Truthful Facility Location with Additive Errors

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by  Iddan Golomb

Truthful Facility Location with Additive Errors
Iddan Golomb∗ ChristosTzamos †
January2, 2017
Abstract
Weaddresstheproblemoflocatingfacilitiesonthe[0,1]intervalbasedonreportsfromstrategic
agents. The cost of each agent is her distance to the closest facility, and the global objective is to
minimizeeitherthemaximumcostofanagentorthesocialcost.
7
1 Asopposedtotheextensiveliteratureonfacilitylocationwhichconsidersthemultiplicativeerror,
0 we focus on minimizing the worst-case additive error. Minimizing the additive error incentivizes
2 mechanismstoadapttothesizeoftheinstance.I.e.,mechanismscansacrificelittleefficiencyinsmall
instances (location profiles in which all agents are relatively close to one another), in order to gain
n
more [absolute] efficiency in large instances. We argue that this measure is better suited for many
a
J manifestationsofthefacilitylocationprobleminvariousdomains.
Wepresent tight bounds formechanismslocatingasinglefacilityinbothdeterministicandran-
2
domizedcases.Wefurtherprovideseveralextensionsforlocatingmultiplefacilities.

]
T
G 1 Introduction
.
s
We consider a setting in which several agents are located on a line, and a central planner intends to
c
[ place a facility at some pointuponthatline. Each ofthe agentsseeks to be as close as possible to the

facility–thecostofanagentisherdistancetothefacility. Theplannerwishestominimizesomeglobal
1
objective–eithertheaverageorthemaximumcostofanagent. Insomecases, agentsmightmisreport
v
9 their preference in order to obtain a better outcome from their perspective. We aim to devise truthful
2 mechanisms, in which no agent can benefit from misreporting, regardless of the reports of the other
5 agents.
0
Intheliterature,themostcommonmeasureforassessingamechanism’sefficiencyistheapproxima-
0
tionratio–theworst-caseratio(overanylocationprofile)betweentheglobalcostofthemechanismand
.
1 thatof theoptimalmechanism. Inthispaper,westriveto minimizetheadditiveerror–theworst-case
0
differencebetweenthesametwovalues.
7
Toillustratethedifferencebetweenthesetwoerrorfunctions,considertheclassicalfacilitylocation
1
: probleminwhichamunicipalitywantstobuildalibraryononeofitsstreetsinordertoservetheresidents
v
ofthestreet. Optimizingovertheapproximationratiomightguarantee,forinstance,thatnoresidentwill
i
X needto walk to the librarymorethan2 times the maximalwalkingdistanceof an agentin the optimal
r assignment. In contrast, optimizing over the additive error might guarantee that no agent will need to
a
walktothelibraryformorethan3kilometersovertheoptimalassignment. Thisexampledemonstrates
thatoptimizingovertheadditiveerrorguaranteesefficiencywithrespecttoatangibleandnaturalmeasure
ofabsolutedistance,andfurtherprovidesatargetfunctionwhichiseasilyunderstood.
Asopposedtothemultiplicativeerrorwhichisscale-invariant,byfocusingontheadditiveerrorwe
sacrificesomeefficiencyinsmallinstances(e.g.,casesinwhichallresidentsareclosetothelibrary),in
whichthesolutionsarealreadyquitegood,soastoachieveanearlyoptimalsolutioninlargerinstances
(wheretheerrorissignificantlyhigher).Indeed,inmanycasessome(large)instancesaremoreimportant
tothedesignerthanother(smaller)instances,andtheadditiveerroraccommodatesforthisneed.
Clearly, when the size of the instance doubles, the additive error also doubles. However, in many
differentmanifestationsofthefacilitylocationproblem,thedomainsarebounded.Werestrictourselves
∗TelAvivUniversity. Email: [email protected]. TheworkofIddanGolombwaspartiallysupportedbytheEuropean
ResearchCouncilundertheEuropeanUnion’sSeventhFrameworkProgramme(FP7/2007-2013)/ERCgrantagreementnumber
337122.ThisworkwasdoneinpartwhileI.GolombwasvisitingtheSimonsInstitutefortheTheoryofComputing.
†MassachusettsInstituteofTechnology.Email:[email protected]
theSimonsInstitutefortheTheoryofComputing.
1
to such cases, i.e., in our setting the network is not R, but rather the [0,1] segment of the line. Some
examplesofsuchsettingsinclude:
– Intheaforementionedexampleoflocatingapubliclibraryonastreet,thereisclearlyarealisticbound
onthelengthofastreet,e.g.,thelengthofthelongeststreetinthecity.
– A group of public officials needs to vote over the amount of resources to invest in a given policy
(political,social,etc.). E.g.,theMinistryofEducationisfacedwiththedecisionofhowmuchmoney
the governmentshould allocate to a reform in the school system. Note that in this case scaling can
bedonewithrespecttothesizeofthetotaleducationbudget,sincedecisionswhichaffectmillionsof
dollarstendtobemoreimportantthanthosewhichaffectthousandsofdollars.
– Agroupofpeoplesittinginaroomwanttocollectivelydecideonthetemperatureoftheairconditioner.
Eachpersonhastheirownidealtemperature,andsomereasonableassumptionscanbeusedtobound
the minimal and maximal temperature values (e.g., it is probably safe to assume that nobody’s true
preferenceis100degreesCelsius).
