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To cite this version:

Lang, Jérôme and Mengin, Jérôme and

Xia, Lirong Voting on multi-issue domains with conditionally

lexicographic preferences. (2018) Artificial Intelligence, 265.

18-44. ISSN 0004-3702

Official URL

DOI :

https://doi.org/10.1016/j.artint.2018.05.004

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Voting

on

multi-issue

domains

with

conditionally

lexicographic

preferences

Jérôme

Lang

a

,

Jérôme

Mengin

b,∗

,

Lirong

Xia

c aCNRS,PSLResearchUniversity,UniversitéParis-Dauphine,France

bIRIT,UniversitédeToulouse,CNRS,UT3,France cRensslaerPolytechnicInstitute,USA

a b s t r a c t

Keywords:

Computationalsocialchoice Voting Winnerdetermination

Lexicographic preferences Maximumsatisfiability

One approach to voting on several interrelated issues consists in using a language forcompact preference representation, from which the voters’ preferences are elicited and aggregated. Such a language can usually be seen as a domain restriction. We considera well-known restriction, namely, conditionallylexicographic preferences, where both therelative importance between issues and the preference between the values of an issue may depend on the values taken by more important issues. The naturally associated languageconsists indescribing conditional importance andconditionalpreferencebytreestogether with conditional preference tables. In this paper, we study the aggregation of conditionally lexicographic preferences for several common voting rules and several classes of lexicographic preferences. We address the computation of the winning alternative for some important rules, both by identifying the computational complexity of the relevantproblems and by showing that for several of them, computing the winner reduces in a verynaturalwaytoa maxsat problem.

1. Introduction

There aremany situationswhere agroup ofagentshas to make acommon decisioninmulti-issuedomains, i.e. aset of possiblyinterrelatedbinaryissues.Atypicalexampleofsuchasituationisthatofmultiplereferenda:thereisaset ofbinary issues (suchasbuildingasportcentre,buildingaculturalcentreetc.),and thegroup mustmakeayes/nodecisiononeach ofthem[1].Anotherexample isgroupproductconfiguration,where thegroup mustagreeon acomplexobjectconsistingof several components,eachtakingoneoftwopossiblevalues.

Votingonseveralinterrelatedissuesisachallengingproblem.Iftheagentsvoteseparatelyoneachissue,thenparadoxes generally arise [1,2]. Such paradoxes rule out this ‘decompositional’ approach, except in the very restricted case when voters have separablepreferences. Asecond wayconsists inusinga sequential votingprotocol: issues are consideredone after another, in a predefined order, and the voters know the assignment to the earlier variables before expressing their preferencesonlaterones(see,e.g.,[3–5]).Thismethod,however,worksreasonablywellonlyifwecanguaranteethatthere

This paper isan invited revision of a paper whichfirst appeared at the 18th International Conference on Principles and Practice of Constraint

Programming,2012.

*

Correspondingauthor.

E-mailaddresses: lang@lamsade.dauphine.fr (J. Lang),Jerome.Mengin@irit.fr (J. Mengin),xialirong@gmail.com (L. Xia). https://doi.org/10.1016/j.artint.2018.05.004

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exists acommonorderoverissuessuchthat everyagentcanexpressher preferencesunambiguouslyon thevaluesofeach issue atthetimesheisaskedtoreportthem.

Athirdclassofmethodsconsistsinusingalanguageinwhichtheagentscanexpresstheirpreferencesoverthespaceof combinedalternativesforallissuessimultaneously.Ifthelanguageisexpressiveenoughtoallowforexpressinganypossible preferencerelation,thentheaforementionedparadoxesareavoided.However,thisexpressivityimpliessignificantelicitation andcomputationcosts:inamulti-issuedomain,thenumberofalternativesisexponentialinthenumberofissues,therefore the numberofpossible preference relationsisdoubly exponential and whichever languagewe useto represent them,the inputwillbeexponentialintheworstcase.Onewayaroundthisproblemistousesomecompactpreferencerepresentation language;thiscomes atthecostofsomedomainrestriction, dependingonthechosenlanguage.

Therefore,whenaggregatingpreferencesonmultipleinterrelatedissues,achoice,oratrade-off,mustbemade between (a) beingpronetosevereparadoxes,(b)requiringaheavycommunicationandcomputationburdenor(c)imposingadomain restriction.

In thispaper, we explorea particular domainrestriction byfocusingon conditionallylexicographicpreferences.Standard lexicographic preferencesare a simple model, wherealternatives arefirst sorted according toone issue, consideredto be the most important one; alternatives that have equal values for that issue are thensorted accordingto the second most

important issue, and so on until the set of alternatives is totally ordered. The psychology literature showsevidence that lexicographicpreferencesareoftenanaccuratemodelforhumandecisions[6,7].Althoughveryintuitive,thismodelisquite restrictive. However, it can be extended by allowing the preferences over the values of an issue, as well as the relative importance oftwo issues,todependonthevaluesofmoreimportantissues.Therelativeimportanceofissuesisnolonger alinearordering,but atree,withthemostimportantissue atitsroot.Apreference relationisconditionallylexicographic if

it isdefined bysucha lexicographic preferencetree. To thebest ofour knowledge,conditionally lexicographic preference trees were first introduced by Fraser [8,9], and have been considered in individual decision making (and especially in constraint satisfaction problems)by Wilson [10,11] and Wallace and Wilson [12], as well as, from the learning point of view, byBooth etal.[13],Brauning etal. [14,15] and byLiuand Truszczy ´nski [16].Notethat Wilson[11] andBrauninget al. [14,15] proposetoextendfurtherthelexicographicpreferencemodelbyallowingseveralvariablestoappearatthesame importance level/nodeinthetree;wedonotconsidersuchtreesinthispaper.

Lexicographic preferences appear to bea reasonable assumption in many domains (see the work of Taylor [17] for a discussionontheplausibilityoflexicographicpreferencesinvariouspoliticalcontexts).However,itimpliesastrongdomain restriction:forq binaryvariables,thereare(2q)!preferencerelationsandonly2q (respectivelyq!×2q)lexicographic

prefer-ence relationsif theimportanceorderisfixed(respectively,not fixed).Asexplainedabove, domainrestrictionsareneeded to escape strongparadoxesand significantcommunicationcost,but therisk,bymaking atoostrongdomainrestriction, is to beon theweakside regardingitsplausibility,sinceonly atiny fractionofthewhole set ofpreference relationsfallsin theclassoflexicographic preference relations.(A similarproblem occurswithseparablepreferences.)Conditionally lexico-graphic preference models still area domainrestriction, but a muchweaker one than standard lexicographic preferences, asthereare exponentiallymoreconditionallylexicographic preferencesthan lexicographic preferences[13].Therefore, this assumptioncanbeseenasareasonabletrade-offbetweenexpressivityandcomplexity.

The aggregation oflexicographic preferencesover combinatorial domainshas received only little attention. Taylor[17] considers theaggregationoflexicographicpreferenceson adomainoftwoattributeswithreal-valueddomains.Each voter has unconditional, single-peaked preferences on each of the two attribute domains. He shows that when voters all have thesameimportanceorder, therealwaysexistsaweakCondorcet winner,butthat thisisno longer thecaseif votersmay diverge on the relativeimportance ofthe two attributes(and hecharacterizes situations wherea weakCondorcet winner exists). Pattanaik [18] generalizes the former results on domainsof more than two attributes, and establishes conditions for the existence of a socially best alternative for a wider class of voting rules including simple majority. Bhadury et al. [19] generalizeTaylor’slatterresultto morethantwo attributes.Encarnacion[20] focuseson preferenceaggregation(with a preference relation as output)rather than voting: he also considers lexicographic preferences with a unconditional im-portance order, commonto allagents, andsingle-peaked preferences onthe domainofeachvariable, and showsthat this domainrestrictionisArrow-consistent.

