Bounded single-peaked width and proportional representation 1

Transcription

1 Boune single-peke with n proportionl representtion 1 Denis Cornz, Luie Gln n Olivier Spnjr Astrt This pper is evote to the proportionl representtion (PR) prolem when the preferenes re lustere single-peke. PR is multi-winner eletion prolem, tht we stuy in Chmerlin n Cournt s sheme [6]. We efine lustere single-pekeness s form of singlepekeness with respet to lusters of nites, i.e. susets of nites tht re onseutive (in ritrry orer) in the preferenes of ll voters. We show tht the PR prolem eomes polynomil when the size of the lrgest luster of nites (with) is oune. Furthermore, we estlish the polynomility of etermining the single-peke with of preferene profile (minimum with for prtition of nites into lusters omptile with lustere singlepekeness) when the preferenes re nrissisti (i.e., every nite is the most preferre one for some voter). 1 Introution Soil hoie theory els with mking olletive hoies on the sis of the iniviul preferene reltions of set of gents (or voters) over set of lterntives (or nites). In this fiel, n tive strem of reserh els with multi-winner eletions, where one ims t eleting suset of nites rther thn single nite. This ours for instne when eleting n ssemly. In suh sitution, omintoril iffiulty rises: while there re only m possile outputs of singlewinner eletion with m nites, there re ( m κ) possile ssemlies of κ representtives. This iffiulty is often overome y orgnizing κ single-winner eletions over κ sueletortes. With this wy of prtitioning the eletion, it my nevertheless hppen tht the elete ssemly fils to represent minorities [4]: ssume tht the representtives of prty re in seon position for the κ single-winner eletions, then the prty will hve no representtive in the ssemly. Proportionl representtion ims t tkling this issue y performing single multi-winner eletion ensuring tht olletively the voters re stisfie enough y t lest one elete nite. This n e hieve for instne y using Chmerlin n Cournt s sheme [6], where one elets suset of κ nites minimizing misrepresenttion sore. The effetive omputtion of suh winning susets of nites hs een stuie y severl uthors. Proi et l. hve shown tht the prolem is NP-hr in the generl se, ut polynomil for fixe κ [12]. Lu n Boutilier provie polynomil pproximtion lgorithm with performne gurntee (for mximizing representtion sore), n show, on ifferent experimentl tsets, tht it lmost lwys returns n optiml solution [10]. Their setting is nevertheless ifferent from proportionl representtion in politil siene: they im t esigning system le to reommen set of options to group, se on the iniviul preferenes of its memers. Suh system oul e use for instne y onferene orgnizer wishing to selet suset of sushis for the gl inner, se on the iniviul preferenes of the prtiipnts over the vrieties of sushis. Clerly, this ontext uthorizes suoptimlity. Coming k to voting proeures, it is nevertheless importnt to note tht the sores only provie n orinl informtion: if n ssemly A hs misrepresenttion sore 1 while n ssemly B hs misrepresenttion sore 1 + ε, one n only onlue tht A is etter thn B, n not tht B is lose to e s goo s A. Furthermore, in politil setting, it is simply not possile to elet n ssemly without gurnteeing tht it is the true winner. To our 1 This pper ppere in the proeeings of ECAI 2012.

