Global Preferential Consistency for the Topological Sorting-Based Maximal Spanning Tree Problem a

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1 Glol Preferentil Consisteny for the Topologil Sorting-Bsed Mximl Spnning Tree Prolem Rémy-Roert Joseph * Astrt. We introdue new type of fully omputle prolems, for DSS dedited to mximl spnning tree prolems, sed on dedution nd hoie: preferentil onsisteny prolems. To show its interest, we desrie new ompt representtion of preferenes speifi to spnning trees, identifying n effiient mximl spnning tree su-prolem. Next, we ompre this prolem with the Preto-sed multiojetive one. And t lst, we propose n effiient lgorithm solving the ssoited preferentil onsisteny prolem. Keywords: Consisteny enforing, Intertive methods, Multiojetive omintoril optimiztion, Preferenes ompt representtion, Spnning tree. 1 INTRODUCTION Given n undireted grph G = (V, E) with V the verties nd E the edges, spnning tree x of G is onneted nd yli prtil grph of G. x is then lwys omposed with V 1 edges. We denote y S ST(G) the spnning trees set of G. For short, we write: e x, with e E, to sy: e is n edge of the spnning tree x. More generlly, we will ssimilte x to its edges set. The lssil prolem of mximum spnning tree ( ST/Σu/OPT) is defined s follow: ST/Σu/OPT: Given n undireted grph G = (V, E) nd utility u(e) ssoited with eh edge e E, the result is fesile spnning tree x of G, mximizing the sum of utilities of edges in x, if suh tree exists. Otherwise, the result is no. Severl onsisteny prolems hve een reently investigted on spnning trees. On the one hnd, we note the onsisteny prolem ssoited with fesile spnning trees of grph [25]. Other investigtions pointed out onsisteny ssoited with weighted spnning trees [8], nd mximum spnning tree [9]. On the other hnd, numerous lol onsisteny prolems omining lssil spnning tree prolems with other onstrints hve een investigted. For exmple, the dimeter onstrined minimum spnning tree prolem (DCMST) [16]. Within non-onventionl preferenes, the sitution is rdilly different. Very few onsisteny spnning tree prolems hve een investigted in literture. We ite lol onsisteny prolem proessed for the roust spnning tree prolem with intervl dt (RSTID) [1]. Yet, the most of omintoril prolems from the rel prtil world require the modeling of impreision or unertinty, multiple divergent viewpoints nd onflits mngement, to wholly ssess the solutions nd to identify the est ompromise ones. These singulrities require more omplex modeling of preferenes [27, 21]. For now some dedes, the OR/CP ommunity srutinizes omintoril prolems enling non-onventionl glol preferenes. Thus, we ttended to the flowering of gret numer of pulitions deling with multiojetive omintoril optimiztion prolems (see [10, 3] for surveys). Nevertheless, very few rtiles delt with omintoril prolems with purely ordinl nd/or intrnsitive preferentil informtion. We mention the reent investigtions in the sope of (i) deision theory with mximl spnning trees nd mximl pths in digrph [18], (ii) gme theory with stle mthings (see [20] for survey), (iii) lgeri omintoril optimiztion [28, 5], or (iv) rtifiil intelligene with some onfigurtion prolems [4, 14] nd with heuristi serh lgorithms [17, 14]. We deide to ring nother stone to this uilding, with the onept of preferentil onsisteny pplied to the topologil sorting-sed mximl spnning trees prolem. The deision prolemti of finding suitle preferred solution is semi-strutured: in the generl se (eyond totl preorders), preferred solution fitted to the deision-mker nnot e only identified from the implemented preferentil informtion. Preferred solutions re not ll equivlent, some re prtilly omprle others re inomprle, nd sometimes, there exists no optiml or mximl solution [27, 21]. To investigte these semi-omputle prolems, we will use the onept of Deision Support System (DSS) to explore the preferred solutions set. This explortion n e hieved other thn y uilding itertively new preferred solutions s usully in multiriteri optimiztion ; For exmple, y desriing this preferred set with the set of vlues present in t lest one preferred solution. The notion of onsisteny, defined in Constrint Progrmming, gthers