– A groupof colleagueswishes to set up a meeting in the upcomingweek, and each one has an ideal
time slotforthe meeting. In somecases itis assumedthatthe utility functionis single-peaked(see,
e.g.,[25]).Inthisexample,timeprovidesanaturalscalefortheproblemanditseemsreasonabletoget
solutionswhereeverybodywaitsatmost5minutesmorethantheoptimal,eveniftheoptimalsolution
is0. Incontrast,iftheoptimalsolutionis3hours,a2approximationcanbeverybad.
Assuming quasi-linear utilities and allowing payments, then the well known Vickrey-Clarke-Groves
(VCG) mechanism is truthful and can achieve the optimal social cost (e.g., [17]). However, in many
real-lifesituations(suchas in the examplespreviouslymentioned)we restrictthe use ofmoneydueto
ethical,legalorotherconsiderations.
Givenasetofvotes,itispolytimetofindtheoptimaloutcome. However,whenrestrictedtotruthful
mechanisms, we show that the optimal result cannot be selected in the general case. In other words,
approximationisusedtocircumventtruthfulnessandnotcomputationalhardness.
1.1 OurContributions
We analyzethe worst-case additiveerrorbetweentruthfulmechanismsandthe optimalmechanismfor
twoobjectives-themaximumcostandtheaveragecostofanagent. Theaveragecostisthesameasthe
morecommonlyusedobjectiveofthesocialcostuptoamultiplicativefactorofn(theamountofagents),
andwechosetofocusontheaveragecostpurelyduetoconvenienceastheresultsareeasiertointerpret
withouttheparametern.
ItInObservation2,weshowthattheerrorfortheaveragecostgrowsatleastlinearlywithn. Using
this normalization, it is clear that since the network is the [0,1] line, the additive error for both the
average cost and the maximum cost of any mechanism is always between 0 and 1. For example, the
trivialmechanismwhichlocatesthefacilityat1/2regardlessoftheagents’reportshasanadditiveerror
of 1/2 for both the social and maximum cost. Our choice of an interval length of 1 is only done for
thesakeonconvenience,andourresultsholdwithoutlossofgeneralityforanyinterval[0,M]withan
obviousmultiplicativeadjustmentoftheratios(e.g.,M/6insteadof1/6).
Inthepaper,weshowthefollowingresults:
– Locatingasinglefacility(Figure1)-We providetightboundsforallvariationsofthisproblem. For
the averagecost, it is well knownthatlocating the facility on the medianreportis both truthfuland
optimal. For the maximumcost, we start bypresentinga randomizedlower boundof 1/6. In order
to do so, we characterize the structure of the optimal mechanism, and show that it must locate the
facilityatoneoffivepossiblepoints(theleftmostandrightmostreports,the centerbetweenthem,0
and1).Wefurthershowthatitmustbesymmetric(anotionweformallydefineinthepaper).Wethen
presenttwolocationprofiles,andutilizethischaracterizationtoexhibitthelowerbound.Wemoveon
todeviseamechanismcalledBLRC,whichisaprobabilitydistributionovertwoknownmechanisms,
thatmatchesthelowerboundof1/6.Incontrast,thebestupperboundofcurrentlyknownmechanisms
was1/4(LRC:left-right-centerby[23]).
Fordeterministicmechanisms,weshowalowerboundof1/4.Wethenpresentasimpledeterministic
mechanism called phantom-half that matches this bound. In contrast, the best upper bound of any
dictatorshipis1/2.
2
Figure1: Ourresultsforsinglefacilityproblemwiththe maximumcostobjective. We showtight
boundsof1/6forrandomizedmechanismsand1/4fordeterministicmechanisms. Thebestresults
forcurrentlyknownmechanismswere1/4and1/2forrandomizedanddeterministicmechanisms
respectively.
Randomized Deterministic
Lower 1/6 1/4
Bound Thm. 5 Thm. 15
Upper 1/6 1/4
Bound BLRC,Thm.12 Phantom-half,Thm.17
Known 1/4 1/2
Mechanisms LRC(from[23]) Anydictatorship
– Extensionsto multiple facilities - We first show a trivialmechanism, EqualSpread, which locates k
facilitieswithequaldistancesfromoneanother(withoutusingtheinstancedata)andhasamaximum
costof 1 . Wethenextendtheconstructionofthedeterministiclowerboundforonefacilitytothe
2k−1
multiple-facilitysetting,andreachalowerboundof 1 ,whichshowsthattheresultforthisproblem
6k
isΘ(1/k).
For the average cost, we present two extensionsto Equal Spread which slightly improve the error -
therandomizedpaired-equal-cost(PEC)whichhasanerrorof 1 (Theorem22)andthedetermin-
4k−2
isticelection-parity-equal-cost(EPEC)mechanismwhichreachesanerrorof 3 (Theorem23). In
8k−4
addition,forthespecialcaseof2facilities,weshowadeterministicmechanismbasedonpercentiles
whichachievesanupperboundof1/5(comparedto1/4byEPEC).
1.2 Related Work
Thetruthfulfacilitylocationproblemhasrichroots. Theinitialfocuswasmainlyoncharacterizingthe
class of truthfulmechanisms - in 1980 Moulin characterizedall deterministic truthfulmechanisms for
locatingafacilityontheline,whenthepreferencesaresingle-peaked[20],andthischaracterizationwas
laterextendedtogeneralgraphsbySchummerandVohra[25].
