A Computational Study of the Kemeny Rule for Preference Aggregation Andrew Davenport and Jayant Kalagnanam I%M T.*.+atson Resear2h Center5 Yor7town Heights5 New Yor75 10?@8 daBenportDEs.iFm.2om5 HayantDEs.iFm.2om

Abstract +e 2onsider from a 2ompEtational perspe2tiBe the proFlem of how to aggregate the ran7ing preferen2es of a nEmFer of alternatiBes Fy a nEmFer of different Boters into a single 2onsensEs ran7ing5 following the maHority Boting rEle. So2ial welfare fEn2tions for aggregating preferen2es in this way haBe Feen widely stEdied sin2e the time of Condor2et M178?O. One drawFa27 of maHority Boting pro2edEres when three or more alternatiBes are Feing ran7ed is the presen2e of 2y2les in the maHority preferen2e relation. The Kemeny order is a so2ial welfare fEn2tion whi2h has Feen designed to ta27le the presen2e of sE2h 2y2les. HoweBer 2ompEting a Kemeny order is 7nown to Fe NPShard. +e deBelop a greedy heEristi2 and an eTa2t Fran2h and FoEnd pro2edEre for 2ompEting Kemeny orders. +e present resElts of a 2ompEtational stEdy on these pro2edEres.

Introduction Preferen2e aggregation 2on2erns how to 2omFine the preferen2e ran7ings of a nEmFer of alternatiBes Fy a nEmFer of different Boters into a single 2onsensEs Mor so2ietyO ran7ing. The aggregation of preferen2es in this way arises in appli2ations sE2h as determining the winners of ele2tions or sports toErnaments MTrE2hon 1@@8O5 mEltiS2riteria and word asso2iation UEeries in dataFases MDwor7 et al. 2001O5 the ran7ing of sEppliers Fy FEyers dEring strategi2 soEr2ing and the 2omFining of sear2h resElts from mEltiple sear2h engines in order to fight spam MDwor7 et al. 2001O. The properties of so2ial welfare fEn2tions for aggregating preferen2es haBe Feen stEdied Fy mathemati2ians sin2e the 18th 2entEry5 and it is well 7nown that 2ompli2ations and paradoTes 2an arise when there are more than two alternatiBes to Fe ran7ed. ArrowYs impossiFility theorem MArrow 1@?1O states that no nonSdi2tatorial so2ial welfare fEn2tion 2an eBer satisfy a set of desiraFle and 2ompelling properties for fair ele2tions simEltaneoEsly on all domains of preferen2es. The deFate on the merits of different types of so2ial welfare fEn2tions is still ongoing MSaari and Zalognes 1@@8O5 howeBer two 2lasses of ran7ing methods haBe Feen widely stEdied. Positional methods5 sE2h as the %orda 2oEnt M%orda 1781O5 assign points to ea2h alternatiBe5 depending on how they are ran7ed Fy ea2h Boter. Candidates are then ordered in the 2onsensEs ran7ing a22ording to the sEm of their assigned points. MaHority ran7ing methods determine an oEt2ome in terms of the maHority ran7ing for the alternatiBes: alternatiBe x

