,grayandblackcorrespondsto blocks,andaredefinedinthelegend.Allstablepartnersof wn mustbegray.Menin blackareallrankedafter mn = µM (wn ).Thedifferenceinrankbetween wn ’sworst andbestpartnerisatmostthenumberofgraymen(here r l =6),minus1,plusthe numberofwhitemenrankedafteragraymanandbefore mn (here x =1).
∀k ≥ 1,uk =max w,i P[mi+k w mi ] P[mi w mi+k ] w findsboth mi and mi+k acceptable
Let w beawoman.Givenasubsetofheracceptablemenandarankingofthat subset a1 w ··· w ap ,weconditionontheeventthatin w ’spreferencelist, a1 w ··· w ap holds.Let mi = a1 be w ’sfavoritemaninthatsubset.Let Ji bearandomvariable,equaltothehighest j ≥ i suchthatwoman w prefers mj to mi .Formally, Ji =max{j ≥ i | mj w mi }.Then,forall k ≥ 1,wehave P[Ji <i + k |
Proof. Ji isdeterminedby w ’spreferencelist.Weconstruct w ’spreferencelist usingthefollowingalgorithm:initiallyweknowherranking σA ofthesubset A = {a1 ,a2 ,...,ap } ofacceptablemen,and mi = a1 isherfavoriteamong those.Foreach j from N to i indecreasingorder,weinsert mj intotheranking accordingtothedistributionof w ’spreferencelist,stoppingassoonassome mj isrankedbefore mi (orwhen j = i isthatdoesnothappen).Thenthestep j ≥ i atwhichthisalgorithmstopsequals Ji .
Toanalyzethealgorithm,observethatateachstep j = N,N 1,...,we alreadyknow w ’srankingofthesubset S = {mj +1 ,...,mN }∪{a1 ,...,ap }∪ {menwhoarenotacceptableto w }.If mj isalreadyin S , w prefers mi to mj , thusthealgorithmcontinuesand Ji <j .Otherwisethealgorithminserts mj intotheexistingranking:bydefinitionofregulardistributions(Definition 1), theprobabilitythat mj beats mi giventherankingconstructedsofarisatmost theunconditionalprobability P[mj w mi ].
Bydefinitionof uj i ,wehave1
uj i ) 1 ≥ exp( uj i )
Summingoverallrankings σS of S thatarecompatiblewith σA andwith Ji ≤ j , P[Ji <j | Ji ≤ j ]= σS compatiblewith Ji ≤j andwith σA
Finally, P[Ji <j ]= N
).
RecallfromLemma 3 that r l 1+ x isanupperboundonthedifferenceof rankofwoman wn ’sworstandbeststablehusbands.Wefirstboundtheexpected valueoftherandomvariable x definedinLemma 3
12H.Gimbertetal.
Lemma5. Givenawoman wn ,definetherandomvariable x asinLemma 3: conditioningon H, x = |{i ≤ l |∃j>l,mj wn mi wn mn }| isthenumber ofmeninabetterblock,whocanberankedbetween wn ’sworstandbeststable husbands.Then E[x] ≤ k ≥1 kuk .
Proof. Startbyconditioningon H,andlet mn = a1 w a2 w ··· w ap be wn ’s rankingofmenwhopreferhertotheirpartnerin μM .Wedrawthepreference listsofeachwoman wi with i<n,anduseAlgorithm 2 tocomputethevalue of l .
Foreach i ≤ l ,weproceedasfollows.If mn wn mi ,then mi cannotbe rankedbetween wn ’sworstandbeststablepartners.Otherwise,weareina situationwhere mi wn a1 wn ··· wn ap .UsingnotationsfromLemma 4, w prefers mi toall mj with j>l ifandonlyif Ji <l +1.ByLemma 4 thisoccurs withprobabilityatleastexp( k ≥l+1 i uk ).Thus
jumpsto i 1andlooksforawitnessagain.Whenthereisnowitness,aprefix separatorhasbeenfound,thus l isthelargestprefixseparator ≤ n 1.Similarly, Algorithm 2 computesthesmallestprefixseparator r whichis ≥ n.Thus,by definitionofblocks,(l,r ]istheblockcontaining wn .
Women
Fig.2. Computingtheblockcontainingwoman wn .Theverticalblackedgescorrespondtothemen-optimalstablematching µM .Thereisalightgrayarc(mj ,wi )if j>i andwoman wi prefersman mj toherpartner: mj wi mi .Theprefixseparatorscorrespondtothesolidredverticallineswhichdonotintersectanygrayarc. Algorithm 2 appliesaright-to-leftgreedymethodtofindthelargestprefixseparator l whichis ≤ n 1,jumpingfromdashedredlinetodashedredline,andasimilar left-to-rightgreedymethodagaintofindthesmallestprefixseparator r whichis ≥ n. Thisdeterminestheblock(l,r ]containing n.(Colorfigureonline)
Definition6. Let X betherandomvariabledefinedasfollows.Let (Δt )t≥0 denoteasequenceofi.i.d.r.v.’stakingnon-negativeintegervalueswiththefollowingdistribution:
Then X = Δ0 + Δ1 + + ΔT 1 ,where T isthefirst t ≥ 0 suchthat Δt =0
TheproofsofthefollowingLemmascanbefoundin[13].
Lemma7. Givenawoman wn ,let (l,r ] denotetheblockcontaining n.Conditioningon H, l and r areintegerrandomvariable,suchthat r n and n 1 l arestochasticallydominatedby X .
Lemma8. Wehave E[X ] ≤ exp( k ≥1 kuk ) k ≥1 k 2 u
2.5PuttingEverythingTogether
Proof(ProofofTheorem 1). Withoutlossofgenerality,wemayassumethat N = M ≤ W andthateachmanismatchedintheman-optimalstablematching μM :toseethat,foreachman m weadda“virtual”woman w ashisleastfavorite acceptablepartner,suchthat m istheonlyacceptablepartnerof w .Amanis