Procaccia and Tennenholtzwere the first to proveboundson the approximationratio of the game-
theoreticfacilitylocationproblem[23]. Theirinitialmodelhasbeenextendedbythese authorsandby
others in many ways, and most of these extensions leads to additional bounds on the approximation
ratio—Thenetworkwasextendedfromthelinetocycles([1],[2]),trees([1],[11])andgeneralgraphs
[1]; Some papers consider building several facilities, where the cost of an agent is her distance to the
closest facility(e.g. [19], [18], [14], [15]); Other paperslookat heterogeneousfacilities, i.e., facilities
servingdifferentpurposes[26];Severalpapersconsiderasettinginwhicheveryagentpossessesmultiple
locationsandpaysthesumofthedistancestoherlocations([23], [19]);Additionalobjectivefunctions
were considered besides the maximum cost and the social cost, for instance the L norm [11] or the
2
minimaxenvy[6](in thelatter, the approximationwas donewith anadditiveerror); Additionalpapers
consider different preferences of the agents, for instance doubly-peaked preferences [12], “obnoxious
facilitylocation”inwhichagentswanttobeasfarawayaspossiblefromthefacility[7],settingswhich
combine agents with ordinary preferences and agents who wish to be far from the facility ([9], [28]);
Anotherdirectionthatwas researchedis thetradeoffbetweenthe approximationratio andthe variance
([24]); Some papers consider differentmethods of voting, for instance by restricting the outcome to a
discretesetofcandidates([8],[27],[10])orbyusingmediators[5]. Whenrestrictingthelocationofthe
facilitytogivencandidates,minimizingthesocialcostoffacilitylocationproblemshasbeenassociated
tothenotionof“distortion”([22],[3],[10],[4],[16]).
Nissimet. al. alsouseadditiveerrorsforfacilitylocation,asanexampleoftheirframeworkwhich
maintainsdifferentialprivacy[21]. However,theirmodeldiffersfromoursettingastheygrantthemech-
anismtheadditionalpowertoimposeagentstoconnecttoaspecificfacility.Impositionallowsforbetter
approximationratios,asseenforinstancein[13].
3
2 Model
Let N = {1,...,n} be a set of agents, where each agenti ∈ N is located at some pointx ∈ [0,1].
i
Thevectorx = (x ,...,x )isknownasa locationprofile. Adeterministicmechanismforlocatingk
1 n
facilitiesis a mappingfromsomelocationprofile(thereportsof theagents) toa setofk pointsonthe
interval(thechosenlocationsofthefacilities),thatis: M :[0,1]n →[0,1]k.
Assuming the facility locations are M(x) = {l ,...,l }, the cost of point p is its distance to the
1 k
closestfacility:cost (M,x)=min |l −p|. Thecostofagentilocatedatx isdefinedasthecost
p 1≤j≤k j i
ofherlocation,cost (M,x). Eachagentaimstominimizehercost.
xi
Arandomizedmechanismisamappingfromalocationprofiletosomedistributionoverk-tuplesof
locations:M :[0,1]n →∆([0,1]k). Thecostofagentiistheexpectedcostofthisagentaccordingtothe
probabilitydistributionreturnedbythemechanism,thatis: costxi(M,x) = E(l1,...,lk)∼M(x)minj|xi−
l |.
j
A truthful mechanism M (also known as a strategyproof mechanism) is one in which an arbitrary
agentcannotsufferfromreportingherreallocation,regardlessofthereportsoftheotheragents: Forall
i∈N,allx ,x′ ∈[0,1]andx ∈[0,1]n−1:
i i −i
cost (M,(x ,x ))≤cost (M,(x′,x )).
xi i −i xi i −i
For randomized mechanisms, these mechanisms are often denoted by the term truthful in expectation
mechanisms,inordertodistinguishthemfromuniversallytruthfulmechanisms(astrongernotionwhich
wedonotuseinthispaper,inwhichanagentcannotregretreportinghertruelocationex-post).
For a location profile x and reported locations x′ the average cost is: AC(M,x,x′) =
1 cost (M,x′). Note that we scale the sum of the agents’ costs by a factor of n. For truthful
n i xi
mechanisms,wecandropthemisreportinthenotationanddenotetheaveragecostbyAC(M,x). Given
alPocationprofilex,theoptimalaveragecostis: AC(OPT,x)=min(l1...lk)∈[0,1]k n1 iminj|lj −xi|.
ForatruthfulmechanismM,theadditiveerrorgivenalocationprofilexisthedifferencebetweentheav-
eragecostofM andtheaveragecostoftheoptimalmechanism:σ (x)=AC(M,x)P−AC(OPT,x).
M
The additive error of a truthful mechanism M is the maximal error over any location profile: σ =
M
maxxσM(x).
Given location profile x and reported locations x′ the maximum cost (max-cost in short) is:
MC(M,x,x′) = max cost (M,x′). Similarly to the average cost case, for a truthful mechanism
i xi
M wedenotethemax-costbyMC(M,x). Givenalocationprofilex,theoptimalmaximumcostis:
MC(OPT,x)= min maxmin|l −x |
j i
(l1...lk)∈[0,1]k i j
For a truthful mechanism M, the additive error given a location profile x is the difference between
the maximum cost of M and the maximal cost of the optimal mechanism: δ (x) = MC(M,x) −
M
MC(OPT,x). The additive error of a truthfulmechanism M is the maximalerror over any location
profile:δM =maxxδM(x).