is ran7ed ahead of alternatiBe y in the 2onsensEs ran7ing if more Boters prefer x to y. Positional methods sE2h as the %orda 2oEnt5 while Feing Bery simple to 2ompEte5 are 2onsidered to Fe highly manipElaFle and fail to satisfy important properties sE2h as the independen2e of irreleBant alternatiBes M2hanges in indiBidEalsY ran7ings of \irreleBant\ alternatiBes oEtside of a 2ertain sEFset shoEld haBe no impa2t on the so2ietal ran7ing of this sEFsetO and the Condor2et 2riterion Mif some alternatiBe is ran7ed ahead of all other alternatiBes Fy an aFsolEte maHority of Boters5 then it shoEld Fe ran7ed first in the 2onsensEs ran7ingO. MaHority ran7ing methods sE2h as Condor2et methods may Fe sEFHe2t to 2y2les in the maHority preferen2e relation when there are more than two alternatiBes to Fe ran7ed5 and thEs may fail to sele2t any winners at all Mthis is 7nown as the Condor2et paradoT MCondor2et 178?OO. The Kemeny rEle MKemeny 1@?@O has Feen proposed as a way of see7ing a 2ompromise ran7ing in the maHority Bote when there are 2y2les present in the maHority preferen2e relation. The Kemeny rEle satsifies the Condor2et 2riterion and a wea7er Bersion of lo2al independen2e of irreleBant alternatiBes. HoweBer the 2ompEtational drawFa27 of the Kemeny REle is that is NPShard to 2ompEte MCohen5 S2hapire and Singer 1@@@] Dwor7 et al. 2001O. This has dis2oEraged the deBelopment of eTa2t algorithms for 2ompEting Kemeny orders. Instead greedy heEristi2s MCohen5 S2hapire and Singer 1@@@O or tra2taFle mEltiSstage methods haBe Feen deBeloped that 2omFine Foth positional and maHority Boting methods M%la27 1@?8] Dwor7 et al. 2001O. A disadBantage of sE2h approa2hes is that their theoreti2al properties are not 7nown5 and hen2e their oEt2omes 2an Fe Enpredi2taFle. FErthermore5 NPShardness is a only worst 2ase 2ompleTity resElt whi2h may not refle2t the diffi2Elty of solBing proFlems whi2h arise in pra2ti2e. In the se2tions whi2h follow we des2riFe new eTa2t and greedy heEristi2 pro2edEres for 2ompEting Kemeny orders5 and present resElts of an initial 2ompEtational stEdy into solBing proFlems Esing these pro2edEres.

Notation Let X Fe a set of m alternatiBes and N Fe a set of n Boters. Ea2h Boter j has wea7 order or ran7ing rj of the alternatiBes in X. Ea2h element rjs is the ran7 of Boter j of alternatiBe s. A ran7ing with no tie for a ran7 is an order on X5 whi2h 2an Fe represented as a seUEen2e (s1,...,sn) 5

where si is the alternatiBe with ran7 i. A preference aggregation function Malso 7nown as a so2ial welfare fEn2tionO maps a set of ran7ings R Malso referred to as a profile of rankingsO into a single 2onsensEs ran7ing. For ea2h pair of alternatiBes s and t ran7ed Fy some profile of ran7ings R5 we define vst as the nEmFer of Boters eTpressing a preferen2e for s oBer t in their j j indiBidEal ran7ings5 that is vst M R O =a j ∈ N : rs < rt a . If vst > vts we refer to s > t as the majority vote and t > s as the minority vote for s and t. +e represent a profile of ran7ings R Fy a weighted5 dire2ted graph we 2all the preference graph. Ea2h alternatiBe in X is represented Fy a node in the graph. %etween ea2h pair of nodes representing alternatiBes s and t is an edge of weight vst. +e represent the stri2t maHority relation Fy a simple5 dire2ted5 weighted graph 2alled the strict majority graph5 whi2h we also refer to more simply as the maHority graph. There is an edge Fetween ea2h pair of nodes representing alternatiBes s and t iff vst > vts. The weight of sE2h an edge is vst− vts. The wea7 maHority relation is represented Fy the weak majority graph5 whi2h has an edge Fetween ea2h pair of nodes representing alternatiBes s and t iff vst ≥ vts. The weight of sE2h an edge is vst − vts.