More recently, Liu and Truszczy ´nski [21] obtained results completing theresults weobtained in ourconference paper [22] (see Section 5 for more discussions) and used answer-setprogramming to encode and solve preference aggregation problemsoncombinatorialproblemswithconditionallylexicographicpreferences(wewillcomebacktothisinmoredetail inSection4).

Dividinoetal.[23] makeuseoflexicographicpreferencesforpreferenceaggregationinmulti-attributedomains.Amajor difference withour workis that they uselexicographicity forthepreferenceaggregationphase and notfor defining eachof theindividualpreferences. Theystartwithaset ofitems (pieces ofinformationon theweb),eachassociated withatuple ofvaluescorrespondingtovariouscriteriasuchasthetimetheitemwas posted,itssource,etc.Each criterioncorresponds to aweak order over items (for instance, for time, the morerecent the better). Finally, given an importance order ⊲ on

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The generic problem of aggregating conditionally lexicographic preferences can bestated asfollows. The set of alter-natives is a multi-issue domain D composed of a finite set of binary issues.1 We have a set of voters, each providing a

conditionally lexicographic preference relation over D under the compact and natural form ofa lexicographic preference tree(LP-treeforshort) [13],whichwewilldefinesoon.Therefore,a(compactly represented)profile V consistsofa collec-tion ofLP-trees. Finally, for a given voting rule r, we askwhether there is asimple wayto compute the winner,namely

r(V), where‘simple’means thatthewinnershouldbecomputedefficientlyanddirectlyfrom V .Thismeans thatwemust avoid producing theentirepreferencerelationsof everyvoter explicitly,whichwould requireexponential space.Formany cases where winner determination is computationally hard,we show that these problems canbe efficiently converted to maxsatproblemsandcanbesolvedby maxsat solvers.

The restofthe paperisorganizedasfollows.Conditionally lexicographicpreferences and theircompact representation byLP-treesaredefinedand discussedinSection2.InSection3westatetheproblemconsideredinthispaper,namelythe application ofvoting rulestoprofiles composedofconditionallylexicographic preferences.Wewill focusonthreefamilies ofrules.First, k-approvalrulesinSection 4:weshowthat formanyvaluesofk,wecangiveaquitesatisfactoryanswerto

ourquestion above,even forourmost generalmodels. Notethat by‘satisfactory’ wedonotnecessarilymean‘computable in polynomial time’. For instance, we will show that in some cases, there isa model-preserving translation between the winner determination problem and a maximum (weighted or unweighted) satisfiability problem. Since efficient maxsat solversexist, in suchcases wecan consider preference aggregation astractable tosome extent. In Section 5 wefocus on the Borda rule (and, to a lesser extent, to some other positional scoring rules sharing some properties with Borda): we show thatwinner determinationcanbesolvedinpolynomialtimeforsomeofthesimplestLP-treemodels,andpropose a translation intoa weighted minimumsatisfiability problemfor some general models. Wealso provide anatural family of scoring rulesfor which similartranslations exist. Thenin Section6we consider theexistence ofaCondorcet winner, and show that even decidingwhethera given alternativeisaCondorcet winner ishard.In Section7, weconsider aparticular setting where all voters have a common, possibly conditional, importance structure among issues, but can have varying, conditional preferences overthe valuesofeach issue. Section8 isdevoted tothespecific caseof LP-treeswithfixed local preferences butpossibly differentstructures.In Section9wediscuss theapplicationof ourresults tocommittee elections, aswellastheimportance ofourdomainrestriction, andtheextension ofourresultsto variableswithnonbinarydomains. Finally, Section10concludes.

2. ConditionallylexicographicpreferencesandLP-trees

Let I= {X1,. . . ,Xq} (q≥2) beaset of issues,where eachissue Xi takes a value ina binarylocaldomain Di, denoted

as {0i,1i}, or as {xi,xi}, or, when there is no ambiguity, as {0,1}. The set of alternatives is D=D1× · · · ×Dq, that is,

an alternative is identified by its values on all issues. Alternatives are denoted by d, e, etc. For any Y ⊆ I we denote

DY=QXiYDi.Anelementof DY iscalledapartialalternative.Let L(D)denote theset ofalllinearorders over D.

Wefirstgiveahigh-leveldescriptionoflexicographicpreferences.Anytwoalternatives d, e arecomparedbylookingat theissues insequence,accordingto theirimportance,until wereachanissue X suchthat thevalue of X ind is different from thevalue of X in e.d ande are thenordered accordingtothelocalpreference relationoverthevaluesof X . Forsuch

lexicographic preference relationswe needboth animportance relation between issues, and localpreference relations over thedomainsoftheissues.Boththeimportancebetweenissuesandthelocalpreferencesmaybeconditionedbythevalues of moreimportantissues.Suchlexicographic preferencerelationscan becompactlyrepresented byLexicographicPreference trees(LP-trees) [13],formallydefinedintheupcomingsubsection.

2.1. Lexicographicpreferencetrees

AnLP-treeL iscomposedoftwo parts:

1. AtreeT whereeachnodet islabelled byanissue,denotedbyIss(t)=Xi,suchthateveryissueappearsonceandonly

onceoneachbranch;eachnon-leafnodeeitherhastwooutgoingedges, labelled bythetwovaluesinthelocaldomain (0i and 1i)respectively,oroneoutgoingedge,labelled by {0i,1i}.

2. A conditionalpreferencetable CPT(t) for each nodet, which is defined as follows. Let Anc(t) denote the set of issues labelling theancestors of t.Let Inst(t) (respectively, NonInst(t)) denote the set ofissues in Anc(t) that have two (re-spectively, one)outgoing edge(s).There is aset Par(t)⊆NonInst(t) such that CPT(t) is composedofthe agent’s local preferencesover DIss(t) forallvaluations ofPar(t). Thatis, if Iss(t)=Xi then forevery valuation u of Par(t),theCPT

containseitheru:0i≻1i oru:1i≻0i.

AnLP-treeLrepresentsalinearorder≻L over D asfollows.Let Xi betheissueassociatedwiththerootofL,andlet≻Xi denotetheassociatedlocalpreferencerelation.WedenotebyLX

i=xi (respectivelyLXi=xi)thesubtreeofLcorrespondingto

1 Theassumptionthatissues(orvariables)arebinaryismadeforthesakeof simplicity,andbecauseitholds inmanypracticaldomains,especially

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L P1 L P2 L P3 L P4 M mm F ff P pp P pp F p:ff p:ff m m P pp F ff M p:mm p:mm F ff M mm P pp P pp M mm f f P pp F ff M mm

Fig. 1. Four LP-trees.

thebranch Xi=xi (respectively Xi=xi).Letd= (d1,. . . ,dq)and e= (e1,. . . ,eq) betwodifferentalternatives.Thend≻Le

if either(1)diXiei,or(2)di=ei and d≻LXi=di e.

Moreover, to eachalternative dD correspondsa singlebranch in thetree: ateachnode labelled with issue Xi with

twooutgoingedges,thebranchcorrespondingtod followstheedgelabelled withdi.Theimportanceorder associatedwithd

inL,denotedbyIO(L,d),istheorderinwhichissuesappearalongthat branch.Weuse⊲todenoteanimportanceorder

todistinguishitfromagents’preferences≻over D. IfinL eachnodehas nomorethan onechild,thenallalternativesare associated withthesameimportanceorder⊲,andwesaythat ⊲istheimportanceorderofL.

The size |L| ofanLP-tree L is thenumber ofits nodesplus the accumulated size ofitsconditional preference tables. Note thatthesize ofL maybeexponentialinq eveninthecaseofalinearstructure.

Example1.Suppose therearethreebinaryissuestobedecidedamongtheinhabitantsofacity. 1. Shouldthecitybuildametro?