the theoretil surrounding of this desriptive pproh of impliit sets. This is retive [26] nd dedutive pproh of solving; In polynomil numer of tions (removings, instntitions nd ktrkings), the user leds to preferred solution. Consequently, fter n introdution on preferene reltions ( 2.1), we mke rief presenttion on ompt representtion of preferenes ( 2.2). We next point out generliztion of the mximum spnning trees prolem: the mximl spnning trees prolem ( 3.1). So, we introdue ( 3.2) preferentil onsisteny, Aepted in: Workshop on modeling nd solving prolems with onstrints (ECAI 2008-W31), Ptrs, Greee, 21 july 2008 : * Université des Antilles et de l Guyne / Institut d'etudes Supérieures de Guyne, Frenh Guyn, Frne, e-mil : remy.joseph@rmil.om 30

2 i.e. templte redefining onsisteny in order to tke into ount of peulirities of omintoril prolems exploiting non-onventionl preferenes, followed y its using on the mximl spnning trees. In generl, most of relevnt omputle prolems supporting the initil deision prolem re intrtle. Aordingly, we point out n esy suitle mximl spnning trees su-prolem ( 4), sed on ompt preferene representtion inspired y topologil sorting ( 4.1). In 4.2, we give n exmple of using in the multiriteri ontext nd we ompre this su-prolem with the Preto-sed multiojetive version. Next, we design glol preferentil onsisteny lgorithm ( 5) dedited to it. We onlude ( 6) with some perspetives. 2 PREREQUISITES IN DECISION THEORY Throughout this rtile, we tke ple t very generl strtion level, where glol preferenes re represented y non omplete, intrnsitive nd even yli inry reltion on the solutions spe, ut enling mximl set (there exist no solution stritly preferred to ny of them). Here re some definitions: 2.1 Preferene reltion Given non-empty finite set S, (risp inry) preferene reltion [23, 27, 21] of n individul on S is reflexive inry reltion on S ( S S nd x S, (x, x) ) trnslting some judgments of this individul onerning his preferenes etween the lterntive elements of S. For every ouple of elements x nd y of S, the ssertion «x y» is equivlent to «(x, y)» nd mens tht «x is t lest s good qulity s y for onsidered individul». A preferene reltion rries out prtition of S S into four fundmentl reltions: (indifferene) x y ( x y nd y x ) for ll x, y S (strit preferene) x y ( x y nd not(y x) ) for ll x, y S (strit version) x y y x for every x, y S (inomprility) x y ( not(x y) nd not(y x) ) for every x, y S Preferene reltions defined on finite set formlly orrespond with the onept of simple direted grphs (shortly digrphs). Aordingly, the grphil representtion of digrphs will llow us to illustrte our investigtion. For every non-empty A S, the restrition of to A is the preferene reltion A defined s follow: A = {(x, y) A A, suh tht: x y}. By use, we do not speify the restrition, the ontext enling to identify the trgeted suset of S. A preferene reltion is: trnsitive iff [x y nd y z] x z, for ll x, y, z S qusi-trnsitive iff [x y nd y z] x z, for ll x, y, z S iff the strit preferene reltion is trnsitive P-yli iff t > 2 nd x 1, x 2,, x t S, [x 1 x 2 x t] not(x t x 1) iff hs no iruit of strit preferene. n equivlene reltion iff it is reflexive 1, symmetri nd trnsitive prtil preorder iff it is reflexive nd trnsitive 1 Mention tht inry reltion is symmetri iff x y y x, for ll x, y S; ntisymmetri iff x y not(y x), for ll x, y S with x y; nd omplete (or totl) iff x y or y x, for ll x, y S nd x y. omplete (or totl) preorder iff it is reflexive, trnsitive nd omplete omplete (or totl) order iff it is reflexive, trnsitive, ntisymmetri nd omplete Given finite non-empty set S strutured y preferene reltion, the mximl set (or effiient set) of S ording to, denoted M(S, ), is the suset of S verifying: M(S, ) = {x S y S, not(y x)}; while the optiml set of S ording to, denoted B(S, ), is the suset of S verifying: B(S, ) = {x S y S, x y}. Of ourse, there exists other hoies of xioms identifying preferred (i.e. est qulity, or est ompromise) solutions from preferene reltion, nd we refer to [11, 24] for deepening. Given preferene reltion on finite set S, nother preferene reltion on S is n extension of if x, y S, x y x y. The reltion is lled liner extension of if is n extension of nd is totl order. We hve the following result (see [23]): preferene reltion on finite set S is P-yli every non empty suset of S hs non empty mximl set ( A S, M(A, ) ) there exists liner extensions of nd they re otined y topologil sorting. 