Majority voting and the Condorcet criterion The Condorcet criterion MCondor2et 178?O states that if some alternatiBe is ran7ed ahead of all other alternatiBes Fy an aFsolEte maHority of Boters5 then it shoEld Fe ran7ed first in the 2onsensEs ran7ing. +hen sE2h an alternatiBe eTists5 it is 2alled the Condorcet winner. Example 1: Consider the preferences expressed by the profile (A,B,C), (A,C,B), (B,A,C) (e.g. the 1st voter prefers A to B to C, the 2nd voter prefers A to C to B, the 3rd voter prefers B to A to C). The preferences for each of the pairwise rankings are: vAB=2,vBA=1,vBC=2, vCB=1, vAC=3, vCA=0. We have a majority of 1 vote for A > B, a majority of 1 vote for B > C and a majority of 3 votes for A > C. %

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c C C A c 0 FigEre 1: The preferen2e graph MleftO and maHority graph MrightO of the profile of ETample 1 A

In ETample 1 alternatiBe A is the Condor2et winner. Note that when there is a Condor2et winner5 it 2an Fe foEnd Fy identifying the node in the wea7 maHority graph with an indegree of bero. HoweBer5 there may not always Fe a Condor2et winner5 as 2y2les may o22Er in the maHority ran7ing of alternatiBes. Example 2: Consider the profile of rankings (A,B,C), (B,C,A), (C,A,B). There is a majority of 1 vote for A > B, a majority of 1 vote for B > C and a majority of 1 vote

for C > A. This profile defines a cycle in the majority relation of A > B, B > C, C > A. %

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1 C C A 1 2 FigEre 2: The preferen2e graph MleftO and maHority graph MrightO for the profile of ETample 2 A

Cy2les do o22Er in pra2ti2e5 eBen for UEite small proFlems. For eTample5 in a stEdy of Olympi2 figEre s7ating 2ontests5 2y2les were foEnd in 2ompetitions inBolBing @ oEt of 2c s7aters MTrE2hon 1@@8O. +hen 2y2les are present in the maHority relation5 the extended Condorcet criterion MTrE2hon 1@@8O giBes a partial ordering of alternatiBes Mwith no 2y2les it giBes a 2omplete orderingO. The eTtended Condor2et 2riterion states that if there is a partition XC5 XD of X sE2h that for any s in C and any t in D if the maHority prefers s to t Mvst > vtsO then s mEst Fe ran7ed ahead of t in the 2onsensEs ran7ing. One appli2ation of the Condor2et and eTtended Condor2et 2riteria is in fighting ddspamYY in indiBidEal ran7ings MDwor7 et al. 2001O. HoweBer it has Feen shown that no positional method for aggregating preferen2es 2an eBer satisfy either of these 2riteria MYoEng 1@7eO. GiBen some partial or 2omplete 2onsensEs ran7ing of a profile of ran7ings R5 we 2an enfor2e the eTtended Condor2et 2riterion on the ran7ing in the following way1: Algorithm 1: 1. Identify all the maTimal strongly 2onne2ted 2omponents in the maHority graph of R. Ea2h strongly 2onne2ted 2omponent of sibe > 1 node identifies a set of nodes whi2h are inBolBed in at least 1 2y2le. 2. For ea2h node n in ea2h 2omponent5 2onsider all target nodes m on all oEtgoing edges of n. If m is not in the same 2omponent as n5 add to the 2onsensEs ran7ing the ordering s > t5 where s is the alternatiBe represented Fy node n and t is the alternatiBe represented Fy node m. AlternatiBes represented Fy nodes within nonStriBial strongly 2onne2ted 2omponents of the maHority graph are inBolBed in 2y2les5 and will not Fe ordered Fy Algorithm 1. Kemeny MKemeny 1@?@O proposed a way of Frea7ing sE2h 2y2les5 Esing a notion of distan2e for orders. GiBen a total order r5 a wea7 order rj and 2 alternatiBes s and t5 we define: !1 if rs < rt and rt j ≤ rs j j δ st M r 5 r O = " #0 otherwise and j j ∆ M r 5 r O = $ $ δ st M r 5 r O s∈ X

1

t∈ X

A similar s2heme is presented in MCohen5 S2hapire and Singer 1@@@O and in MDwor7 et al. 2001O.