2. Shouldthecitycentrebepedestrian?

3. Shouldtherebeafeeforcarsenteringthecity?

The threeissues are M (metro) with domain {m,m}, P (pedestrian) with domain {p,p}, and F (fee) with domain {f,f}. Therearefourvoters.EachvoterhasconditionallylexicographicpreferencesrepresentedbytheLP-treesL P1, L P2, L P3and

L P4 respectivelyasdepictedinFig.1.

ConsidertheLP-tree L P1.LetP1bethenodeattheextremityoftheleftbranch.WehaveIss(P1)=P,Anc(P1)= {M,F}, Inst(P1)= {M},NonInst(P1)= {F},and Par(P1)= ∅ becausethepreferencesover {p,p} donotdependonthevalueofF. On the otherhand, if F1 isthe nodeattheextremityof therightbranch, thenPar(F1)= {P}.The linearorderrepresented by

L P1 is

mf p

mf p

m f p

m f p

mp f

mp f

mp f

mp f

.

Moreover, IO(L P1,mf p)= [MFP]and IO(L P1,m f p)= [MPF].

We say that an LP-tree has unconditional local preferences when the preference relation on the value of every issue is independent from the values of all other issues. In other words, each issue in the tree only contains unconditional preferences. Thisis thecasefortrees L P3 and L P4 but not for L P1 nor L P2.In L P1, for instance,thepreference between

p and p depends on the value of M. We note that when the preferences are unconditional it is still possible that the importance relationbeconditional,i.e. thetreemayhavebranches.

Likewise,wesaythattheimportancerelationisunconditional whentheorderoftheissuesisthesameinallbranchesof thetree(or,equivalently,whenthetreehas asinglebranch).ThisisthecaseoftreesL P2 and L P4.Notethat L P4 hasboth unconditionalimportanceand unconditionallocalpreferences.See[13] foradiscussionon varioussub-classesofLP-trees.

ApropertythatmakesLP-treesappealingforsocialchoiceisthatitiseasytocomputetherankofalternatives:

Observation1.Consider an alternative d and an LP-tree L. For each issue Xi, letti be the node labelled with Xi in the

branch ofLcorrespondingtod;wecandefinethelevel of Xi w.r.t.d andthelocalrankofd w.r.t.to Xi asfollows:

• level(L,d,Xi) isthelevel ofti in thetree, that is,thedistance fromt to theroot plus one.The level ofthe rootis1

andthelevel ofallleavesisq.

• 1(L,d,Xi) is the local rank of the value of d for issue Xi in the local preference at ti. Because issues are binary,

1(L,d,Xi)=0 if didi at node ti (given the values of d forthe issues that appear above ti), and 1(L,d,Xi)=1

otherwise.

Let rank(L,d)betherankofd inthelinearorderL definedbyL,then: rank

(

L

,

d

) =

1

+

q

X

i=1 2q−level(L,d,Xi)

1(

L

,

d

,

X i

)

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Example2.Consider L P1 definedinExample1.

rank

(

L P1

,

mf p

) =

1

+

22

×

0

+

21

×

0

+

20

×

0

=

1

;

rank

(

L P1

,

m f p

) =

1

+

22

×

1

+

21

×

1

+

20

×

1

=

8

;

rank

(

L P1

,

m f p

) =

1

+

22

×

0

+

21

×

1

+

20

×

1

=

4

.

Notethatrank(L,d) canbecomputedbyasimpletop-downtraversalofL alongthebranchcorrespondingtod,intime linearinthenumberofissues.

Conversely,Algorithm1computesthealternativewithrankk inthelinearorder≻L correspondingtoanLP-treeL:the

base2representationofk−1 isused toguideatop-downtraversalofthetreefromtheroottoaleaf.Therunningtimeis again linearinthenumberofissues.

Algorithm 1: F ind Alternati ve(L,k).

1 Letk−1= (kq−1...k0)2andL∗= L;

2 for i=q1 downtoi=0 do

3 Let Xi betherootissueofL∗.Letthelocalpreferencesbexixi;

4 if ki=1 then 5 L∗← LXi=xi ;dixi ; 6 else 7 L∗← LXi=xi ;dixi ; 8 end 9 end 10 return d.

3. ApplyingvotingrulestoLP-trees

A (voting)profile V over a set of alternatives D is a collectionof n votes V1,. . . ,Vn, each being a linearorder on D.

An(irresolute)votingrule r mapsevery profile V toanonemptysubset of D:r(V) is thesetof co-winners for V under r.

A scoring function S is a mapping from L(D)n×D to R. Often, a voting rule r is defined in such a way that r(V) is

the set of alternatives maximizing some scoring function Sr. In particular, a positionalscoringrule is defined by a scoring

vector v= (v(1),. . . ,v(m)), where m is the number of alternatives (in this paper m=2q): for any vote V

iL(D) and

any cD, let Sv(V,c)=v(rankV(c)), where rankVi(c) is the rank of c in Vi. Then for any profile V = (V1. . . ,Vn), let

Sv(V,c)=Pnj=1Sv(Vj,c).The winneris thealternativemaximizing Sv(V,·).In particular, thek-approval ruleAppk (with

km),isdefined bythescoringvector v(1)= · · · =v(k)=1 and v(k+1)= · · · =v(m)=0.Let SkApp denote scoringvector fork-approval.TheBorda ruleisdefinedbythescoringvector(m−1,m−2,. . . ,0).Let SBorda denotethescoringvectorfor Borda,that is, SBorda(Vi,c)=m−rankVi(c).

Let NV(c,d) denote the number of votes in V thatrank c ahead of d. An alternative c is the Condorcetwinner for a

profile V if forevery d6=c,a(strict) majorityofvotesin V prefers c to d,that is,if NV(c,d)>n/2.If forevery d6=c we

have NV(c,d)n/2,withequalityforatleastoned,thenc isaweakCondorcetwinner.

3.1. Votingrestrictedtoconditionallylexicographicpreferences

Applying a voting rule to profiles consisting of arbitrary preferences on multi-issue domains is highly unpractical, be-cause the specification of such preferences requires exponential space if no domain restriction is made. Does it become significantly easierwhen werestrict to conditionallylexicographic preferences? Thisis thekey problemaddressed inthis paper.Ofcoursetheanswerdependson thevotingruleused.

A conditionallylexicographicprofile is a collection of n conditionally lexicographic preferences over D. As conditionally lexicographic preferencesarecompactlyrepresentedbyLP-trees,wedefineanLP-profile V asacollectionofn LP-trees.

Given anLP-profile V and avoting rule r, anaive way of finding theco-winners would consist in determining the n

linear orders inducedby theLP-treesand thenapplying r to these linearorders. However, thiswouldbe very inefficient, bothinspacecomplexity andtimecomplexity.Therefore,wewouldlike toknowifitisfeasible,and efficient,to compute the winners directly from the LP-trees. More specifically, we ask the following questions: (a) given a voting rule, how difficultisittocomputetheco-winners(or,else,oneoftheco-winners)forvariousclassesofLP-trees?(b)forscore-based rules,how difficultisitto computethescoreoftheco-winners?Formally,foreachvotingruler definedasthemaximizer ofscoringfunctionS,weconsiderthefollowingdecisionandfunctionproblems.

EVALUATION(for r):

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Question Doesthereexistan alternatived suchthat S(V,d)>T ? WINNER(for r):

Input LP-profile V

Output r(V)

Note that if evaluation is NP-hardand the scoreofan alternativecan becomputedin polynomial time,then winner cannot bein P unless P = NP:if winner werein P, then evaluation couldbesolvedinpolynomial timebycomputinga winner anditsscore.Forthevotingrulesstudiedinthispaper,if notmentionedspecifically, evaluation isw.r.t. thescore functions wepresent whendefiningtheserules.

3.2. LP-treesandpropositionallogic

Some of the positive results in the sequel are obtained by translating LP-trees into some particular types of logical formulas.Webrieflyrecallheresomebasicnotionsofpropositionallogicandsomesatisfiabilityproblemsthatwillbeused inthesequel.