2 knowlege, the only generl ext pprohes propose for proportionl representtion in Chmerlin n Cournt s sheme re se on integer progrmming s the ones y Potthoff n Brms [11] n y Blinski [1] (in this ltter referene, the formultion ws tully propose for the κ-mein lotion prolem, whih is equivlent to the proportionl representtion prolem [12]). The solution of these IP formultions might of ourse tke exponentil time in the worst se. Very reently, Betzler et l. propose n extensive investigtion of prmeterize omplexity results for the prolem [4]. Besies, they estlishe tht the prolem eomes polynomil when the preferenes re single-peke [5]. Single-pekeness is the most populr omin restrition in soil hoie theory. In single-winner eletions, it mkes it possile to overome Arrow s impossiility theorem (tht sttes tht no voting rule n simultneously fulfill set of si xioms). In prtiulr, there lwys exists Conoret winner (i.e., nite who is preferre to ny other nite y mjority of voters) if preferenes re single-peke. Suh preferenes re typilly enountere in politil siene. Intuitively, preferenes re single-peke when 1) ll voters gree on left-right xis on the nites refleting their politil onvitions, n 2) the preferenes of ll voters erese long the xis when moving wy from their preferre nite to the right or left. Nevertheless, this onition on preferenes n e it restritive when severl nites shre similr opinions (e.g. they elong to the sme prty) sine it is unlikely tht the preferenes of ll voters re single-peke on this suset of nites. We therefore stuy new omin restrition, lustere single-pekeness, where singlepekeness hols on susets of nites (prties or more generlly lusters), n not within lusters. The nites elonging to the sme luster re rnke onseutively in the preferenes of ll voters, though not neessrily in the sme orer. Given prtition of the nites into lusters suh tht the preferenes re lustere single-peke, the with of the prtition is the size of the lrgest luster minus one. Note tht, for given set of iniviul preferene reltions, severl prtitions into lusters n e omptile with lustere single-pekeness: we ll single-peke with the minimum with mong ll possile prtitions of nites into lusters. We show tht the single-peke with is omputle in polynomil time if preferenes re nrissisti, n tht oune single-peke with mkes it possile to esign polynomil time solution lgorithm for the proportionl representtion prolem. Note tht the sme strutures hve een stuie y Elkin et l. [7], uner nother terminology (in prtiulr, lusters re lle lone sets). Their min onern is not to stuy how lustere single-pekeness n e use to etermine the winner of n eletion, ut they show interesting onnetions with PQ-trees, n use them to esign n lgorithm to ompute prtition of the nites into (s mny s possile) lusters. The links etween their work n ours will e etile in Setion 4. The pper is orgnize s follows. We first formlly introue the proportionl representtion prolem n lustere single-pekeness (Set. 2). Then we present ynmi progrmming proeure for solving the proportionl representtion prolem when the preferenes re lustere single-peke (Set. 3). A key prmeter for the effiieny of the proeure is the with of the prtition into lusters. We therefore stuy the omplexity of etermining the single-peke with of set of iniviul preferene reltions (Set. 4), n show the polynomility of the prolem for nrissisti preferenes. 2 Preliminries 2.1 Proportionl representtion Let V e set of n voters n C set of m nites. Let P e n m n preferene profile mtrix over C, tht is, eh nite ppers extly one in eh olumn. So the set of olumns of P is the set V n eh olumn v is the preferene reltion of voter v. We enote y r(v, ) the rnk of nite in the preferenes of voter v, n y x v y the preferene for y over x. A non-eresing misrepresenttion funtion µ : {1,..., m} N is efine suh tht

3 µ(r(v, )) is the misrepresenttion vlue of for v. The proportionl representtion prolem ims t etermining suset S C of κ nites suh tht the totl misrepresenttion sore is minimize. In Chmerlin n Cournt s sheme, the soring funtion s : 2 C N is efine s follows: s(s) = min µ(r(v, )) S v V The proportionl representtion prolem n then e simply written: min S =κ s(s). The following exmple illustrtes the vlue of using Chmerlin n Cournt s sheme. Exmple 1. Consier proportionl representtion prolem with 6 voters 1, 2, 3, 4, 5, 6 (inies of the olumns) n 4 nites,,,, n the following preferene profile mtrix: P = Assume tht the misrepresenttion funtion is µ(r) = r 1. If κ = 2, then the possile susets n sores re (for simpliity stns for {, }): The optiml solution is suset with sore 1. With suh solution, only one voter is not represente y her preferre nite (ut y her seon hoie). Assume now this multi-winner eletion is ivie into two single-winner eletions, nmely n eletion L etween n for voters 1, 2, 3, n n eletion R etween n for voters 4, 5, 6. The winner of eletion L (resp. R) is (resp. ). Consequently, the winning solution is, whih is the worst one oring to the misrepresenttion sores! 