Givena locationprofilex, we denotethe leftmostandrightmostpointsin xbyx andx respec-
L R
tively.Thecenterpointbetweenthesetwopointsiscalledx = xL+xR.
C 2
We now show that lower bounds proven for few agents can also be extended to location profiles
witharbitrarilymanyagentssincetheerrorgrowsatleastlinearlywiththeamountofagents. Thefirst
observationshowsthatanylowerboundforboththemaximumandaveragecostprovenwithalocation
profilewithnagentsalsoholdsforanyprofilewithq·nagentsforanarbitrarypositiveintegerq (i.e.,
foranyprofilewithanevennumberofagents).
Observation1. LetM beatruthfulmechanismwhichhasanadditiveerrorofǫforanyprofilewithq·n
agentsforsome globalcostfunction(eitherthemaximumorthe averagecost)with q ≥ 1. Thenthere
exists some truthfulmechanismM′ whichhasan additiveerror ofǫ forthe same globalcostfunction,
foranyprofilewithnagents.
Proof. LetM′bethemechanismwhichsimulatesM basedonthelocationprofilewhichduplicateseach
reportqtimes. Thatis,forsomeinputx =(xA,...,xA),M′createstheprofile
A 1 n
x = (xB = xB = ··· = xB = xA,xB = xB = ··· = xB = xA,··· ,xB =
B 1 2 q 1 q+1 q+2 2q 2 (n−1)q+1
xB =···=xB =xA)andlocatesthefacilitiesonM′(x )=M(x ).
(n−1)q+2 nq n A B
4
Intheseprofilesinwhichtheprofileisduplicated,theadditiveerrorofthesemechanismsisthesame
(andthereforeinthegeneralcase,thecostofM isatleastaslargeasthatofM′):forthemaximumcost
itholdsthat:
MC(M′,x )=max|xA−M′(x )|
A i A
i
=max|xB −M(x )|=MC(M,x ).
j B B
j
Fortheaveragecost,boththenumeratorandthedenominatoraremultipliedbyq:
n
1
AC(M′,x )= |xA−M′(x )|
A n i A
i=1
X
qn
1
= |xB −M(x )|=MC(M,x ).
qn j B B
j=1
X
Clearly,thecostoftheoptimalmechanismisthesameforbothofthesecasesaswell.
It is only left to provetruthfulnessunderM′. This holdsdue to the lemma by Lu et al. in [18] in
whichtheyshowthatanystrategyproof(thatis,truthfulaccordingtoourterminology)mechanismisalso
partialgroupstrategyproof.Amechanismisdefinedtobepartialgroupstrategyproofifforanygroupof
agentsonthesamelocation,eachindividualcannotbenefitiftheymisreportsimultaneously. Formally,
given a non-emptyset S ⊂ N, profile x = (x ,x ) where x = (x,...,x) for some x, and some
S −S S
misreportedlocationsx′ itholdsthatforanyi∈S:
S
cost (M(x ,x )≤cost (M(x′ ,x )).
i S −S i S −S
If there had been some beneficial deviation of some x to x′ in M′ then there must have been some
i i
deviationofthecoalition(x ,x )to(x′,x′)inM,incontradictiontopartialgroupstrategyproofness.
i i i i
Thenextobservationshowsthatanylowerboundthemaximumcostprovenwitha locationprofile
with n agents also holds for any profile with n + 1 agents. Clearly this observation can be applied
repeatedlytoshowalowerboundforanarbitraryamountofagents.
Observation 2. Let M be a truthful mechanism which has an additive error of ǫ for any profile with
n+1agentsforthemaximumcost.ThenthereexistssometruthfulmechanismM′whichhasanadditive
errorofǫforthethemaximum,foranyprofilewithnagents.
Proof. LetM′bethemechanismwhichsimulatesM basedonthelocationprofileforwhichthelastagent
is placed together the penultimate agent. That is, for some inputx = (xA,...,xA), M′ creates the
A 1 n
profilex = xB =xA,xB =xA,...,xB =xA,xB =xA andlocatesthefacilitiesonM′(x ) =
B 1 1 2 2 n n n+1 n A
M(x ).
B
(cid:0) (cid:1)
Proof of truthfulness and the additive error follows the same lines as the previous observation –
TruthfulnessholdsduetotopartialgroupstrategyproofnessinM. ThemaximumerrorofM andM′is
thesamesincetheoutputtedlocationisthesame(M′(x )=M(x ))andthesetofthelocationsinthe
A B
inputs(thedistinctlocationsinx ,x )arethesame.
A B
3 Locating a Single Facility
3.1 Randomized Mechanisms
Forthesinglefacilityproblem,ProcacciaandTennenholtzintroducedtheLRC(left-right-center)mech-
anism,whichchoosesx andx withprobability 1 each,andchoosesx withprobability 1 [23].They
L R 4 C 2
furthershowedthatLRCistruthful-in-expectationandachievesatightboundforthemultiplicativeratio.
ItiseasytoseethatLRCachievesanadditiveerrorof 1 (forthelocationprofilex=(0,1)).
4
Westartbyshowingarandomizedlowerboundof 1. Wematchthisboundwithamechanismwhich
6
extendsLRC(calledBalanced-LRCorBLRCinshort).