The BalEe of δ st M r 5 r O indi2ates whether there is a disagreement in the relatiBe ran7ing of s and t Fetween r j and rj. ∆ M r 5 r O measEres the total nEmFer of disagreements Fetween r and rj5 and is referred to as the Kendall tau distance MDia2onis 1@88O. For 2omplete orders5 the Kendall taE distan2e is the ddFEFFle sortYY distan2e: it giBes the nEmFer of pairwise adHa2ent transpositions to transform one order into another. The distan2e Fetween a 2omplete order r and a profile of ran7ings R is giBen Fy d(r, R) = $j=1..n ∆(r, rj). A Kemeny order for a profile R is an order r whi2h minimibes the distan2e d(r, R) 5 i.e. an order whi2h has the minimEm nEmFer of disagreements with the pairwise ran7ings in the profile R. It has Feen shown that any Kemeny order satisfies the eTtended Condor2et 2riterion MTrE2hon 1@@8O. j

Algorithm design Problem formulation

GiBen a Mwea7O maHority graph GR of a profile of ran7ings R5 we wish to 2ompEte a total order of the alternatiBes in R ta7ing into a22oEnt as many of the maHority preferen2es eTpressed Fy the edges in GR as possiFle. If GR is a2y2li25 a topologi2al sort of the nodes in the graph will identify a Kemeny order of the alternatiBes. +hen GR 2ontains 2y2les5 it Fe2omes ne2essary to remoBe some set of edges from GR sE2h that the resElting graph is a2y2li2. To find a Kemeny order5 the sEm of the weights of the edges remoBed from the graph mEst Fe minimibed. This proFlem is 7nown as the minimEm weight feedFa27 edge set proFlem MFesta5 Pardalos5 and Resende 1@@@O5 and is 7nown to Fe NPS hard. FeedFa27 set proFlems haBe re2eiBed 2onsideraFle attention in re2ent years MFesta5 Pardalos5 and Resende 1@@@O5 espe2ially with respe2t to approTimation algorithms and tra2taFle 2ases. HoweBer5 as far as we are aware5 there has Feen no wor7 on eTa2t approa2hes for finding optimal solEtions to the weighted Bariant of the feedFa27 edge set proFlem in dire2ted graphs. In the se2tions whi2h follow5 we oEtline a new greedy heEristi2 and eTa2t Fran2h and FoEnd approa2h for finding Kemeny orders5 Fased on formElating the proFlem as minimEm weight feedFa27 edge set proFlem.

Solution representation

+e Ese a solution graph to represent orderings of alternatiBes that are estaFlished dEring a sear2h pro2edEre when 2onstrE2ting a Kemeny order. Ea2h alternatiBe in the set X is represented Fy a node in the solEtion graph. A dire2ted edge Fetween a pair of nodes representing alteratiBes s and t indi2ates that in the solEtion s is ran7ed ahead of t. A solEtion graph whi2h is simple5 dire2ted5 a2y2li2 and where all nodes haBe degree of |X|−1 represents a 2omplete ordering of the alternatiBes in X. The goal of the sear2h pro2edEre is to 2onstrE2t a solEtion graph representing a 2omplete ordering whi2h is a Kemeny order. A 2omplete ordering

2an Fe oFtained from the solEtion graph Fy performing a topologi2al sort of the nodes in the graph. +e often want to 2ompEte a maHority graph whi2h ta7es into a22oEnt the orderings that haBe Feen made in a solEtion graph5 eBen if some of these orderings 2orrespond to minority Botes. ThEs all edges in the solEtion graph haBe 2orresponding edges in the maHority graph representing the same orderings5 regardless of the maHority Botes for these alternatiBes. In the 2ase where an edge in the solEtion graph 2orresponds to an ordering with a minority Bote5 the reBerse edge in the dire2tion of the maHority Bote will not Fe present in this modified maHority graph.