GivenasetofpropositionalsymbolsP,logicalformulas canbebuiltwiththeusualconnectives∧(conjunction),∨ (dis-junction)and ¬ (negation).Inseveraltranslations,wewilluseonepropositionalsymbolforeachissue, andsowewilluse thesamenotationforanissue Xi andthecorrespondingpropositionalsymbol,sothatthesetofpropositionalsymbolswill

often beI itself.

Theliterals aretheelementsofPand theirnegations;thatis,thesetofliteralsis{X,¬X|X∈ P}.Aclause hasthe gen-eralformC=l1∨l2∨ . . . ∨l|C|,whereeachli isaliteral.Whenalltheliterals,exceptpossiblyone,arenegatedpropositions,

itisaHorn clause. Adualofaclauseisacube,whichisaconjunctionofliterals.

A valuation of P assigns aBoolean value 0 (false) or1 (true) to eachsymbol in P. A valuation satisfies aclause if it

makesatleastoneliteraltrue,and sat istheproblemofdecidingifagivensetofclausesissatisfiable,that is,ifthereisat least onevaluation thatsatisfiesallclauses;suchavaluation iscalledamodel oftheset ofclauses. sat isan NP-complete problem.

The maxsat problem is a generalization of sat: given a set of clauses, we want to find a valuation maximizing the number of satisfied clauses. In the partialweightedmaxsat problem, some clauses have an associated positive weight, the others are left unweighted; the goal is to maximize the sum of theweights of the satisfied weighted clauses, while satisfying alltheunweightedclauses. These are NP-completeproblems, but the2016 maxsat Evaluation, partof the19th International Conference on Theory and Applications of Satisfiability Testing, shows that some solvers are able to solve industrialbenchmarksinafewminutes,withuptohundredsofthousandsofvariables,andmillionsofclauses(ofsize 3).2 Some of the most competitive solvers, at the time of writing, are described in e.g. [25–27]. In the generalizedmaxsat problem, the input is not restricted to clauses: given a set of propositional formulas, we want to find a valuation that satisfiesasmanyoftheformulas aspossible.

weightedminsatisanother variantof sat: givenaset ofclauses, eachassociatedwithapositive cost, wewantto find a valuation with minimum cost, where the cost of a valuation is the sum of the weights of the clauses that it satisfies. Althoughitis NP-complete,thisproblemtoocanbesolvedusingefficientsolvers;someworkbytranslatingtheseinstances into weightedmaxsatinstances(e.g. [28,29]),whereasmorerecentonesusebranch-and-boundtechniques directlyonthe minsatinstances(see,e.g. [30,31]).[30] reportsonexperimentswhererandom minsat instanceswithonehundredvariables anduptofivetimesmoreclausesweresolvedinafewminutes.Accordingto[31],theperformancesof minsat solversseem tobecomparabletothat of maxsat solvers,someparticularproblemsbeing moreefficientlysolvedwithone ortheother. 4. k-Approval

Recall that it is possible to compute, in time linear in the numberof issues, the k-th ranked issue ofany LP tree for any k (cf. Algorithm1).Therefore, given an integerk<2q and anLP profile V ,we can computethe topk alternatives of

allLP-treesinV ,and storetheminatabletogetherwiththeirk-approvalscores.As wehaveatmostkn suchalternatives, constructing thetabletakestimein O(knq).Hencewehavethefollowingresult.

Proposition2.Foranyconstantk,givenanLP-profileV withn voters,thek-approvalco-winnersforV canbecomputedintimein O(knq).

Example 3.Let V be the profile of Example 1. The winners for 1-approval (that is, plurality) are mf p,m f p, m f p and m f p – their1-approvalscoreis1. Thewinnersfor2-approval arem f p andm f p,witha2-approvalscoreof 2.

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Asimilarresultalsoholdsforcomputingthe(2qk)-approvalco-winnersforany constantk.However,unlessq is very small,there islittle practicalinterestin using(2qk)-approvalfora fixed(small) valueofk, sinceinpractice, weexpect kn≪2q,sothatalmosteveryalternativewouldbeaco-winner.

We now focuson k-approval voting for values of k that are defined via proportions of the number 2q ofalternatives.

Let Mq be apositive integer and defineRat(M) astheset ofallrational numbersof theform j/2M for some positive

integer M and j∈ {1,. . .2M−1}.Equivalently,Rat(M) isthesetofallnumbers

α

intheinterval(0,1)withafinitebase2

representation oflengthatmost M.Forany

α

∈Rat(M),

α

-proportion-approval isanothernameforthe

α

2q approval rule.

Wefirstconsider 12-proportion-approval,or,equivalently,2q−1-approval.Thisrulerequirestocountthetop50% alterna-tives,whichcorrespondstothe‘better’halfofthetree.Inthiscase,eachvoteronlyhastocommunicatehermostimportant issue and itspreferred value. Applying 2q−1-approvalhere isthereforebothintuitive and cheap in communication,and it turnsoutthat theco-winnerscanberepresentedcompactlyand computedveryeasily.

Theorem3.Winnerfor2q−1-approvalandLP-profilescanbecomputedintimeO(nq).

Proof. Analternatived isamongthefirsthalfofalternatives inanLP-treeLj ifandonlyiftherootissueofLj isassigned

toitspreferred value.Webuildatablewiththefollowing2q entries{11,01,. . . ,1q,0q}:foreveryLj weadd1 tothescore

of 1i (resp. 0i) if Xi is theroot issue of Lj and the preferred value is 1i (resp. 0i). When this is done, for each Xi, we

instantiate Xi to1i (resp.0i)ifthescoreof1i islargerthanthescoreof0i (resp. viceversa);ifthescoresareidentical,we

do notinstantiate Xi.Weendupwithapartial instantiation,whoseset ofmodels(satisfying valuations)isexactlytheset

ofco-winners. ✷

Example4.LetV betheprofileconsistingofthe4LP-treesofExample1:P isthemostimportantissuefortwovoters,both with preferred value p; M isthemostimportantissue for onevoter withpreferred valuem,and F isthe mostimportant issue foronevoterwithpreferredvalue f :therefore,the4-approvalwinner ismf p.IfweaddtothisprofileafifthLP-tree with mostimportant issue M andpreferred value m thenM willnot beinstantiated,and thetwo cowinners willbemf p

and m f p.

While 12-proportion-approvaltakesintoaccountonlythemostimportantissue ofeachvotertogetherwithhispreferred value, 14- and 34-proportionapproval takeintoaccount themost importanttwoissuesofeachvoter; 18-, 38-, 58- and 78 take intoaccount themostimportantthreeissuesofeachvoter,andsoon.

Wenowgiveapracticalwaytocompute the

α

-proportion-approval co-winners,usingatranslation oftheprobleminto aninstanceof generalizedmaxsat,withaone-to-onecorrespondencebetweenthesolutionsofthevotingproblemandthe solutionsofitstranslation(whichisusuallyreferredtoasa“model-preservingtranslation”).Usingareversetranslation,we willthenprovethat,forany

α

∈Rat(M)\ {12},

α

-proportion-approvalisinfact NP-hard.

Toillustrate theidea wefirst givetheconstructionfortwo special cases:

α

=1/4 and

α

=3/4.Considerfirst thecase

α

=1/4 and some LP-tree Lj whose top issue is Xi1 with preferred value xi1∈ {0i1,1i1}. Assume that the second most

important issue given xi1 is Xi2, with preferred value xi2. Then, the 142q best alternatives are those that have precisely values xi1 and xi2 for Xi1 and Xi2 respectively. Wecan encode these alternatives bya formulali1∧li2, whose modelsare

preciselythe 142q bestalternativesforL

j:ifxi1=1i1 thenweletli1=Xi1,andotherwiseli1= ¬Xi1;li2 isdefinedsimilarly.