2.2 Clustere single-pekeness Definition 1. Let C = (C 1,..., C q ) e n orere prtition of C into q non-empty susets (lle lusters). Preferene profile mtrix P is lustere single-peke with respet to C if for ll v V there exists n inex p in {1,..., q} suh tht: i < j < p x v y v z p < j < i x v y v z for ll x C i, y C j n z C p. For voter v, we ll C p the pek of v, whih mens tht ny nite in C p is preferre to ny nite in C \ C p. This efinition oinies with usul single-pekeness when C i = 1 for ll i. The only nite in C p is then the most preferre one. Exmple 2. Coming k to Exmple 1, it n e esily seen tht the preferenes re not singlepeke w.r.t. xis (,,, ), y onsiering Figure 1 where eh urve represents preferene rnking of voter, nmely voters 1, 2, 6. For eh urve n eh nite on the X-xis, the vlue on the Y-xis is the rnk in the orresponing preferene rnking (the etter the rnk the higher the point). Preferenes re single-peke w.r.t. n X-xis iff ll urves hve single pek. This is not the se in the left grph sine the urve of voter 6 (in ol) spikes own for n then spikes up for. More generlly, it n e shown tht the preferenes in Exmple 1 re not single-peke, whtever permuttion of nites on the X-xis is onsiere. However, the preferenes re lustere single-peke with respet to ({, }, {}, {}), enote y (,, ) for simpliity. Note tht n re jent in ll preferene rnkings, whih is neessry onition to e lustere (ut not suffiient for lustere single-pekeness!). A preferene profile is lustere single-peke with respet to n orere prtition (C 1,..., C q ) iff it is single-peke when onsiering eh suset C i s single nite. In the exmple, introuing luster {, } mounts to onsiering n s single nite. The preferene profile mtrix eomes then the one inite on the right-hn sie of Figure 1. In the grph on the right, one n oserve tht the preferenes eome then single-peke, i.e. they re lustere single-peke with respet to (,, ).

4 Dynmi Progrmming Figure 1: Clustere single-pekeness. We now present ynmi progrmming lgorithm tht generlizes the one propose y Betzler et l. for single-peke preferenes [4]. Let P (i, C, k) enote the suprolem where ll nites in C C re me mntory n one selets k C nites in C 1... C i. For the onveniene of the reer, we riefly rell the reursion sheme of the proeure propose y Betzler et l., with n lterntive proof. Assume tht the preferenes re single-peke with respet to xis (x 1,..., x m ) (i.e. lustere single-peke with respet to (C 1,..., C m ), where C i = {x i } i). Let z(i, k) enote the optiml sore for prolem P (i 1, {x i }, k), where one selets x i n k 1 nites mong {x 1,..., x i 1 } (the i 1 leftmost nites on the xis). The uthors use the following reursion: z(i, k) = min j [k 1 i 1] { z(j, k 1) } mx{0, µ(r(v, x j )) µ(r(v, x i ))} v V The optiml sore for suset of κ nites is then min i {κ m} z(i, κ). The vliity of the reursion n e estlishe y showing tht seleting suset of k nites, inluing x j n x i (mntory nites), in {x 1,..., x j, x i } (prolem P (j 1, {x j, x i }, k)) mounts to seleting k 1 nites, inluing x j, in {x 1,..., x j } (prolem P (j 1, {x j }, k 1)). Inee, it reues to omputtions on the sme minor of the preferene profile. Definition 2. Any preferene profile mtrix tht epits the iniviul preferenes of suset V V of voters over suset C C of nites is lle minor n enote y P(V, C ). The voters n e prtitione into two sets: the set V [1,j 1] of voters whose pek x p is in {x 1,..., x j 1 }, n the set V [j,m] of voters whose pek x p is in {x j,..., x m }. Both prolems P (j 1, {x j, x i }, k) n P (j 1, {x j }, k 1) mount to omputtions in the sme minor: Prolem P (j 1, {x j, x i }, k): ll voters in V [j,m] n e elete from the preferene profile mtrix sine their preferre nite mong {x 1,..., x j, x i } is either x i or x j, tht re mntory, n therefore the preferenes of these voters ply no role in the etermintion of the optiml solution to P (j 1, {x j, x i }, k). Furthermore, ll voters in V [1,j 1] prefer x j to x i sine their pek is to the left of x j, n therefore nite x i plys no role sine x j is mntory. Consequently, the prolem reues to seleting k 1 nites, inluing x j, oring to minor P(V [1,j 1], {x 1,..., x j }).