2.2 Compt representtions of preferenes in omintoril prolems In omintoril prtil pplitions, solutions re impliit: desried y set S of elementry omponents of set E ( S P(E)). Then, it is neessry to imgine ompt representtion of preferenes for their eliittion (quisition) nd their proessing; euse these opertions with n expliit representtion the listing of the ouples x, y S suh tht x y eing usully intrtle. Thus, in lssil omintoril optimiztion, the preferenes re represented y utility funtion u from P(E) to R to mximize: x y u(x) u(y). In multiriteri optimiztion sed on the Preto dominne, preferenes re represented y vetor of utility funtions (u 1,, u p), ggregted y the Preto dominne: x y [ i {1,, p}, u i(x) u i(y)]. This hierrhil ggregtion will e noted pσu>pareto. And more generlly, every ggregtion of fmily of p utility funtions y rule AR will e noted pσu>ar. In rtifiil intelligene, numerous ompt representtions of preferenes ppered: from CP-nets [4, 14] to onstrints desriing the preferentil neighourhood of the solutions (lled preferentil onstrints in [13]), y going through soft onstrints [3, 19] nd dynmi CSP [26]. In the following, ny ompt representtion of preferene reltion is denoted I( ). We will present in 4.1 the ompt representtion used here for our mximl spnning trees su-prolem. 3 PREFERENTIAL CONSISTENCY AND MAXIMAL SPANNING TREES 3.1 Mximl spnning trees prolems Consider the prolem of finding stisfiing (in the mening of Newell & Simon [15]) mximl spnning tree. Denoted y 31

3 DS(ST/CBPR/MAX), this semi-strutured prolem is formulted in the following wy: DS(ST/CBPR/MAX): Given n undireted grph G = (V, E) nd ompt representtion I( ) of preferene reltion on P(E), the result is fesile spnning tree whih is: (i) mximl for (S ST(G), ), if suh solution exists, nd (ii)suited with the system of vlues of the user. Otherwise, the result is no. Remrk 1. DS nd CBPR men respetively deision support nd risp inry preferene reltion. The ondition (ii) mens the user vi n intertive proess will tret the lk of equivlene nd the inompleteness etween mximl solutions. This definition of prolem involves tht the stisfiing solution, must e lso mximl in (S ST(G), ). In other words, the only degree of freedom let to the DSS user is the hoie of suited solution mong the mximl ones. This definition refers for exmple to ontexts where preferenes hve een given y the different tors of the deision prolem, next ggregted in glol possily inomplete nd intrnsitive preferenes on the solutions P(E) vi ompt representtion I( ); Now, n individul: the user, eing le to ring effiiently forgotten preferentil informtion t different times of the deision proess, is in hrge of finding the suited solution mirroring t est glol preferenes. At this semi-strutured prolem is ssoited the omputle prolem of finding mximl spnning tree, denoted ST/CBPR/MAX, the definition of whih orresponds with the DS(ST/CBPR/MAX) one, fter ersing the property (ii). In suh generl frmework, these omputle prolems re hrd. To e onvined, it is suffiient to onsider the peulir se where the used ompt representtion of preferenes is the Preto-sed multiriteri one. Hene, the memership prolem ssoited with this multiojetive spnning trees prolem is NP-omplete [6, 12]. 3.2 Preferentil onsisteny for mximl spnning trees In Constrint Progrmming [19], onsisteny is prt of more generl prolemti lled desription. The im of onsisteny is the desription of the fesile set of onstrint system y wy of vlues or omintions of vlues elonging to t lest one fesile element. Consisteny prolemti n e extended, in the frmework of omintoril prolems exploiting non-onventionl preferenes, so s to tke into ount of preferentil informtion. Simply, onsisteny will not rely on fesiility ut on est qulity or est ompromise. Hene, we won t remove inonsistent vlues in the mening tht they elong to no fesile solution, ut rther euse they elong to no preferred solution. In this se, we spek out