Considernowthecase

α

=3/4 andsomeLP-treeLj whosetopissueis Xi1 withpreferredvalue xi1 and whosesecond

most important issue givenxi1 is Xi2, with preferred value xi2 in thiscase: the 342q best alternatives arethose for which

Xi1=xi1 and those for which Xi1=xi1 and Xi2=xi2. These alternatives can be encoded as the models of the formula li1∨li2,whereli1 andli2 arethesameasdefined aboveforthecase

α

= 14.

Moregenerally, wedefine, for LP-treeL over a set I of q issues, and afraction

α

∈Rat(M), aformula φ (L,

α

) which

characterizes thealternatives that areamong the

α

2q best alternatives ofL.We notethat, sinceqM,

α

2q isaninteger

number. Let Xi be the root issue, let x+i be the preferred value for Xi, and xi be its less preferred value; let li=Xi if

x+i =1i,andli= ¬Xi if x+i =0i,then:

φ (

L

,

α

) =

li

∧ φ(

L+

,

2

α

)

if 0

<

α

<

1

/

2 li if

α

=

1

/

2 li

∨ φ(

L−

,

2

α

1

)

if 1

/

2

<

α

<

1

where L+ (respectively L−) is the subtree of L corresponding to x+i (respectively xi ), with the local preference tables being simplifiedbyremovingallconditionalpreferenceswhere Xi=xi (respectively Xi=x+i ).

ThetranslationsforLPtreeL P1fortwovaluesof

α

areshownonFig.2.

Proposition4.Let

α

∈Rat(M).LetV beanLP-profileoverasetI ofqM issues.Let8V = {φ(Lj,

α

)| LjV}bethemultiset

offormulaeassociatedwith V (notethatthesameformulacanappearinseveralcopiesin8V).Thenthe

α

-proportion-approval

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M mm F ff P pp P pp F p:ff p:ff m m φ (L P1,38)=M∧φ(L P11,34) φ (L P11,34)=F∨φ(L P21,12) φ (L P21,12)=¬P φ (L P1,58)=M∨φ(L P31,14) φ (L P31,14)=P∧φ(L P41,12) φ (L P14,12)=F L P11 L P21 L P31 L P41 – P=p α=3/8:φ =M∧(F∨¬P)

worstacceptablealternative:m f p

α=5/8:φ =M∨(PF)

worstacceptablealternative:mp f

Fig. 2. LP tree L P1and its translations into formulas for 3-approval and 5-approval.

Proof. The resultfollows fromthefactthat analternatived isinthe

α

2q best alternativesofsome LP-treeL

j ifand only

if thecorrespondingvaluationsatisfiesφ (Lj,

α

). ✷

In orderto usestate-of-the-art maxsat solvers tocompute

α

-proportion-approval co-winners, wecan easily transform

8V intoasetofclausesofquadraticsize:givensome LP-treeLj,weintroduceanewpropositionalvariable Lj,anddefine

aset ofclauses4j inwhichevery clausehas theform C∨ ¬Lj,so thataninterpretationcansatisfy Lj andalltheclauses

in4j only ifitsatisfiesalltheconstraints encodedin4j.Thedefinitionof4j isrecursive,basedontheformulaφ (Lj,

α

),

oftheformli1⋄1(li2⋄2(. . . (liM−1⋄M−1liM). . .)):

• 4j isinitialized to{liM∨ ¬Lj};

• atthenextstage,if⋄M−1 isaconjunction,then4j becomes{liM−1∨ ¬Lj,liM∨ ¬Lj};if ⋄M−1 isadisjunction,then4j becomes{liM−1∨liM∨ ¬Lj};

• moregenerally,atthe (M−k)thstage, if ⋄ik isa conjunction,thenwe add lik∨ ¬Lj to 4j;otherwise, when ⋄ik isa disjunction,thenweaddlik asnewdisjuncttoeveryclausein4j.

Example 5.For L P1 and

α

=3/8, φ (L P1,3/8)=M∧ (F ∨ ¬P): the set of clauses is initialized to {¬P ∨ ¬L1}; it then becomes {F ∨ ¬P∨ ¬L1}, and finally {M∨ ¬L1,F∨ ¬P∨ ¬L1}. For

α

=5/8, φ (L P1,5/8)=M∨ (P ∧F) thus the set of

clausesisinitializedto {F∨ ¬L1},thenbecomes{P∨ ¬L1,F∨ ¬L1},and finally{M∨P∨ ¬L1,MF∨ ¬L1}. For an LP-profile V , we obtain a set of “hard” clauses 4(V)=S

LjV4j, and the partialweightedmaxsat instance

consistsinfindingvaluationsthat satisfyallclausesin4(V),andamaximumnumberofthe Lj’s.

With thistranslation, the numberofhard clauses foreach voteris thenumberof 1’sin

α

, and thesize ofthe largest clause becomes one plus the number of 0’s in

α

. It follows that the size of 4j for each voter is in O(M2), and the

translation, with n voters, is in O(M2n); in terms of the parameters used in general to describe maxsat instances, the number ofclauses isin O(nM), theirsize isbounded by M,and the numberofvariables isq.Giventheperformances of thebestcurrent maxsat solvers,thisapproachmaybeapplicablewhentherearehundredsofthousandsofvoters,forsome

M≤10.

Weconclude thissectionbyaproof that winner for

α

-proportionapproval is NP-hard,which means thatwe maynot

expectmuchbetterthan theabovetranslation. Infact,ourproofstemsfromthereversetranslation.Wewillshowthatthe model of aclass of logical formulas can be encoded by

α

-proportionapproval winners. Note that a formula φ (L,

α

) for

some LP-treeL andsome

α

alwayshastheform

li1

1

(

li2

2

(. . . (

liM−1

M−1liM

) . . .))

(2)

where thelij’sareliterals,eachover adifferentvariable, and whereeach⋄i isaconnective ∧or∨,that onlydepends on thebase 2representation of

α

:let

α

= (0.

α

1. . .

α

M)2,then⋄i is∧ if

α

i=0,and⋄i is∨ if

α

i=1.Forany formulaφ that

has theformasin(2),weconstructanLP-treeL(φ) asfollows:

• theimportancestructureisunconditional:[Xi1⊲Xi2⊲. . . ⊲XiM−1⊲XiM ⊲Others],where Xij istheissue correspond-ingtotheliterallij,andwhere‘Others’refers toallotherissues.

• thelocalpreferenceforissue Xij isunconditionalandis1ij≻0ij iflij=Xij,or0ij≻1ij iflij= ¬Xij.

Then, given a set of formulas 8having a common structure corresponding to some fraction

α

∈Rat(M),

α

6=1/2, we

can define theprofile V = {L(φ)| φ ∈ 8} of LP-treeswith unconditional importance and unconditional local preferences: the

α

-proportionapproval co-winnerscorrespond tothevaluationsthat satisfyamaximum numberofformulas of8.We

now provethat generalizedmaxsatremainsNP-completewhenrestrictedtothistypeofformulas.

Formally, given a set of propositional symbols P and afraction

α

= (0.

α

1. . .

α

M)2∈Rat(M), let F(P,

α

) bethe set of

formulas of the form asin (2). We define max(

α

)sat to be the problem that consists in finding a valuation satisfying a maximum numberof formulas ofagiven set 8⊆F(P,

α

),for agiven set of propositionalvariables P.In therestofthis

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Lemma5.Forany

α

∈Rat(M)\ {1/2}, max(

α

)satisNP-hard.

Proof. By induction. The base cases are

α

=3/4 and

α

=1/4. Consider first

α

=3/4= (0.11)2: F(P,34) is the set of

2-clauses (disjunctions oftwo literals).So max(34)sat is max2sat, known to be NP-complete [32]. Consider next thecase

α

=1/4= (0.01)2: F(P,14) isthesetof2-cubes (conjunctions oftwoliterals).Let 8⊆F(P,14):then8′= {¬φ | φ ∈ 8}is equivalentto aset of2-clauses,andsatisfyingasmanyformulas of8aspossibleamountstosatisfyingasfewformulas of 8′ aspossible.So max(14)satisequivalentto min2sat,knowntobe NP-completetoo[33].