5 Prolem P (j 1, {x j }, k 1): for ll voters in V [j,m], nite x j is neessrily the most preferre one in {x 1,..., x j }. Sine nite x j is mntory, ll voters in V [j,m] n e elete from the preferene profile mtrix. The prolem reues then to seleting k 1 nites, inluing x j, oring to minor P(V [1,j 1], {x 1,..., x j }). The two prolems P (j 1, {x j, x i }, k) n P (j 1, {x j }, k 1) re thus equivlent, whih estlishes the vliity of the reursion. We now show how this reursion sheme n e extene to hnle lustere single-peke preferenes. Assume tht the preferenes re lustere singlepeke with respet to n orere prtition (C 1,..., C q ). Let z(i, C i, k) enote the optiml sore when nites in C i C i re mntory, nites in C i \ C i re forien, n one selets k C i nites in C 1... C i 1. In our setting, the reursion n e written s follows: { z(i, C i, k) = min min z(j, C j, k C i ) j [1 i 1] C j Cj C j mx{0, min µ(r(v, y)) min µ(r(v, x))} y C j x C i v V where z(i, C i, k) = + if C i > k or C 1... C i 1 < k C i. The optiml sore for suset of κ nites is then: min i [1 q] min z(i, C C i, κ) i Ci C i } (1) The proof of the reursion is similr to the one in the single-peke se. It mounts to estlishing the equivlene of prolems P (j 1, C j C i, k) n P (j 1, C j, k C i ), y onsiering prtition of V into the set V [1,j 1] of voters whose pek is in {C 1,..., C j 1 } n the set V [j,q] whose pek is in {C j,..., C q }: Prolem P (j 1, C j C i, k): ll voters in V [j,q] n e elete from the preferene profile mtrix sine their preferre nite mong C 1... C j 1 C j C i is either in C i or in C j. All voters V [1,j 1] prefer nite in C j to nite in C i sine their pek is to the left of C j. Consequently, the prolem reues to seleting k C i nites, inluing nites in C j, oring to minor P(V [1,j 1], C 1... C j 1 C j ). Prolem P (j 1, C j, k C i ): for ll voters in V [j,q], the most preferre nite in C 1... C j 1 C j neessrily elongs to C j. The voters n therefore e elete from the preferene profile. The prolem reues then to seleting k C i nites, inluing nites in C j, oring to minor P(V [1,j 1], C 1... C j 1 C j ). Both prolems re thus equivlent, whih estlishes the vliity of the reursion. Algorithm 1 esries the ensuing ynmi progrmming proeure. Algorithm 1: Dynmi progrmming for i = 1,..., q o for C i C i with C i κ, C i o z(i, C i, C i ) = v V min x C i µ(r(v, x)) for i = 2,..., q o for C i C i with C i κ, C i o for k = C i + 1,..., min{κ, C i + i 1 j=1 C j } o ompute z(i, C i, k) y Eqution 1 return min min z(i, i [1 q] C i Ci, C i, κ) C i Exmple 3. For simpliity, set {, } is enote y in this exmple, n {, } {, } y. Consier proportionl representtion prolem with 6 nites,,,, e, f hving lustere single-peke preferenes with respet to (,, e, f). Let us stuy how mny triples of

6 nites re exmine y the proeure when omputing z(4, f, 3). Given r susets S 1,..., S r of nites, let us enote y opt{s 1,..., S r } suset in rg min i s(s i ). The following omputtion is performe y the proeure: ( { }) f, fopt{, }, fopt{, }, z(4, f, 3) = s opt f, fopt{e, e, e, e} Therefore 5 susets re exmine (three of the four opt opertions hve een performe uring the previous itertions) while there re 10 susets of rinlity 3 inluing f. For smll single-peke with, the omputtionl svings eome of ourse more n more signifint when the size of the instne inreses. Atully, the following omplexity nlysis shows tht the ynmi progrmming proeure is polynomil for oune singe peke with. Eqution 1 requires inee omputtionl time within O(nqt2 t ) where t = mx i C i 1. Furthermore, the numer of ompute terms z(i, C i, k) is upper oune y q2t κ. Therefore the running time of the proeure is within O(nq 2 t2 2t κ), whih mounts to O(nm 3 ) for oune singlepeke with t (we rell tht q m n κ m). Theorem 1. The proportionl representtion prolem over oune single-peke with preferenes is polynomil. The omplexity nlysis shows tht mx i C i is key prmeter for the effiieny of the lgorithm. Note tht there lwys exists n orere prtition for whih the preferenes re lustere single-peke: in the worst se, it is suffiient to onsier the prtition (C). It is nevertheless interesting from n lgorithmi viewpoint to hve n orere prtition where eh suset inlues few nites. Two ses n our: either the prtition is known in vne (for instne, when the nites inite their ffilition to politil prty n the preferenes of the voters re onsistent with the isplye ffilitions) or it is unknown. In oth ses, it is esirle to e le to ompute n orere prtition omptile with lustere single-pekeness n suh tht mx i C i is minimize. In the next setion, we show the polynomility of this prolem for nrissisti preferenes [3, 13]. 