preferentil onsisteny. Without going into detils, prolems onsisting in ersing preferentilly inonsistent vlues, from onstrint system nd ompt representtion of preferene reltion, re lled preferentil onsisteny prolems. As in onstrint stisftion, severl levels of preferentil onsisteny n e defined, ording to whether ll or prt of preferentilly inonsistent informtion is deleted. We nmed glol preferentil onsisteny the removing of ll the preferentilly inonsistent informtion. Remrk 2. In non-onventionl preferene ontext, eh used hoie xiom (e.g. optimlity, mximlity, domintion, ) identifies speifi hoie set (optiml set, mximl set, domintion set, ) whih re generlly pirwise different (see 2.1). This other prmeter speilizes preferentil onsisteny. Thus, we spek out OPT-onsisteny for preferentil onsisteny using optimlity s hoie xiom, MAX-onsisteny for preferentil onsisteny using mximlity, nd so on. To etter understnd preferentil onsisteny, in the following, we study in detils the se of mximl spnning tree prolem. Consider then the following generl omputle prolem, of preferentil onsisteny for mximl spnning trees of grph: GPC(ST/CBPR/MAX): Given n undireted grph G = (V, E) nd ompt representtion I( ) of preferene reltion on P(E), list the edges in E elonging to mximl spnning tree for, if suh edges exist. Otherwise return no. An edge e is lled MAX-onsistent for (G, I( )) if there exists t lest one mximl spnning tree for (S ST(G), ) ontining e. Otherwise, it is lled MAX-inonsistent for (G, I( )). In this rtile, we do not dwell on the omputtionl omplexity of this prolem. But there re gret hnes it is t lest s diffiult s ST/CBPR/MAX, with the sight of investigtions in onstrint progrmming [2, 19]. Yet, in order to etter ppreite the using of this kind of omputle prolem in DSS, we turn towrds n effiiently solvle su-prolem of ST/CBPR/MAX. 4 THE ST/TOSORT-VSMAX/MAX PROBLEM 4.1 Compt representtion nd TOSORT-VSMAX ondition From now, to point out n edges set, for exmple {, }, we dopt the nottion. Given n undireted grph G = (V, E) nd P- yli preferene reltion E on E, we onsider the inry reltion K on P(E) defined s follow: x, y P(E), x K y liner extension {e 1,, e E } of E on E, verifying: e i E e j i < j for ll 1 i, j E, nd for every 1 j E, e j x (x {e 1,, e j 1}) {e j} ontins yle β α γ undireted grph G = (V, E) preferene reltion E on E Figure 1. Exmple of n undireted grph nd totlly ordered preferene reltion on its edge set 2. Exmple 1. The Figure 1 illustrtes the se of omplete undireted grph G on 3 verties, with totl order E on E. Then, the inry reltion K verifies (Figure 2), in ddition with reflexive 2 To void surhrges of the grphil representtion, the reflexive rs re not drwn. 32

4 rs, tht: A K B, for every A M = {, } nd B P(E) \ M, euse the only liner extension of E is itself. Figure 2. The reltion K elorted from (G, E) of Figure 1. The Figure 3 onsiders n undireted grph G = (V, E), with V = {α, β, γ, δ} nd E = {,,, d, h}; nd P-yli reltion E on the edges set E of G verifying, in ddition of reflexive rs: E h, E, E d, d E, h E, E h, d E h. Then, the inry reltion K estlishes iprtition {M, P(E) \ M} of P(E) with M = {x P(E) suh tht: d x or h x} nd stisfies the following reltions: (A, B) M (P(E) \ M), A K B nd (A 1, A 2) M M, A 1 K A 2. β α d Figure 3. Exmple of n undireted grph nd P-yli preferene reltion on its edge set 32. γ h δ undireted grph G = (V, E) preferene reltion E on E Definition 1. A preferene reltion on P(E) is lled TOSORT- VSMAX for the ouple (G, E) iff: (x, y) S ST(G) S ST(G) with x y, x K y x y ( the reltion is n extension of K ) x K y x y or x y Remrk 3. The word TOSORT in the nottion TOSORT-VSMAX points out the reltion K: the reltion E n e topologilly sorted the reltion E is P-yli there exists non-empty mximl set of edges for every non-empty edges suset of E there exist totl orders extending E. And the seond word VSMAX points out oth onditions of this definition the extension ondition nd the trnsltion of the indifferene of K into indifferene nd inomprility of whih define very strong version of mximlity. Exmple 2. The Figure 4 illustrtes preferene reltion on P(E) stisfying TOSORT-VSMAX for the ouple (G, E) of Figure 1. This