Thefollowingtwodefinitionssetthefoundationsfortheproofofthelowerbound.
5
Definition3(5-pointmechanism). Foranarbitrary locationprofilex, a 5-pointmechanismM is one
whichcanonlyassignapositiveprobabilitytoasubsetofthefollowing5points:0,x ,x ,x ,1.
L C R
WedenotetheprobabilitiesthatM locatesthefacilitiesonthesepointsasp ,p ,p ,p ,p respec-
0 L C R 1
tively.
Definition4(Symmetric5-pointmechanism).
A5-pointmechanismM istermedsymmetricifforanylocationprofilexsuchthatx = 1−xitholds
thatp =p andp =p .
L R 0 1
Theorem5. Anyrandomizedtruthfulinexpectationmechanismforlocatingonefacilityhasanadditive
errorofatleast 1 forthemaximumcost.
6
Proof. Wepresentalowerboundfortwoagents. First, weshowthatanytruthfulinexpectationmech-
anism for 2 agentscan be replacedwith a 5-pointtruthfulin expectationmechanismwhile conserving
theadditiveerror. Then,weshowthatwecanfurtherrestrictourselvestosymmetric5-pointmechanism
without increasing the additive error. This characterizationis used by a pair of location profiles and a
transitionbetweenthem,yieldingthelowerbound.Finally,duetoObservation2,thislowerboundcanbe
extendedtoanarbitraryamountofagents-foranynumberofagentsntheredoesnotexistamechanism
withaloweradditiveerrorthan 1.
6
Lemma 6. If there exists a truthful in expectation mechanism M which has additive error α for the
maximumcostfor2agents,thenthereexistsatruthfulinexpectation5-pointmechanismM′whichalso
hasadditiveerrorαforthemaximumcostfor2agents.
Proof. The proof will transformM to M′ by movingthe probabilitythat M allocated to any point z,
totheleftandrightneighborsofz fromthesetA = {0,x ,x ,x ,1},withoutchangingtheexpected
L C R
location. The resultingmechanismwill clearly be a 5-pointmechanism, and the followingclaimswill
show that it also preserves truthfulness and the additive error. We start by showing the effect of this
transformationonanarbitrarydeterministicmechanismM .
1
Forsomearbitrarypointz,letz ,z beitsleftandrightneighbors,respectively,fromthesetA,that
1 2
is:
z =argmax(x≤z)
1
x∈A
z =argmin(x≥z)
2
x∈A
LetZ be therandomvariablewhichcantake twopossible values,z andz , suchthatE[Z] = z (that
1 2
isPr(M (x) = z ) = z2−z ,Pr(M (x) = z ) = z−z1 ). Forsomelocationprofilex,letM bethe
2 1 z2−z1 2 2 z2−z1 1
deterministicmechanismwhichlocatesthefacilityatpointz,andletM betherandomizedmechanism
2
whichlocatesthefacilityaccordingtotherandomvariableZ.
Claim7. Foranyp∈[0,z ]∪[z ,1],thecostofpunderM isthesameasitscostunderM ,thatis:
1 2 1 2
cost (M ,x)=cost (M ,x).
p 1 p 2
Proof. Assumewithoutlossofgeneralitythatp≤z <z . ThecostofpunderM is: cost (M ,x)=
1 2 1 p 1
|p−M (x)|=z−p.
1
Ontheotherhand,thecostunderM is:
2
z −z z−z
cost (M ,x)=E|Z −p|= 2 (z −p)+ 1 (z −p)
p 2 1 2
z −z z −z
2 1 2 1
z −z z−z
2 1
= (z −p)+ [(z −z )+(z −p)]
1 2 1 1
z −z z −z
2 1 2 1
z−z
1
=(z −p)+ (z −z )=z−p
1 2 1
z −z
2 1
Thus,bothcostsarethesame.
Claim 8. For any p ∈ [0,1], the cost of p under M is less than or equal to its cost under M :
1 2
cost (M ,x)≤cost (M ,x).
p 1 p 2
6
Proof. ThecostunderM isclearlycost (M ,x)=|p−z|.
1 p 1
ForM itholdsthat:
2
cost (M ,x)=E|Z−p|≥|E[p−Z]|=|p−E[Z]|=|p−z|
p 2
This holds due to Jensen’s inequality (for any convex function f and random variable Z: f(E[Z]) ≤
E(f(Z))),sincetheabsolutevalueisaconvexfunction.
Thesetwo claimsand truthfulnessof M collectivelyshow thatif M was truthful, thenso is M :
1 1 2
letxbealocationprofile,andletx′ betheprofileafterdeviationofsomex ∈ {x ,x }tox′. Then:
i L R i
costxi(M2,x)=costxi(M1,x)≤costx′i(M1,x)≤costx′i(M2,x).
Claim9. ThemaximumcostofxunderM isequaltothemaximumcostunderM .
1 2
Proof. Assumewithoutlossofgeneralitythatz ≤ x . Therefore,x incursthemaximumcostunder
C R
M , and this cost is x −z. Due to Claim 7, this is also the cost of x under M . Since in M the
1 R R 2 2
probabilityissplitonlybetweenpointsx ,x (ifx < z ≤ x )orpoints0,x (if0 ≤ z ≤ x ),then
L C L C L L
clearlytheagentwiththemaximumcostinM isx .