Let

α

= (0.

α

1. . .

α

M)2= 12

α

1+ 14

α

2+ . . . + 21M

α

M. Suppose first that

α

1=0. We give a polynomial reduction from max(2

α

)sat to max(

α

)sat. Note that 2

α

= (

α

1.

α

2. . .

α

M)2= (0.

α

2. . .

α

M)2 has a base-2 representation of length strictly smaller thanthat of

α

. Let8⊆F(P,2

α

),letP′= P ∪ {X0},where X0 isanewvariable(not belongingtoP),and define

8′= {X0∧ φ | φ ∈ 8}:then8′⊆F(P′,

α

).Now, satisfyingasmany formulas of8aspossibleisequivalenttosatisfyingas

many formulasof8′ aspossible,sinceonlypositiveoccurrencesof X0 appearin8′.

Finally, suppose that

α

1=1: we will prove that max(2

α

−1)sat can bereduced to max(

α

)sat. Note that in thiscase

2

α

−1= (0.

α

2. . .

α

M)2 has a base-2 representation of length strictly smaller than that of

α

. Let 8⊆F(P,2

α

−1), let P′= P ∪ {X0},where X0 isanewvariable–notinP,anddefine8′= {X0∨ φ,¬X0∨ φ | φ ∈ 8}⊆F(P′,

α

):8′isobtained from 8 by two copies of each φ ∈8, and appending at the beginning a disjunction, one with X0 and one with ¬X0. Suppose that m isavaluation over P thatsatisfies

µ

formulas of8:letm′ beanextension ofm withsomevalue for X0, then m′ satisfies |8| +

µ

formulas of 8′. Conversely, if a valuation m′ over P′ satisfies λ formulas of 8′, then λ≥ |8|

becausem′ mustsatisfyoneof X0 and¬X0, andtherestrictionofm′ toP satisfiesλ− |8|formulas of8.So satisfyingas

many formulasof8aspossibleisequivalenttosatisfyingasmanyformulasof8′ aspossible.

Asimpleinductiononthelengthofthebase-2representation of

α

thenprovestheresult:thebasecasescorrespondto alength2(

α

=1/4 or3/4). ✷

Theorem6.Foranyfraction

α

∈Rat(M),

α

6=1/2, winnerfor

α

-proportion-approvalisNP-complete,evenifalltreeshave uncon-ditionalimportanceandunconditionallocalpreferences.

Proof. We have seen that

α

-approval winner is equivalent to max(

α

)sat, and we have just proved that the latter is

NP-hard.ThefactthattheseproblemsareinNPstemsfromthefactthat generalizedmaxsatisin NP. ✷

5. Borda

Inthissection,weprovideatranslationofwinnerdeterminationfortheBordaruleintoa weightedminsatproblem.We also show that computing the Borda winner given anLP-profile is hard, exceptin the simple casewhere the preferences and theimportancerelationsareunconditional.

Recall that the Borda score of an alternative d w.r.t. LP-tree L is SBorda(L,d) =m−rank(L,d), with m=2q and rank(L,d)=1+Pq

i=12q−level(L,d,Xi)1(L,d,Xi).Hence,theBordascoreofd forprofile V = (L1,. . . ,Ln)is: SBorda

(

V

,

d

) =

n

X

j=1

"

2q

1

q

X

i=1 2q−level(Lj,d,Xi)

1(

L j

,

d

,

Xi

)

#

=

n

X

j=1 q

X

i=1 2q−level(Lj,d,Xi)

(

1

− 1(

L j

,

d

,

Xi

)).

The Borda winner isthe alternatived that maximizes thisscore.As weshall see shortly,thisoptimization problem is hardinthegeneral case.Letusstartwithacaseinwhichthewinnercanbecomputedinpolynomialtime.

If the importance structure is unconditional, then level(Lj,d,Xi) does not depend on d; let us write it more simply

level(Lj,Xi).ItcanbecomputedinpolynomialtimebyasimpleexplorationofthetreeLj.Similarly,ifthelocalpreferences

areunconditional,then1(Lj,d,Xi)doesnotdependonthewholevectord eitherbutonlyondi,thuswewrite1(Lj,di)=

1(Lj,d,Xi).When,foreveryvoter,theimportancestructureandthelocalpreferencesareunconditional,wecanthuswrite:

SBorda

(

V

,

d

) =

n

X

j=1 q

X

i=1 2q−level(Lj,Xi)

(

1

− 1(

L j

,

di

))

This means thatwe canchooseinpolynomialtimethewinning valuefor eachissueindependently:itis thevaluedi that

maximizes Pn

j=12q−level(Xi,Lj)(1− 1(Lj,di)).

Theorem7.If,foreveryvoter, theimportancestructureandthelocalpreferencesareunconditional,then winnerforBordacan becomputedinpolynomialtime.

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M m≻ ¯m 4:M F 2:MF ff P 1:M∧ ¬P pp P pp 2: ¬MP F p:ff ¯ p: ¯ff 1: ¬MPF 1: ¬M∧ ¬P∧ ¬F ¯ m m

Fig. 3. LP tree L P1and its translation into weighted formulas.

As we will seeshortly, if welift either one of theunconditionality conditions, computing theBorda winners becomes NP-hard.However,wefirstdescribehowthisproblemcanbeconvertedtoa weightedminsatproblem.

Weconsider one propositional variable Xi for eachissue Xi∈ I.Let L bean LP-treeand t anode ofL.Let Xi bethe

issue associated witht, andl be the levelof t inL (wherelevel 1correspondsto theroot and levelq to theleaves). Let

v be theinstantiation of Inst(t) in that branch, wecan associate withit a conjunction V of literals as follows: for every variable Xi in Inst(t), V contains either Xi if v has value 1i for Xi, or ¬Xi if v has value 0i for Xi. Then for each rule

u:xx′ inthetableassociated with Xi att,letU bethesimilarlogicaltranslationofu;wedefinetheformulaψt,u,with

weight2ql for minsat,asfollows:

• 2ql:UV X

i ifx=1i;and

• 2ql:UV ∧ ¬X

i if x=0i,

As an example, LP tree L P1 is recalledon Fig. 3 with thecorresponding weighted formulas. Alternativem¯p¯¯f satisfies the formula ¬M∧ ¬P∧ ¬F , with weight1: this alternativewill beranked 8−1=7th; alternative m¯fp satisfies¯ M and M∧ ¬P ,withweight4+1=5,soitisranked8−5=3rd.

Proposition8.LetLbeanLPtree,let9(L)= {ψt|t node ofL},andletd beanalternative.Thesumoftheweightsoftheformulas

of9(L)thataresatisfiedbyd istheBordascoreofd w.r.t.L.

Proof. The formulas that are satisfied byd are all on thebranch correspondingto d in L. Considering anode ti on this

branch labelledwithvariable Xi,thelevel ofti ispreciselylevel(L,d,Xi),so theweightis2p−level(L,d,Xi), andtheformula

is satisfiedif and only if thepreferred value for Xi –given thevalues of d fortheissues above ti in thisbranch –isthe

value givenbyd, sotheoverallweightis q

X

i=1

2p−level(L,d,Xi)

(

1

− 1(

L

,

d

,

X

i

))

whichisexactlytheBordascoreofd w.r.t.L. ✷

GivenanLP-profile V= (L1,. . . ,Ln), weconsider themultisetofweighted formulas 9(V)= ∪j9(Lj)(notethat there

maybeseveralcopiesofthesameweightedformula)andwelookforavaluationmaximizingthesumoftheweightsofthe formulasin9(V)thatitsatisfies.Sincetheformulasin9(V)areconjunctionsofliterals,8(V)= {(w: ¬φ)| (w: φ)∈ 9(V)}

is a set of weighted clauses, and a valuation minimizes the weights of the clauses satisfied in 8(V) if and only if it

maximizes thesumoftheweightsofthecubessatisfiedin9(V),and thisisaninstanceof weightedminsat.