illustrtion shows TOSORT-VSMAX reltion my inlude strit preferene iruits. d h powerset P(E) of the edges set of G set of spnning trees of G For the Figure 3, the fesile spnning trees set is S ST(G) = {d, h, d, h, dh, d, h, dh}; Aordingly, every TOSORT- VSMAX preferene reltion on P(E) stisfies: ( x, y) { d, h} SST ( G) \ M, x y. d nd h re either indifferent or inomprle Figure 4. An exmple of TOSORT-VSMAX preferene reltion 2 on the powerset of E of Figure 1. The preferene reltion E on E is lled the ompt representtion of the TOSORT-VSMAX reltion on P(E). Here re some properties: Properties 1. Given ouple (G, E) mde up n undireted grph G = (V, E) nd P-yli reltion E on the edges set E, then: () Every TOSORT-VSMAX preferene reltion for (G, E) identifies the sme mximl set s the reltion K indued y E. () The existene of fesile spnning trees wrrnties the existene of non empty mximl set for (S ST(G), K). The proof is immedite. The reltion K is the minimum informtion to know in order to identify the mximl set of TOSORT-VSMAX preferene reltions. Now, we onsider the following su-prolem of ST/CBPR/MAX: ST/TOSORT-VSMAX/MAX: Given n undireted grph G = (V, E) nd ompt representtion E of TOSORT-VSMAX preferene reltion on P(E), return mximl spnning tree for, if suh solution exists. Otherwise return no. We denote S ST/TV/MAX(G, E) the set of possile mximl spnning trees outputted y n lgorithm solving this prolem. Theorem 1. The ST/TOSORT-VSMAX/MAX prolem n e solved in polynomil time in the input size (G, E). Sketh of Proof: One lgorithm onsists in elorting liner extension {e 1,, e E } of E on E ( the TOPOLOGICAL SORT prolem 3 [7, 22]); Next in ssigning utility u(e) to eh edge e of E in order to stisfy the following ondition: u(e i) > u(e i+1), 1 i E 1; for exmple, u(e i) = E i. And, t lst in solving the lssi spnning tree prolem ( ST/Σu/OPT) with the instne (G, u). The resulting 3 In the rest of this rtile, we will hve to use prtiulr lgorithm solving this prolem. We will onsider the following one: inresingly nd greedily numer the mximl edges mong the not yet numered edges of E. The designed list of edges is then liner extension of E. powerset P(E) of the edges set of G set of spnning trees of G 33

5 mximum spnning tree is then lso mximl solution for ST/TOSORT-VSMAX/MAX. 4.2 Multiojetive spnning tree prolems sed on topologil sorting Now we onfront this prolem to the lssil mximum spnning tree prolem, nd its Preto-sed multiojetive version. Exmple 3. The lssil prolem of mximum spnning tree ( ST/Σu/OPT) n e polynomilly trnsformed into the ST/TOSORT- VSMAX/MAX prolem. Indeed, for ny spnning tree x of G, the sum of utilities of edges in x defines totl preorder u on P(E): (x, y) P(E)², x u y e x ue e y ue The reltion u is TOSORT-VSMAX, nd its ompt representtion u E is the preorder indued y u: e, e E, e u E e u(e) u(e ). The ouple (G, u E) is then n instne of ST/TOSORT-VSMAX/MAX, nd its solution set S ST/TV/MAX(G, u E) = B(S ST(G), u). This ssertion is esily provle y ersing the topologil sorting prt of the sketh of proof of Theorem 1. The ST/TOSORT-VSMAX/MAX prolem n e used to model nd solve multiriteri prolems. So, the multi-ttriute utility funtion n e ggregted first to produe glol preferenes on the edges, nd next to prtilly rnk sets of edges. Here is n exmple: Exmple 4. The ST/PARETO>TOSORT-VSMAX/MAX prolem onsiders n undireted grph G = (V, E) nd ouple (p, u) mde up positive numer p nd multi-ttriute utility funtion u from E {1,..., p} to R. p is the numer of onsidered riteri nd u(e, k) is the utility of the edge e ording to the riterion k. In this prolem, the preferene informtion (p, u) is ggregted with Preto dominne, in order to define glol preferene reltion EP on eh edge: e, e E, e EP e for every 1 k p, u(e, k) u(e, k) Next, this preferene reltion on the edges is ggregted with the K reltion, to otin olletive opinion PK etween the susets of E. Then we onsider the instne (G, (2, u)) mde up the undireted grph G = (V, E) of the Figure 3, nd the iriteri utility funtion u given y the following tle: Tle 1. Exmple of iriteri utility funtion u(edge, riterion) on the edges of the undireted grph of the Figure 3. edges d h riterion riterion preferene