2 R
To concludethe proof of the lemma, it is only necessary to repeatthe above processfor any point
z ∈/ Awhichischosenwithapositiveprobability.
Wenowshowthatwecanrestrictourselvestosymmetric5pointmechanisms.
Lemma10. Foranytruthfulinexpectation5-pointmechanismM whichhasanadditiveerrorα,there
1
existsatruthfulinexpectationsymmetric5-pointmechanismM whichalsohasanadditiveerrorofα.
2
Proof. LetM beanarbitrarytruthful5-pointmechanismwhichachievesanadditiveerrorα. Wewill
1
defineamechanismM ,anduseittoconstructatruthfulinexpectationsymmetricmechanismM .
3 2
We start by defining M . For an arbitrary profile x denote the probabilities M chooses for
3 1
0,x ,x ,x ,1asp1,p1,p1,p1,p1 respectively. LetM bea5-pointmechanismwhichdoesthefol-
L C R 0 L C R 1 3
lowing:Foranarbitraryprofile1−x,M locatesthefacilitybasedonpoints0,1−x ,1−x ,1−x ,1
3 R C L
withthefollowingprobabilities: p3 = p1, p3 = p1, p3 = p1, p3 = p1 andp3 = p1 (noticethatthe
0 1 L R C C R L 1 0
leftmostandrightmostpointsin1−xare1−x and1−x respectively). InsomesenseM canbe
R L 3
seenasthe“anti-symmetric”mechanismofM .
1
M is truthful– Assume towardsa contradictionotherwise, that is, there existssome x anda ben-
3
eficialdeviationto x′. Then,thedeviationfrom1−xto 1−x′ wouldhavebeenbeneficialin M , in
1
contradictiontothefactthatM istruthful. Additionally,followingthesamelogic,themaximumcost
1
ofM isequaltothatofM (foranyprofilexinM witherrorα,theprofile1−xhasanerrorofαin
3 1 3
M ).
1
Let M be the mechanism which runs M and M with probability0.5 each. M is a probability
2 1 3 2
distributionovertruthfulmechanisms,thereforeitisnecessarilytruthful. Also,themaximumcostisthe
sameinM ,M ,anditisthereforealsothemaximalcostinM . Finally,bytheconstructionofM itis
1 3 2 2
clearlyasymmetric5-pointmechanism.
Nowthatwehaveproventhatwecanfocusonlyonsymmetric5-pointmechanisms,wepresenttwo
locationprofilesandatransitionbetweenthemwhichconcludestheproof.
LetxA =(1,2)andxB =(0,2)betwolocationprofiles.LetM beasymmetric5-pointmechanism
3 3 3
whichachievestheoptimaladditiveerror.
For profile xA, let the probability of locating the facility on the left (that is 1/3) be pALR and the
probabilityoflocatingthefacilityinthecenterpoint(thatis1/2)bepA. Sincetheinstanceissymmetric,
C
the probability of locating the facility on the right point (2/3) will also be pA and the probabilityof
LR
locatingthefacilityat0andlikewiseat1ispA =1/2−pA − pAC.
01 LR 2
7
Wehavethatthecostoftheagentlocatedattheleftisequalto:
1 2 pA pA
cost (A)=pA + + LR + C
1/3 01 3 3 3 6
(cid:18) (cid:19)
pA pA
=pA + LR + C
01 3 6
=1/2−pA −pA/2+pA /3+pA/6
LR C LR C
1 pA 2pA
= − C − LR
2 3 3
WenowconsidertheprofileB. Theadditiveerroroftheprofileisequalto
2 1 1
E[max{|X|,|X− |}]− =E|X − |=cost (B).
3 3 3 1/3
However,sincethemechanismistruthfulitholdsthat
cost (A)≤cost (B).
1/3 1/3
IfweassumetowardsacontradictionthatM canachieveanadditiveerroroflessthan 1, wehavethat
6
cost (B)< 1 andconsequentlycost (A)<1/6.Thisimpliesthat 1−pAC −2pALR < 1 whichgives
1/3 6 1/3 2 3 3 6
pA+2pA >1. ThisiscontradictionsinceinprofileAprobabilitiesmustsumto1.
C LR
Wenowpresentanewmechanism,andshowthatitcanachieveamatchingupperboundof1/6:
Definition11(BLRCmechanism).
Thebalancedleft-right-center(BLRC)mechanismlocatesthefacilityatthepoint 1 withprobability 1,
2 3
anddeploystheLRCmechanismwithprobability 2. Moreexplicitly,BLRClocatesthefacilityaccording
3
tothefollowingdistribution:
1 1 1 1 1
, w.p. ; x w.p. ; x w.p. ; x w.p.
L R C
2 3 6 6 3
Theorem12. BLRCistruthfulinexpectation,andachievesanadditiveerrorofatmost 1 forthemaxi-
6
mumcost.
Proof. BLRC is based on two truthful-in-expectationmechanisms– LRC was provento be truthfulin
expectationin[23],andlocatingthefacilityonthefixedpoint1/2isclearlytruthfulastheagentshave
noinfluenceovertheresult. Thereforetakingadistributionoverthesetwomechanismsisalsotruthfulin
expectation.
Proofoftheapproximationratiowillbebasedontwolemmas:
Lemma 13. Let x,y bepointssuchthat: 0 ≤ x < y ≤ 1, andletx,x′ be the followingtwo location
profiles: x=(x,y)andx′ =(0,y−x). Thenδ (x)≤δ (x′).