Onourexampleweget

8(

L P1

) =

½

4

: ¬

M

,

2

: ¬

M

∨ ¬

F

,

1

: ¬

M

P

,

2

:

M

∨ ¬

P

,

1

:

M

∨ ¬

P

∨ ¬

F

,

1

:

M

P

F

¾

Theorem9.ForanyprofileV ofLP-trees,thereisasetofweightedclausesCofsizepolynomialinthesizeofV ,suchthatthesetof Bordaco-winnersforV isexactlythesetofthevaluationsofCwiththelowestweight.

Some minsat solvers use a translation of minsat instances into maxsat ones. Using the same approach, we can also directly translate the winner problemfor Borda intoa weightedmaxsatinstance, but with moreclauses: each weighted conjunction of literals φ = (w:l1∧ . . . ∧lk) that appears in 9(V) is translated into a set of k weighted clauses 2(φ)=

{(w:l1),(w: ¬l1∨l2),. . . ,(w: ¬l1∨ ¬l2∨ . . . ∨lk)}.Let2(L)denotetheunionofthe2(φ)’sthatwegetforatreeL.For

L P 1 weget:

2(

L P1

) =

4

:

M

,

2

:

M

,

2

: ¬

M

F

,

1

:

M

,

1

: ¬

M

∨ ¬

P 2

: ¬

M

,

2

:

M

P

,

1

: ¬

M

,

1

:

M

P

,

1

:

M

∨ ¬

P

F 1

: ¬

M

,

1

:

M

∨ ¬

P

,

1

:

M

P

∨ ¬

F

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Lj L′j L L′ Xi1 Xi2 di1 Xi3 A B Xi2 ¯ di1 Xi3 di2 A B Xi3 ¯ di2 A di3 B B ¯ di3 A Xi1 Xi2 Xi3 A B A B A B

A:1≻0; all other issues: 0≻1 1≻0 1≻0 A,B:1≻0 for all issues for all issues Other: 0≻1 Fig. 4. Reduction of 3sat to evaluation with Borda score and unconditional preferences.

For a profile V , 2(V) then denotes the union of thesets of weighted clauses obtained for eachvoter in V . To see why this maxsat formulationisequivalenttoourBorda scoremaximizationproblem,consider analternatived, and aweighted conjunctionφ = (w:l1∧ . . .lk)∈ 9(V):

• ifd|=l1∧ . . .lk,thend satisfiesallclausesin2(φ);and

• ifd6|=l1∧ . . .lk,thenthereisexactlyoneclausein 2(φ)thatisnotsatisfiedbyd (if j isthesmallestindex suchthat

d6|=lj,thend satisfiesallclausesin2(φ)except¬l1∨ . . . ∨ ¬lj−1∨lj).

Therefore,the“score”ofd for 2(V)is

X

(w:C)∈9(V) d|=C w

|

C

| +

X

(w:C)∈9(V) d6|=C w

(|

C

| −

1

) =

X

(w:C)∈9(V) w

(|

C

| −

1

) +

X

(w:C)∈9(V) d|=C w

.

Itisthereforeaconstantindependent of d,plusthe“score”ofd for9(V);thus maximizingthesumoftheweightsofthe

satisfied cubes in 9(V) is equivalentto maximizingthe sumof theweights of thesatisfied clauses in 2(V). Since2(V)

contains atmost q clauses of size at most q for each node in an LP-tree in V , its size is in O(q2|V|). Whether using a maxsatsolveronthistranslationwouldbemoreefficientthanusingarecent minsat solveron8(V)remainstobetested.

Weshow nextthatwe cannotexpectamuchbetter wayofcomputingtheBordawinner inthegeneral case,sincethe problemisNP-complete.

Theorem10.evaluationisNP-completeforBorda,evenifasingleoneofthefollowingrestrictionsholds:

1. thelocalpreferencesofallvotersareunconditional;or

2. theimportancestructureisthesameforallvotersandisunconditional.

Proof. evaluation is in NP, because the Borda scoreof each alternativecan be computedin polynomial time. We prove NP-hardnessinthetwosubcases1and 2bytwodifferentreductionsfrom3sat:ina3sat instance,wearegivenaformula

F overbinary variables X1,. . . ,Xq, which is a conjunctionof t 3-clauses, F =C1∧ . . . ∧Ct. We are asked whether there

exists avaluationofthevariables X1,. . . ,Xq underwhich F istrue.

Wewillnowdescribehowwecanreducea3SAT instance totwo evaluation instances,onewithaprofilewith uncon-ditionalpreferences,theotherwithaprofilewithcommon,unconditionalimportance.

Reductionwithunconditionallocalpreferences. Thereareq+2 issues:I= {A,B}∪ {X1,. . . ,Xq}. Wefirstdefinethefollowing

2t+2 LP-treesfromwhichtheprofileisconstructed.

• For each jt, we define two LP-trees Lj and L′j with the following structures. Suppose Cj contains variables

Xi1,Xi2,Xi3 (i1<i2<i3), and di1,di2,di3 are the valuations of Xi1,Xi2,Xi3 that correspond to literals in Cj: for

in-stance, if Cj=X1∨ ¬X3∨X5 then di1=1,di2=0,di3=1. In the importance order of Lj, the first three issues are

Xi1,Xi2,Xi3. The fourth issue is A and the fifth issue is B if and only if Xi1=di1, Xi2=di2, or Xi3=di3; otherwise

thefourth issue is B and the fifth issue is A. The other issues are ranked belowin a fixed order, independent of j.

The local preferences in Lj are 0≻1 for all issues except A, for which it is 1≻0. In L′

j the importance order is

[Xi1⊲Xi2⊲Xi3⊲AB⊲Others],andthelocalpreferencesare1≻0 forallissues.Thetreesaredepictedontheleft

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X1 Xq A Lj 0≻1 di1di2di3: ½ 1≻0 if di1di2di3|=Cj 0≻1 otherwise L′ j 1≻0 L 0≻1 1≻0 L′ 1≻0 1≻0

Fig. 5. Reduction of 3sat to evaluation with Borda score and common, unconditional importance.

• TherearetwoadditionalLP-treesLandL′withthesameimportanceorder[A⊲B⊲Others];inLthelocalpreferences

are1≻0 forallissues;inL′ thelocal preferencesare1≻0 for A and B,and0≻1 forallotherissues.

Letd=dab beanalternative,wherea∈ {0,1}isitsvalueforissue A,b∈ {0,1}itsvalueforissue B,andd′itsprojection onto {X1,. . . ,Xq}.Because thepreferencesconcerningeachofthe Xi’sareoppositeinLj andL′j,theBordascoreofd for

theprofile{Lj,L′j} isaconstant K1independentof j andd,plusthescoreforissues A and B,rankedfourthandfifth out ofq+2 issues.Ifwelet1j=1 ifd′|=Cj,and1j=0 otherwisewehave:

SBorda

({

Lj

,

L′j

},

d

)

=

K1

+ 1

j

(

a2q−2

+ (

1

b

)

2q−3

) + (

1

− 1

j

)((

1

b

)

2q−2

+

a2q−3

) +

a2q−2

+

b2q−3

=

K1

+

2q−3

[1

j

(

2a

+

1

b

) + (

1

− 1

j

)(

2

(

1

b

) +

a

) +

2a

+

b

]

TheBordascoreofd forprofile {L,L′}isaconstant K2 plusthescoreforissues A and B,rankedfirstandsecond:

SBorda

({

L

,

L′

},

d

) =

K2

+

2

× (

a2q+1

+

b2q

) =

K2

+

2q+1

(

2a

+

b

).