Greedy heuristic and branch and bound search

+e haBe deBeloped a simple greedy heEristi2 and an eTa2t depthSfirst Fran2h and FoEnd pro2edEre for 2ompEting Kemeny orders5 Fased on 2onstrE2ting a solEtion Fy identifying orderings of pairs of alternatiBes and adding the 2orresponding edges to the solEtion graph. +e Ese the following heEristi2 to sele2t and order a pair of alternatiBes at ea2h node of the sear2h in Foth of these approa2hes: sele2t the pair of alternatiBes whi2h has the greatest differen2e Fetween the maHority and the minority Bote for the different pairwise orderings and order the alternatiBes following the maHority Bote. The greedy pro2edEre simply follows this heEristi2 and applies propagation rEles after ea2h ordering is made5 terminating when the solEtion graph represents a 2omplete order. The Fran2h and FoEnd pro2edEre performs a depthSfirst Fa27tra27ing sear2h5 following the heEristi2 at ea2h node and applying lower FoEnding pro2edEres in addition to the propagation rEles to prEne the sear2h. In the se2tions whi2h follow we dis2Ess the propagation rEles and lower FoEnding te2hniUEes.

Propagation

A solEtion is FEilt Ep in2rementally Fy adding edges to the solEtion graph. In order to ensEre that the solEtion graph remains a2y2li2 we maintain transitiBe 2losEre of the graph after edges are added or deleted. Sin2e the graph is dire2ted and a2y2li25 we 2an Ese an effi2ient in2remental algorithm that maintains transitiBe 2losEre in 2 time O M n O for n nodes in the graph. The first step for Foth approa2hes is to rEn Algorithm 1 to satisfy the eTtended Condor2et 2riterion of the ordering. If there are no 2y2les in the maHority graph5 this step is sEffi2ient to identify a Kemeny order. Otherwise5 as edges are added to the solEtion graph dEring sear2h5 2y2les may Fe Fro7en in the maHority graph that ta7es into a22oEnt solEtion graph orderings Mand new 2y2les may Fe 2reatedO. +e rEn Algorithm 1 as a propagation step on this maHority graph. If there are new strongly 2onne2ted 2omponents in the maHority graph as a resElt of edges added to the solEtion graph5 fErther orderings of alternatiBes may Fe added to the solEtion. Applying this propagation step eBery time an edge is added to the solEtion graph ensEres that we only

eBer Fran2h on orderings that are inBolBed in some 2y2le in the maHority graph asso2iated with some solEtion graph.

Lower bounding

A simple lower FoEnd LB1 on the Kemeny distan2e of a Kemeny order is the sEm of the minority Botes for eBery pair of alternatiBes: LB1 =

$

∀s∈ X

$

minMvst 5 vts O

∀t∈ X 5 s ≠ t

This FoEnd will Fe eUEal to the Kemeny distan2e of the Kemeny order when there are no 2y2les in the maHority graph: in this 2ase we 2an follow the maHority Bote in the Kemeny order for the ran7ing of all pairs of alternatiBes. +hen there are 2y2les5 the FoEnd 2an Fe strengthened dEring sear2h Fy ta7ing into a22oEnt the orderings of the pairs of alternatiBes that haBe Feen fiTed Fy the sear2h pro2edEre. A stronger lower FoEnd 2an Fe 2ompEted Fy determining a set of edge disHoint 2y2les in ea2h strongly 2onne2ted 2omponent of the maHority graph Masso2iated with some solEtion graphO. Ea2h edge disHoint 2y2le 2ontains at least one edge representing a maHority Bote Fetween a pair of alternatiBes whi2h the Kemeny order mEst disagree with. The impa2t of disagreeing with the maHority Bote on the distan2e of the Kemeny order for some pair of alternatiBes s and t is vst − vts5 whi2h is the weight of the edge Fetween the nodes 2orresponding to the pair of alternatiBes in the maHority graph. +e 2an add5 for ea2h edge disHoint 2y2le5 the minimEm positiBe weight of the edges in the 2y2le to the FoEnd LB1 to form a stronger lower FoEnd. +e simplify the lower FoEnd 2ompEtation Fy 2onsidering edge disHoint 2y2les 2ontaining only c nodes. This is motiBated Fy the oFserBations that MaO it is harder to 2ompEte a Kemeny order for profiles where there are no ties in the ran7ings Fetween any pairs of alternatiBes2 and MFO in a maHority graph where there are no ties5 there eTists an edge Fetween eBery pair of nodes in eBery nonStriBial strongly 2onne2ted 2omponent. +hen this is the 2ase then eBery set of nodes inBolBed in a 2y2le of length greater than c mEst 2ontain a 2y2le of eTa2tly length c on a sEFset of these nodes. An illEstration of this proposition is giBen in FigEre c. Consider a 2y2le of length e Fetween nodes A5 B5 C and D. +hen there are no ties5 there mEst Fe an edge in the maHority graph Fetween nodes B and D. If this edge is dire2ted from node B to D then we haBe the cS2y2le inBolBing nodes A5 B5 D Mleft5 FigEre cO. If the edge is oriented the reBerse way5 we haBe the cS2y2le inBolBing nodes B5 C5 D Mmiddle5 FigEre cO. Finally5 we 2onsider the 2ase where a 2y2le 2ontains more than e nodes. The right graph in FigEre c shows a 2y2le inBolBing ? nodes A5 B5 C5 D5 E. Consider the edge Fetween nodes A and D 2