BLRC BLRC
Proof. For the profile x, the optimal mechanism locates the facility at x+y and its cost is y−x. The
2 2
maximumcostofBLRCinthiscaseis: 1·(y−x)+1y−x+1·(max{|y−1/2|,|x−1/2|}).Therefore
3 3 2 3
theadditiveerroris: δ (x)= max{|y−1/2|,|x−1/2|}.
BLRC 3
Fortheprofilex′ thecostoftheoptimalmechanismremains y−x,butthecostofBLRC is: 1(y−
2 3
x)+ 1 · y−x + 1 · 1. Therefore:δ (x′)=1/6.
3 2 3 2 BLRC
For any x,y ∈ [0,1] it holds that 1 ≥ max{|y − 1/2|,|x − 1/2|}, therefore: δ (x′) −
2 BLRC
δ (x)= 1(1 −max{|y−1/2|,|x−1/2|})≥0.
BLRC 3 2
Lemma 14. For any z ∈ [0,1], let x = (0,z). It holdsthatthe additive error ofBLRC for x is 1/6:
δ (x)= 1.
BLRC 6
Proof. Giventhelocationprofilex,theoptimalmechanismlocatesthefacilityatz/2atamaximumcost
ofz/2.ThecostofBLRC is: MC(BLRC,x)= 1(z)+ 1 · z + 1 · 1.
3 3 2 3 2
Therefore,theadditiveerroris: δ (x)= 1 · z + 1 ·(1 − z)= 1 · 1 = 1.
BLRC 3 2 3 2 2 3 2 6
Lemma13holdsforanarbitrarylocationprofilexwithx =x,x =y,andLemma14holdsforan
L R
arbitraryz,andinparticularz =y−x,sotheerrorofBLRCwithtwoagentsisnomorethan1/6.Since
themaximumcostalwaysoccursinatleastoneofthetwoextremereports,combiningthetwolemmata
issufficienttocompletetheproofforanarbitrarynumberofagentsaswell. (cid:4)
8
3.2 DeterministicMechanisms
Wemoveontoprovealowerboundof1/4andamatchingupperbound.Recognizethatanydictatorship
hasanadditiveerrorof1/2,byconsideringx=(0,1).
Theorem15. AnydeterministictruthfulmechanismM hasanadditiveerrorofatleast 1 forthemaxi-
4
mumcost.
Proof. Wefirstdealwiththecasewheretherearetwoagents,andlaterextendtoanarbitrarynumberof
agents.
Let x = (0,1). Assume without loss of generality that M(x) = 1/2+ǫ for some ǫ ≥ 0. Let
x′ = (0,1/2+ǫ). The optimum for x′ is achieved at 1/4+ǫ/2. Therefore, if the mechanism is to
achieveanadditiveerroroflessthan1/4,itmustlocatethefacilityin(0,1/2+ǫ/2). Butifthiswere
the case, then the agent at 1/2+ǫ could benefit by misreporting to 1, which would move the facility
preciselyto1/2+ǫ,contradictingtruthfulness.
Forthecase ofmorethantwoagents,considertheprofileinwhichallotheragentsarepreciselyat
1/2,andrepeattheprocessabove.
ThislowerboundconstructionworksinsimilarlinestothelowerboundofTheorem2.2in[23].
Wenowpresentamatchingupperbound,atruthfulmechanismadditiveerrorof 1.
4
Definition16(Phantom-half). Foranyprofilexwherex ,x aretheleftmostandrightmostlocations,
L R
M locatesthefacilityonthemedianofthefollowing3points: x ,x ,0.5.
L R
Thenameofthismechanismisinspiredbythenotionofphantom-voters(inthiscase-thephantom
voteris0.5)asintroducedin[20].
Theorem17. Phantom-halfistruthfulandhasanadditiveerrorof 1 forthemaximumcost.
4
Proof. Itiseasytoseethatbymisreporting,anyagentcaneithernotaffectthelocationofthefacility,or
moveitfartherawayfromthem.
Fortheadditiveerror:
– Ifx ,x ≤1/2:Theoptimalmechanismlocatesthefacilityat xL+xR foramaximumcostof xR−xL.
L R 2 2
Phantom-halflocatesthefacilityatx foracostofx −x . Theadditiveerroris: xR−xL ≤ 1/2 = 1.
R R L 2 2 4
– Ifx ≤ 1/2,x ≥1/2: Assumewithoutlossofgeneralitythat1/2−x ≥ x −1/2. Theoptimal
L R L R
mechanism locates the facility at xL+xR for a maximum cost of xR−xL. Phantom-half locates the
2 2
facilityat1/2foracostof1/2−x . Theadditiveerroris: (1/2−x )− xR−xL = 1−xL−xR. Since
L L 2 2
x ≥1/2,x ≥0itholdsthat: 1−xL−xR ≤ 1.
R L 2 4
– Ifx ,x >1/2:Thisiscompletelysymmetrictothefirstcase.
L R
4 Extension to Many Facilities
Westartbyshowingatrivialdeterministicmechanism,EqualSpread,whichachievesanerrorofnomore
than 1 forboththeaveragecostandthemaximumcost. Wethenshowalowerboundof 1 forthe
2k−1 6k
maximumcost,meaningtheboundisΘ(1).
k
AfterwardsweshowtwoextensionsofEqualSpread-PEC,arandomizedmechanismwhichreaches
anerrorof 1 forbothmax-costandaveragecost,andEPEC,adeterministicmechanismwhichreaches
4k−2
anerrorof 3 fortheaveragecost.