4 Single peke with We ll with of n orere prtition (C 1,..., C q ) the vlue mx i C i 1. Given preferene profile mtrix, we ll single-peke with the minimum with mong ll orere prtitions omptile with lustere single-pekeness. This n e seen s istne mesuring ner-single-pekeness (the single-peke with is inee equl to 0 for single-peke preferenes). Note tht this shoul not e onfuse with other istne mesures tht hve een propose in the literture, suh s the numer of voters to remove to mke profile single-peke [9]. Exmple 4. Consier the preferene profile mtrix P represente in Figure 2, where the preferenes re not single-peke. It is esy to hek tht they re nevertheless lustere single-peke with respet to orere prtition (, efg,, h) (see the left prt of the figure, where the susets of the prtition re enirle), whose with is {e, f, g} 1 = 2. However the preferenes re lso lustere single-peke with respet to (, f, eg,,, h) (right prt of the figure). The singlepeke with of this preferene mtrix is thus 1. Bllester n Heringer [2] reently showe tht single-pekeness n e lost just euse of the existene of two voters n four nites, or three voters n three nites. Conversely, they showe tht if profile is not single-peke there must exist set of two voters (resp. three) whose preferenes over four nites (resp. three) re not single-peke. More preisely, the uthors hrterize single-pekeness with the following two onitions:

7 f f g g e h f e P = g e h f e g P = g e e g f f h h h h Figure 2: Single-peke with. Worst-restrition: Given triple V V of voters n triple C C of nites, let L(V, C ) e the set of ll nites rnke lst in C y t lest one voter in V. The worst-restrition onition hols if L(V, C ) < 3 for ll triples V n C. α-restrition: the α-restrition onition hols if there o not exist two voters v n v n four nites w, x, y, n z suh tht their preferenes over w, x n z re opposite (w v x v z n z v x v w) n the voters gree out the preferene for y over x (y v x n y v x). Interestingly, these onitions mount to foriing five minors in the profile P (Lemm 1). In this formlism, we propose here shorter proof of the hrteriztion result of Bllester n Heringer. Our proof is se on the polynomil lgorithm propose y Esoffier et l. [8] to etermine if profile is single-peke with respet to some xis. This lgorithm runs in time O(mn) improving on the O(mn 2 ) lgorithm propose y Brtholi n Trik [3]. Before stting Lemm 1, let us present the lgorithm of Esoffier et l. It works reursively n tkes s rguments the left prt (x 1,..., x i ) n the right prt (x j,..., x m ) of the xis uner onstrution. A thir rgument is the suset C of nites whih remins to e positione on the xis. This lgorithm returns n xis omptile with P or proves tht the preferenes re not single-peke (y rising ontrition etween voters). The reursion is me possile y the ft tht single-pekeness over P implies single-pekeness over ny of its minors. It hevily uses the property tht nites rnke lst in the preferenes re neessrily t the extremities of the xis. At eh step of the lgorithm, one nite x or two nites x n y re rnke lst in P(V, C ) n will e positione in x i+1 or x j 1 on the xis. There is ontrition if nite hs to e ple in two ifferent positions (oring to the preferenes of two voters). These positions epen on the wy x n y re positione with respet to x i n x j in the preferenes of ll the voters. The whole proeure is etile in Algorithm 2. The initil ll is Mke-xis(C, (), ()). Before presenting Lemm 1 (on whih our lgorithm to ompute single-peke with strongly relies), we nee to introue the notion of isomorphi minors. A minor P is isomorphi to P if there exists ijetion φ suh tht P n P re ientil up to olumn permuttion if one renmes every nite x in P s φ(x). For instne, preferene profile mtrix P elow is isomorphi to P (tke φ() =, φ() =, φ() = n permute the olumns). P =, P = Definition 3. A minor is lle forien if it is isomorphi to one of the following profiles: T 1 =,T 2 =, F 1 =, F2 =, or F3 =.