This sitEation will always o22Er when we haBe an odd nEmFer of Boters and all Boters ran7 all alternatiBes.

whi2h is not an edge of the ?S2y2le. If this edge is dire2ted from nodes A to D then we haBe the cS2y2le inBolBing nodes A5 D5 E. Otherwise5 the dire2tion from D to A forms a eS2y2le inBolBing nodes A, B, C, D for whi2h we haBe shown there mEst eTist a cS2y2le on a sEFset of these nodes. A similar argEment 2an Fe Esed to show that ea2h 2y2le inBolBing n nodes Mn>3O mEst 2ontain a 2y2le inBolBing either c or n − 1 nodes. %

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FigEre c: IllEstration of cS2y2les in a maHority graph. +e formElate the proFlem of 2ompEting edge disHoint cS 2y2les in a strongly 2onne2ted 2omponent of the maHority graph as an iteratiBe minS2ost maTSflow proFlem MAhEHa5 Magnanti5 and Orlin 1@@cO. An eTample of the networ7 formElation we Ese to solBe this proFlem is giBen in FigEre e for the maHority graph illEstrated in the left of FigEre c. A

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FigEre e: An eTample networ7 formElation for the maHority graph in the left of FigEre c. The networ7 has foEr layers of nodes5 l1,...,l4. In ea2h layer there are |X| nodes5 where ea2h node 2orresponds to a node in the maHority graph. There is a dire2ted edge Fetween ea2h pair of nodes in layers li and li+1 in this networ7 if there is a dire2ted edge Fetween the 2orresponding pair of nodes in the maHority graph. The 2ost on ea2h edge Fetween the layers is some 2onstant K Me.g. the nEmFer of BotersO minEs the weight Fetween the 2orresponding pair of nodes in the maHority graph Mwe want to find 2y2les with the largest possiFle minimEm edge 2ost in the maHority graphO. All nodes in layer l1 are 2onne2ted to the soEr2e node and all nodes in layer l4 are 2onne2ted to the sin7 node of the networ7. These 2onne2tions all haBe a 2ost of 1. All minimEm edge 2apa2ities are set to 0. The maTimEm 2apa2ities of all the edges Fetween nodes in the layers are to set to 1. DEring ea2h iteration we solBe a minS2ost maTSflow proFlem to determine a set of paths throEgh the networ7 starting and ending at some 2ommon node in layers l1 and l45 representing a starting and ending point for a cS2y2le in the maHority graph. For eTample5 the highlighted edges