We now describe thepreference profile. It contains 4t LP-trees: V = {Lj,L′j:1≤ jt}V′, where V′ is composedoft

copiesof{L,L′}.Let K=K1+K2,then:

SBorda

(

V

,

d

) =

t K

+

2q−3

X

j

[1

j

(

2a

+

1

b

) + (

1

− 1

j

)(

a

+

2

2b

)) +

2a

+

b

] +

t2q+1

(

2a

+

b

).

In particular, if a=0 or b=0, then 2a+1−b+2a+b≤5 and a+2−2b+2a+b≤5, thus,independently from the value of1j,theterminthesumisbounded by5forevery j;thus SBorda(V,d)≤t K +5t2q−3+2t2q+1.Or,equivalently:if

SBorda(V,d)>t K+5t2q−3+2t2q+1,thenitmustbethecasethata=b=1.

Now,consider athresholdT=t K+5t2q−3+3t2q+1.Suppose firstthat d issuchthat S

Borda(V,d)≥T>t K +5t2q−3+ 2t2q+1. Then itsvalues for A and B must bothbe 1, and S

Borda(V,d)=t K+2q−3P

j[2× 1j+ (1− 1j)+3]+3t2q+1=

t K+2q−3P

j[4+ 1j]+3t2q+1. Now,since SBorda(V,d)≥T , itmustbethecasethat 1j=1 forevery clause Cj, thusthe

projectionofd ontoD satisfies F ;thereforeF issatisfiable.

Fortheconverse,suppose that F issatisfiable:letd′ beavaluationthatsatisfies F ,and considerthealternatived′1A1B:

SBorda(V,d′1A1B)=T ,where T=t K+5t2q−3+3t2q+1.

Reductionwithacommon,unconditionalimportancestructure. There areq+1 issues: I= {X1,. . . ,Xq}∪ {A}. Let O= [X1⊲ X2⊲· · · ⊲XqA]denote thefixedimportancerelationfortheLP-trees.Weconstructaprofile V consistingof4t LP-trees

withconditionalimportancerelationO,definedasfollows:

• For each jt, V contains two LP-trees Lj and L′j with unconditional local preferences for every issue except A:

supposeCj containsvariables Xi1,Xi2,Xi3 (i1<i2<i3),inbothLP-trees,Par(A)= {Xi1,Xi2,Xi3}. TheCPTsaredefined

asdepictedonFig.5:

– inLj, forevery assignment(di1,di2,di3) of{Xi1,Xi2,Xi3}, theCPTentryfor A isdi1di2di3:1≻0 ifand only ifCj is

satisfiedby(di1,di2,di3);thelocalpreference is0≻1 forevery otherissue.

– inL′j,theCPTfor A isthesameasinLj,andthelocalpreferenceis1≻0 foreveryotherissue.

V contains2t additionalLP-trees:t copiesofL,wherethelocalpreferenceis1≻0 forissue A,and 0≻1 fortheother

q issues;andt copiesofL′,wherethelocalpreferencesare1≻0 forallissues.

Theadditional2t LP-treesareusedtomakesurethatweonlyneedtofocusonalternativeswhose A-componentis1. Letd′= (d1,. . . ,dq)beavaluationof X1,. . . ,Xq.Then:

(14)

SBorda({L,L′},d′0A)=Pqi=12q+1−i(1+0)=2(2q−1):there areq+1 issues, issues X1,. . .Xq have ranks 1toq,and

whateverthevalueofdi,itcountsfor1 inone ofthetwotrees,0 intheother;0A isnotthepreferredvalueinL and

L′,soitdoesnotadd anythingtotheBordascoreofd′0A.

SBorda({L,L′},d′1A)=2(2q−1)+2,because1A isthepreferredvaluefor A inLand L′.

SBorda({Lj,L′j},d′0A)=2(2q−1)+2(1− 1j)(whereagain 1j=1 ifd′|=Cj,and1j=0 otherwise).

SBorda({Lj,L′j},d′1A)=2(2q−1)+2× 1j.

Thus SBorda(V,d′0A)=4t(2q−1)+2t2K(F,d′),where K(F,d′) isthe numberof clauses in F thatare satisfied by d′;

whereas SBorda(V,d′1A)=4t(2q−1)+2t+2K(F,d′). Hencethereexists analternativewhose Bordascoreisatleast T =

4t(2q−1)+4t ifand onlyifthe3sat instanceisayes instance.Thiscompletestheproof. ✷

BeyondBordaandk-approval. We endthissection bydiscussing thegeneralizationof theconstructionsof equivalent satis-fiability problemsfor k-approvaland Bordato other scoringrules.The commonpoint ofthetwo constructionsisthat the set ofweights has the niceproperty that only asmall number offormulas was needed toencode thescores obtainedby the alternatives:moreprecisely, thenumberofformulas generated forasingleLP-tree isatmostthenumber ofnodesof the LP-trees,and each formulahas a size in O(q), thus the total size oftheconstruction isin O(q.Pn

j=1|Lj|), whilethe

numberofalternatives(2q)can,ingeneral,beexponentiallylarger.

• Inthecaseofk-approval,thisisdue tothefactthatthenumberofdifferentweightsinthescoringvectorisverysmall (onlytwo).

• In the case of Borda, this is due moregenerally to the fact that the weights are regular enough so that they can be generatedbyasuccinctbasis.

The construction can thus begeneralized to scoring vectors that usea small number of weights, or more generally a “smallbasis”.Wegiveanexample:s= (s1,. . . ,s2q)where

sj

=

2

(

2q−2

j

+

1

)

if j

2q−2 1 if 2q−2

+

1

j

2q−1 0 if j

>

2q−1

Thecorrespondingsetofweightedformulas–assumingforsimplicitythat L P hasanunconditionalstructure X1⊲X2X3

and localpreferencesxixi foreveryi –is

{

2q−2

:

X

1

X2

X3

;

2q−3

:

X1

X2

X4

; . . . ,

2

:

X1

X2

Xq

;

1

:

X1

X2

;

1

:

X1

}

For instance,forq=4 we gets= (8,6,4,2,1,1,1,1,0,0,0,0,0,0,0,0) andtheset ofweighted formulascorrespondingto L P is

{

4

:

X1

X2

X3

;

2

:

X1

X2

X4

; . . . ,

2

:

X1

X2

Xq

;

1

:

X1

X2

;

1

:

X1

}

6. Condorcet

WefirstobservethattheexistenceofaCondorcetwinnerisnotguaranteedforLPprofiles: Example6.Considerthefollowingprofilewithtwoissues X andY ,andthreevoters:

1: XY,x≻ ¯x,y≻ ¯y: herpreferencerelationisxyxy¯≻ ¯x y≻ ¯x¯y.

2: YX,¯xx,y≻ ¯y: herpreferencerelationisx y¯ ≻xy≻ ¯xy¯≻x¯y.

3: YX,¯xx,¯yy: herpreferencerelationisx¯y¯≻xy¯≻ ¯x yxy.

Itcanbecheckedeasilythatthisprofilehasno Condorcetwinner.

Forany typeofLPprofiles,decidingwhetheragivenalternativec isaCondorcet winnerisin coNP:acertificate thatc isnotaCondorcet winnerisanalternatived thatbeatsc,whichcanbecheckedbycomparingd andc foreveryLP-treein theprofile.

ThefollowingresultshowsthatcheckingifagivenalternativeisaCondorcetwinner ishard.

Theorem11.ForLPprofiles,decidingwhetheragivenalternativeisaCondorcetwinneriscoNP-hard,evenif,foreveryvoter,thelocal preferencesareunconditionalandtheimportancetreeisunconditional.

Figure

Fig. 1. Four LP-trees.
Fig. 2. LP tree L P 1 and its translations into formulas for 3-approval and 5-approval.
Fig. 3. LP tree L P 1 and its translation into weighted formulas.
Fig. 5. Reduction of 3sat to evaluation with Borda score and common, unconditional importance.
+3

Références

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