8 Algorithm 2: Mke-xis(C,(x 1,..., x i ),(x j,..., x m )) if C = then return (x 1,..., x i, x j,..., x m ) if C = {x} then return (x 1,..., x i, x, x j,..., x m ) L nites rnke lst in P(C, V ) y t lest one voter if L = {x} then y nite in C \ {x} / x v y, v / if L 3 then return not single-peke for v = 1,..., n o if L = {x, y} then let x v y (w.l.o.g) if x i v x v x j v y or x i v x v y v x j then if no ontrition then x i+1 x ; x j 1 y else return not single-peke if x j v x v x i v y or x j v x v y v x i then if no ontrition then x i+1 y ; x j 1 x else return not single-peke if L = {x} then if x = x i+1 then Mke-xis(C \ {x},(x 1,..., x i, x i+1 ),(x j,..., x m )) else Mke-xis(C \ {x},(x 1,..., x i ),(x j 1, x j,..., x m )) Mke-xis(C \ {x, y},(x 1,..., x i, x i+1 ),(x j 1, x j,..., x m )) Lemm 1. P is single-peke iff it hs no forien minor. Proof (sketh) Neessity: it suffies to hek tht none of the five forien minors is singlepeke, sine the single-pekeness property is lose uner tking minors. Suffiieny: run Algorithm 2 n suppose tht it returns not single-peke. If it stops t Line 5, then P hs minor T 1 or T 2. Otherwise it stops t Line 10 or 13 n P hs minor F 1, F 2 or F 3. The rest of the setion is evote to the prolem of etermining n orere prtition of minimum with mong the ones tht re omptile with lustere single-pekeness. Note tht Elkin et l. [7] stuie losely relte prolem, nmely fining n orere prtition (C 1,..., C q ) mximizing q. Both prolems re not equivlent, s shown y the following exmple. Exmple 5. Consier the preferene profile mtrix P: x y v P = x v y Both prtitions (v,, x, y) n (v,,, xy) mximize q n re omptile with lustere single-pekeness, ut (v,, xy) is the only prtition tht minimizes the single-peke with. However, for nrissisti preferenes [3, 13], one n show tht the lgorithm propose y Elkin et l. for their prolem returns n orere prtition of minimum with. Nevertheless, our pproh proves tht there is unique (up to reversl) orere prtition mximizing q. Preferenes re si to e nrissisti when eh nite is most preferre y some voter. In politis, s soon s the nites re voting, this ssumption seems resonle. In the reminer, we prove the following result: Theorem 2. Fining the single-peke with is polynomil if P is nrissisti. For eh voter v V n nites, C we enote I v (, ) := { C : = or = or v v or v v } the set of nites etween n in the preferenes of voter v.

9 By onvention, I v (, ) = {}. A suset I of C is lle n intervl of P if for eh v V, one n hoose two nites, I suh tht I = I v (, ). This efinition oinies with the notion of lone set stuie y Elkin et l. [7]. Notie tht the set of intervls I of P is not lose uner tking susets. Nevertheless, it is lose uner intersetion [7]. Given, C, the miniml intervl w.r.t. inlusion tht ontins n is thus uniquely efine: we enote it y I(, ). The following lemm will prove useful in orer to esign n lgorithm le to ompute prtition omptile with lustere single-pekeness. For simpliity, if P is isomorphi to P for φ, we write I(x, y) for I(φ(x), φ(y)). Lemm 2. The following properties hol: If T 1 is minor of P, then I(, ) = I(, ) = I(, ); If T 2 is minor of P, then I(, ) n I(, ) inlue I(, ); If F 1 is minor of P, then I(, ), I(, ), I(, ), I(, ) n I(, ) inlue I(, ); If F 2 is minor of P, then I(, ), I(, ), I(, ), I(, ) n I(, ) inlue I(, ); If F 3 is minor of P, then - I(, ), I(, ), I(, ) n I(, ) inlue I(, ), - I(, ), I(, ), I(, ) n I(, ) inlue I(, ). Proof (sketh) Let v e the voter of the first olumn of T 1. Sine I v (, ), it follows tht I(, ). Thus I(, ) n I(, ) I(, ). The seon olumn gives I(, ) n I(, ) I(, ), n the thir olumn gives I(, ) n I(, ) I(, ). Finlly I(, ) = I(, ) = I(, ). The proofs for the four other forien