in FigEre e illEstrate a path of length c starting at node A in layer l1 and ending at node A in layer l4. SE2h a path 2orresponds to the cS2y2le (A, B, D, A) in the maHority graph in the left of FigEre c. In ea2h iteration we sele2t and fiT the 2ommon starting and ending nodes ns and ne in layers l1 and l4 Me.g. node A in FigEre eO. The maTimEm nEmFer of cS2y2les max3 that 2an Fe foEnd from these nodes is the minimEm of the oEtSdegree of node ns and the inSdegree of node ne. +e set the maTimEm 2apa2ity of the edge from the soEr2e node to the node ns and node ne to the sin7 node to max3. +e set the maTimEm 2apa2ities of all remaining edges from the soEr2e node to the layer l1 nodes to 0. Similarly5 we set the maTimEm 2apa2ities of all remaining edges Fetween nodes in layer l4 and the sin7 node to 0. The solEtion to the minS2ost maTSflow proFlem on this networ7 spe2ifies a set of paths throEgh the networ7 that 2orrespond to a set of cS2y2les in the maHority graph starting and ending at the fiTed node. The BalEe of the flow is the nEmFer of disHoint cS2y2les foEnd in the 2orresponding maHority graph. +e determine5 for ea2h 2y2le foEnd Fy the networ7 flow solEtion5 the lowest 2ost of all the edges in the 2y2le and add Mthe 2ost 2onstant K minEsO this lowest 2ost to LB1. EnEmerating the paths 2an Fe a2hieBed Fy solBing mEltiple shortest path proFlems5 one for ea2h path in the networ7 flow solEtion. Ea2h shortest path 2ompEtation 2an Fe performed with time 2ompleTity O(n+m) for a dire2ted a2y2li2 graph with n nodes and m edges. Finally5 at the end of ea2h iteration5 we delete all edges from the networ7 whi2h were foEnd in the minS 2ost maTSflow solEtion5 to ensEre that a single edge is only assigned to one cS2y2le. +e haBe also oFserBed that the order in whi2h we perform the networ7 flow iterations 2an haBe an impa2t on the nEmFer of cS2y2les that are foEnd. For ea2h node in the maHority graph we determine the minimEm and maTimEm of their inSdegree and oEt degree. +e sort the nodes Fy de2reasing minimEm of these degrees5 Frea7ing ties Fy de2reasing maTimEm of these degrees5 and perform the networ7 flow iterations in this order. The time to 2ompEte the minS2ost maTSflow solEtion of the networ7 dominates the other parts of the pro2edEre. This 2an Fe 2ompEted in time O M m log U M m + n log n OO for n nodes of maTimEm degree U and m edges MAhEHa5 Magnanti5 and Orlin 1@@cO. Sin2e we may haBe Ep to n iterations5 the time 2ompleTity of this lower FoEnding pro2edEre is O M nm log U M m + n log n OO .

Experiments

some 2onsensEs proFaFility p. +e generate proFlems with an odd nEmFer of Boters5 where ea2h Boter ran7s all alternatiBes with no ties.

Results

The first resElts we present eTplore the effe2t of Barying the 2onsensEs proFaFility and the nEmFer of Boters on proFlem solBing diffi2Eltyc. CPU time resElts for 1? alternatiBes are presented in FigEre ? Mfor all eTperiments5 ?00 proFlems are solBed at ea2h data pointO. As we in2rease the 2onsensEs proFaFility5 the CPU time reUEired to find a Kemeny order de2reases dramati2ally. +hen the 2onsensEs proFaFility is oBer 0.75 most proFlems are solBed withoEt any sear2h. In2reasing the nEmFer of Boters also ma7es proFlems easier to solBe at higher 2onsensEs proFaFilities5 FEt harder to solBe at lower proFaFilities. +e do not haBe an eTplanation for the phenomena at low proFaFilities. FigEre i shows the differen2e in UEality Fetween the solEtion foEnd Fy the greedy heEristi2 pro2edEre and optimal solEtion foEnd Fy Fran2h and FoEnd for the same set of proFlems. As we in2rease the nEmFer of Boters and the 2onsensEs proFaFility5 the greedy heEristi2 finds Fetter solEtions. +e haBe seen similar resElts for larger proFlems. The en2oEraging aspe2t of these resElts is that althoEgh finding a Kemeny order 2an Fe Bery diffi2Elt for proFlems with Bery little 2onsensEs5 it Fe2omes mE2h easier for proFlems with a reasonaFle amoEnt of 2onsensEs that we woEld hope to find in real appli2ations. One UEestion whi2h arises is: what are the range of proFlems we 2an solBe to optimality Fy generating solEtions with a greedy algorithm and 2omparing their Kemeny distan2e to the minS2ost maTS flow Kemeny distan2e lower FoEndj +e 2ompare here oEr greedy heEristi2 pro2edEre with the greedy heEristi2 presented in MCohen5 S2hapire5 and Singer 1@@@O whi2h has an approTimation fa2tor of 2e. FigEres 7 and 8 present resElts eTploring the deBiation Fetween the minS 2ost maTSflow lower FoEnd and the greedy heEristi2 solEtions for finding Kemeny orders for proFlems with ?0 alternatiBes. For 2? Boters5 we 2an proBe optimality of most proFlems with 2onsensEs proFaFility greater than 0.7 Fy 2omparing the solEtion of oEr greedy heEristi2 pro2edEre with the lower FoEnd. +ith only ? Boters5 this 2onsensEs proFaFility in2reases to aroEnd 0.@. The greedy pro2edEre of Cohen et al. does less well at high proFaFilities5 FEt finds Fetter solEtions at low proFaFilities. +e haBe also seen similar resElts for proFlems with different nEmFers of alternatiBes.