8k−4
Finally,weshowthatforthecaseof2facilities,wecanimprovetheboundofEPECfortheaverage
costfrom1/4to1/5bychoosingthe“fifths”mechanism,whichlocatesthefacilitiesonthe0.2and0.8
percentiles.
Definition18(EqualSpreadmechanism). TheEqualSpreadmechanismlocatesthekfacilitieson i
2k−1
foroddvaluesofi(suchthat1≤i≤2k−1).
9
Theorem19. Equalspreadisadeterministic mechanismwhichachievesanadditiveerror ofnomore
than 1 forboththeaveragecostandthemaximumcost.
2k−1
Proof. Trivial-Clearly,foranypointp ∈ [0,1],pislocatedatadistanceofatmost 1 totheclosest
2k−1
facility.
Theorem20. AnydeterministictruthfulmechanismM forlocatingk facilitieshasanadditiveerrorof
atleast 1 forthemaximumcost.
6k
Proof. Theproofseparatesk−1agentsfarawayfromoneanother,andthenusestheproofofTheorem
15,exceptonasmallerinterval.
Letx=(x )k+1bethefollowinglocationprofile:letx =0,x = 2 andforevery3≤i≤k+1:
j j=1 1 2 3k
letx = i−1. An optimalmechanismcan locate k−1 facilities on{x ,x ,...x }, andlocate one
i k 3 4 k+1
facilityonthemidpointofx ,x (thatis,on 1 ),foramaximumcostof 1 .
1 2 3k 3k
Any mechanism with an error lower than 1 must also designate one facility to the intervalin the
6k
vicinityofeveryoneofthek−1agentsonx ...x : Ifthereexistssome2 ≤ i ≤ k−1suchthat
3 k+1
there are no facilities in the segment (i−1/2,i+1/2) or no facilities on (k−1/2,1] then there exists an
k k k
agentlocatedatleast 1/2 fromafacility,thereforetheerrorwouldbeatleast: 1/2 − 1 = 1 .
k k 3k 6k
Additionally,anymechanismwithanerrornotlargerthan 1 mustalsodesignateonefacilityserve
6k
the first two agents (x ,x ): If there are no facilities in the segment [0, 1] then the error is at least
1 2 k
1 −0> 1 .
k 6k
Assumewithoutlossofgeneralitythattheremainingfacilityisputatpoint 1 +ǫforsomeǫ ≥ 0.
3k
Letx′bethelocationprofileinforanyj 6=2:x′ =x andx′ = 1 +ǫ. Theaforementionedarguments
j j 2 3k
alsodictatethatforanyM witherrorlessthan 1 ,inx′thenx′,x′ arealsodesignatedonefacility(that
6k 1 2
is,thereisafacilityinthesegment[0, 1]). Inprofilex′ thecostoftheoptimalmechanismis 1 +ǫ/2.
k 6k
ForanyM witherrorlessthan 1 thefacilitymustbelocatedinthesegment(ǫ/2, 1 +ǫ/2).Butifthis
6k 3k
werethecase,thenagentx′ couldbenefitbymisreportingto 2 (andthenthefacilitywouldhavebeen
2 3k
locatedpreciselyonit),contradictingtruthfulness.
WeproposearandomizedmechanismcalledPECwhichextendsthisbasicidea.
Definition 21 (Paired-Equal-Cost(PEC) mechanism). The paired-equal-costmechanism locates the k
facilitiesinthefollowinglocations,for0≤i≤2k−1:
{0, 2 , 4 ,..., 2k } w.p. 1
M(x)= 2k−1 2k−1 2k−1 2
({ 1 , 3 , 5 ,...,1} w.p. 1
2k−1 2k−1 2k−1 2
Theorem 22. PEC is a truthful in expectation randomized mechanism which has an additive error of
1 fortheaveragecost.
4k−2
Proof. PECisclearlytruthfulsincethereportshavenoeffectovertheoutcome.
We show the additive errorby boundingthe additiveerror of an arbitraryagentlocated at pointx
i
fromaboveby 1 .Let j ≤x < j+1 forsomej1. Therefore,theclosestfacilitytotheagentwill
4k−2 2k−1 i 2k−1
beeither j or j+1 ,eachwithprobability0.5. Denotex = j +ǫforsome0 ≤ ǫ< 1 . The
2k−1 2k−1 i 2k−1 2k−1
expectedcostoftheagentwillbe:0.5 |x − j |+| j+1 −x | =0.5(ǫ+ 1 −ǫ)= 1 .
i 2k−1 2k−1 i 2k−1 4k−2
(cid:16) (cid:17)
For the deterministic case we introduce election-parity-equal-cost(EPEC). Like in PEC, facilities
there are two options - locating facilities on { i } for even or odd values of i. In order to decide
2k−1
whethertochoosetheevenoroddvalues,themechanismcountstheamountofagentswhoprefereach
option(basedontheirreports),anddecidesbasedonmajority(tiescanbebrokenarbitrarily).
Theorem23. EPECistruthful,andachievesanerrorof 3 fortheaveragecost.
8k−4
1Itiseasytoverifythatforxi=1thecostis 4k1−2.
10