minors go long the sme lines. We propose greey lgorithm to ompute the lusters of n orere prtition omptile with lustere single-pekeness. This lgorithm proees y ontrting nites so tht no forien minor remins in the preferene profile mtrix. Contrting two nites, C onsists in ontrting I(, ). Contrting n intervl I onsists in ollpsing ll nites in I into single luster nite. This mounts to hoosing representtive in I n removing from P ll the other nites in I. For instne, ontrting n in P yiels luster {,, e} (sine I(, ) = {,, e}) n profile P : P = e e P = Notie tht the preferene profile mtrix P otine y ontrting n intervl of P is minor of P. Note lso tht if I, J I re two intervls of P, then the two minors otine from P either y ontrting I then J, or y ontrting J then I oinie (even if I n J overlp). Besies, if P is minor of P n F is minor of P, then F is lso minor of P. The greey proeure is etile in Algorithm 3. The termintion follows from the ft tht ontrting nites nnot rete new forien minors. Exmple 6. Consier the preferene profile mtrix P in Figure 2 n pply Algorithm 3, ssume tht it etets: f h g g the minor n then the minor g g e e g h f

10 Algorithm 3: Greey lgorithm let P e minor of P isomorphi for φ to: T 1. Contrt φ() n φ() T 2. Contrt φ() n φ() F 1. Contrt φ() n φ() F 2. Contrt φ() n φ() F 3. Contrt φ() n φ(), or φ() n φ() pply these ontrtions (non-eterministilly) until no forien minor remins. The first minor is isomorphi to T 2 for φ() = g, φ() = n φ() =. Therefore nites n re ontrte. The seon minor is isomorphi to F 1 for φ() = f, φ() = e, φ() = h n φ() = g. Therefore nites e n g re ontrte. Tking nite (resp. e) s the representtive of luster {, } (resp. {e, g}), the preferene profile eomes: P = f e f h e e f h h There is no more forien minor in the preferene profile, n thus the greey proeure stops. The lusters re {, } n {e, g}. This lgorithm is polynomil sine the forien minors n e enumerte in O(m 3 n 3 ) for T 1, T 2, n O(m 4 n 2 ) for F 1, F 2, F 3. The lusters ientifie y the lgorithm elong to n orere prtition omptile with lustere single-pekeness. The orere prtition itself n e ompute y pplying Algorithm 2 on the finl preferene profile. Coming k to Exmple 6, Algorithm 2 returns xis (h,,, e, f, ) on P, whih orrespons to the orere prtition (h,,, eg, f, ) sine e (resp. ) is the representtive of {e, g} (resp. {, }). This is n orere prtition of minimum with for this profile. However, in the generl se, the with of the returne orere prtition is not gurntee to e miniml. We now show how to refine the greey proeure to get n orere prtition of minimum with when preferenes re nrissisti. To this en, we introue notion of similrity etween nites tht enles us to ientify neessry n suffiient ontrtions for lustere single-pekeness. Definition 4. Two nites n re si to e similr if they elong to the sme luster in ll orere prtitions w.r.t. whih P is lustere single-peke. It results from Lemm 2 tht the following properties hol: If T 1 (T 2 ) is minor of P, then n ( n ) re similr; If F 1 (F 2 ) is minor of P, then n ( n ) re similr; If F 3 is minor of P, then if I(, ) I(, ), then n re similr; if I(, ) I(, ), then n re similr. These properties imply tht ll ontrtions ut one (F 3 ) in the greey lgorithm over nites whih elong to the sme luster in ny orere prtition of minimum with. The only se of forien minor tht nnot e remove from P y ontrting similr nites is thus F 3 when I(, ) n I(, ) interset properly, i.e. I(, ) I(, ) n I(, ) I(, ). We ll suh forien minors miguous. If one fins n miguous minor M, t lest two nites in M