Problem generation

+e eBalEated the algorithms on randomly generated proFlems. For ea2h proFlem5 we first generated a total order representing a 2onsensEs ordering of m alternatiBes. +e then generated a preferen2e graph where ea2h one of n Boters agrees with the 2onsensEs ordering regarding the ran7ing of eBery pair of alternatiBes with

c

All eTperiments were rEn on a 1GHb PentiEm III 2ompEter Esing Ckk implementations of the algorithms e The proFlem stEdied in MCohen5 S2hapire5 and Singer 1@@@O also allows Boters to eTpress degrees of preferen2es Fetween alternatiBes.

mean CPU time Mse2O

i0

Conclusions

? Boters 1? Boters 2? Boters c? Boters

?0 e0 c0 20 10 0 0.?

0.i

0.7 0.8 2onsensEs proFaFility

0.@

1

FigEre ?: Mean CPU time to find optimal solEtions for proFlems with 1? alternatiBes5 Esing Fran2h and FoEnd.

mean l deBiation

?

? Boters 1? Boters 2? Boters c? Boters

e c 2 1 0 0.?

0.i

0.7 0.8 2onsensEs proFaFility

0.@

1

FigEre i: Mean l deBiation of greedy heEristi2 pro2edEre solEtions from optimal5 1? alternatiBes. greedy heEristi2 2ohen et al

mean l deBiation

2? 20 1? 10 ? 0 0.?

0.i

0.7 0.8 2onsensEs proFaFility

0.@

1

FigEre 7: Mean l deBiation of greedy solEtions from lower FoEnd5 ?0 alternatiBes5 ? Boters.

mean l deBiation

10

greedy heEristi2 2ohen et al

8 i e 2 0 0.?

0.i

0.7 0.8 2onsensEs proFaFility

0.@

1

FigEre 8: Mean l deBiation of greedy solEtions from lower FoEnd5 ?0 alternatiBes5 2? Boters.

+e haBe presented new eTa2t and heEristi2 algorithms for aggregating preferen2es following the Kemeny rEle. ResElts of a 2ompEtational stEdy indi2ate that 2ompEting Kemeny orders appears to Fe 2ompEtationally eTpensiBe when there is Bery little 2onsensEs Fetween Boters. HoweBer we haBe foEnd that when there is a reasonaFle degree of 2onsensEs5 a Kemeny order 2an Fe foEnd within a short amoEnt of time. FErthermore5 2ompared to mEltiSstage te2hniUEes that 2omFine Foth positional and maHority Boting methods5 the Kemeny orders foEnd Fy these algorithms haBe well stEdied theoreti2al properties. FErther resear2h into improBing lower FoEnding te2hniUEes may signifi2antly eTtend the range of proFlems for whi2h Kemeny orders 2an Fe foEnd.

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