11 must e in the sme luster. Nevertheless the single-peke with of n orere prtition epens on the hoie of the nites to ontrt. Furthermore this hoie oes not only epen on the mximum numer of nites involve in the possile intervl ontrtions. For instne, onsier the preferene profile mtrix P of Exmple 5 whih hs the following minor: v The smllest ontrtion implie y the given minor woul e to ontrt n (2 nites in the intervl). But (v,, xy), where n re not in the sme luster, is the only minimum with orere prtition omptile with lustere single-pekeness. For this reson, the greey lgorithm my fil to provie lusters elonging to n orere prtition of minimum with. However when preferenes re nrissisti, no miguous minor n exist in the preferene profile mtrix. Assume inee tht there exists minor M isomorphi to F 3 for φ. Sine P is nrissisti, nite φ() is the most preferre one for some voter v, n onsequently: φ() v φ() v φ() or φ() v φ() v φ(). Therefore we hve I(, ) I(, ) or I(, ) I(, ). The minor is thus unmiguous. To otin n optiml greey lgorithm for nrissisti preferenes, ontrtion relte to F 3 must then e moifie s follows: let v e voter whose most preferre nite is φ() if φ() v φ() v φ() then ontrt φ() n φ() else ontrt φ() n φ(). Furthermore, the greey lgorithm uses neessry n suffiient ontrtions to mke the profile lustere single-peke, n thus prtition (C 1,..., C q ) of minimum with is lerly unique uner mximizing the numer q of lusters. 5 Conlusion An interesting open question is whether there exists generl polynomil lgorithm to ompute the single-peke with (not neessrily in the nrissisti se). Apting the P Q-trees se lgorithm of Elkin et l. [7] to our prolem oul work. Besies, the onept of minors oul e tool for fining short vliity proof. v 6 Aknowlegements We wish to thnk the nonymous reviewers of ECAI n COMSOC for their fruitful omments on n erlier version of the pper. This reserh hs een supporte y the ANR projet ComSo (ANR-09-BLAN-0305). Referenes [1] M. Blinski. On fining integer solutions to liner progrms. In Pro. IBM Sientifi Computing Symposium on Comintoril Prolems, pges , [2] M. A. Bllester n G. Heringer. A hrteriztion of the single-peke omin. Soil Choie n Welfre, 36(2): , [3] J. Brtholi III n M. A. Trik. Stle mthing with preferenes erive from psyhologil moel. Opertions Reserh Letters, 5(4): , 1986.

12 [4] N. Betzler, A. Slinko, n J. Uhlmnn. On the omputtion of fully proportionl representtion. Aville t SSRN: [5] D. Blk. The Theory of Committees n Eletions. Cmrige University Press, [6] J. R. Chmerlin n P. N. Cournt. Representtive eliertions n representtive eisions: Proportionl representtion n the Bor rule. The Amerin Politil Siene Review, 77(3):pp , [7] E. Elkin, P. Fliszewski, n A. Slinko. Clone strutures in voters preferenes. ArXiv e- prints, , [8] B. Esoffier, J. Lng, n M. Öztürk. Single-peke onsisteny n its omplexity. In Proeeings of 18th Europen Conferene on Artifiil Intelligene (ECAI 2008), pges IOS Press, [9] P. Fliszewski, E. Hemspnr, n L. A. Hemspnr. The omplexity of mnipultive ttks in nerly single-peke eletortes. In TARK, pges , [10] T. Lu n C. Boutilier. Bugete soil hoie: From onsensus to personlize eision mking. In IJCAI, pges , [11] R. Potthoff n S. Brms. Proportionl representtion: roening the options. Journl of Theoretil Politis, 10(2):pp , [12] A. Proi, J. Rosenshein, n A. Zohr. On the omplexity of hieving proportionl representtion. Soil Choie n Welfre, 30(3): , [13] M. A. Trik. Reognizing single-peke preferenes on tree. Mthemtil Soil Sienes, 17(3): , Denis Cornz University Pris Duphine LAMSADE Ple u Mrhl e Lttre e Tssigny Pris eex 16, Frne Emil: enis.ornz@uphine.fr Luie Gln University Pris Duphine LAMSADE Ple u Mrhl e Lttre e Tssigny Pris eex 16, Frne Emil: luie.gln@uphine.fr Olivier Spnjr University of Pierre et Mrie Curie LIP6 4 Ple Jussieu Pris Ceex 05, Frne Emil: olivier.spnjr@lip6.fr