reltion EP on E Figure 5. The preferene reltion EP on E provided y ggregtion of u with the Preto dominne 32. d h By ggregting u with the Preto dominne, we otin the preferene reltion EP on E given y the Figure 5. At lst, y solving the ST/TOSORT-VSMAX/MAX prolem on this instne (G, EP), we get the mximl set M(S ST(G), PK) = {d, h, d, h, d, h} Remrk 4. Insted of using the Preto dominne to otin the glol preferene reltion EP on the edges, we n pply ny ggregtion rule AR on u. The only ondition on AR is to provide preferene reltion EP hving t lest the P-yliity property. In the multiriteri deision-mking ommunity [10], the multittriute utility funtion u(e, k), with (e, k) E {1,..., p}, is usully ggregted with simple sum per riterion, to produe fmily of p individul utilities on the powerset of edges. Next, this fmily is ggregted, generlly with the Preto dominne, into glol preferene, noted in this se Σ P, on the sets of edges. Exmple 5. By running n lgorithm solving the ST/pΣu>PARETO/ MAX prolem on the instne (G, (2, u)) desried in the Exmple 4, we otin the mximl set M(S ST(G), ΣP ) = {h, d, h, d, h}, whih is stritly inluded in M(S ST(G), PK). The following theorem desries the reltionship etween the lssil hierrhil ggregtion pσu>pareto nd ours PARETO>TOSORT-VSMAX: Theorem 2. Given n undireted grph G = (V, E), nd ouple (p, u) mde up positive numer p nd multi-ttriute utility funtion u from E {1,..., p} to R; then every mximl solution for ST/pΣu>PARETO/MAX is lso mximl solution for ST/PARETO>TOSORT- VSMAX/MAX. Formlly: x S ST (G), x M(S ST (G), Σ P ) x M(S ST (G), PK ) (1) Before showing this theorem, here is lemm whih desries property of the reltion K: Lemm 1. Given ouple (G = (V, E), E) nd n element x P(E), then the reltion K is trnsitive nd: y P(E) suh tht x K y x is optiml in (P(E), K) x is mximl in (P(E), K) Moreover, if x S ST(G), then: y S ST(G) suh tht x K y x is optiml in (S ST(G), K) x is mximl in (S ST(G), K) Proof: The demonstrtion of the optimlity (first equivlene) is immedite. Wht out mximlity (seond equivlene)? If x is optiml, then x is mximl. Now, wht out the ontrry se? If x is mximl in (P(E), K) then, there 2 ses: If there exists z suh tht x K y, then x is optiml ording to the first equivlene. Otherwise ( if suh z does not exist), then w P(E), x K w w P(E), not(x K w) nd not(w K x). Consequently, there is no optiml element in (P(E), K). This ssertion is equivlent to sy tht for every liner extension {e 1,, e E } of E on E, nd for every suset z of E, there exists 1 j E verifying tht: e j x nd (x {e 1,, e j 1}) {e j} is yli. This is possile, if nd only if (V, E) is tree nd E is not in P(E). This is ontrdition. 34

6 Hene, t lst, x is optiml in (P(E), K) x is mximl in (P(E), K). The trnsitivity of K is diret onsequene of the first equivlene. Next, oth the lst equivlenes re true euse (P(E), K) verifies the Arrow hoie xiom [23]: For ny A, B P(E) nd A B, If B(B, K) A then B(B, K) A = B(A, K) (every restrition of P(E) onserves the optimlity). Proof (Theorem 2): First of ll, oth the following ssertions re flse: () x, y S ST(G), x PK y x ΣP y () x, y S ST(G), x ΣP y x PK y Indeed, for the ssertion (), it is suffiient to tke the undireted grph of Figure 3, with the iriteri utility funtion of Tle 1. The ssertion () is flse euse PK only rries out the dihotomy etween the mximl set nd its omplementry. So, the preferenes etween two non mximl elements re unknown. We prove now the formule (1). So, we reson y ontrdition: Suppose there exists n x S ST(G) mximl for ΣP, ut not for PK. This proposition is equivlent with the following one, ording to Lemm 1: x S ST(G) suh tht: [ y S ST(G), not(y ΣP x)] nd [ y S ST(G), not(x PK y)] By definition, not(x PK y) e 1 E \ x, nd e 2 L(x {e 1}) verifying e 1 EP e 2. Now, if we tke the spnning tree y defined s follow: y = x {e 1} \ {e 2}, then we hve, euse of the definition of e 1 EP e 2: (, ) (, ) (, ) (, ) 1 i p, u e i + u e1 i u e i + u e2 i, nd 1 k p, e x\ { e1} e x\{ e1} u( e, k) + u( e1, k) < u( e, k) + u( e2 e x\ { e1 } e x\{ e1}, k). y ΣP x. This ontrdits the mximlity of x in (S ST(G), ΣP). Hene the result. In the following, we propose n lgorithm solving: GPC(ST/TO- SORT-VSMAX/MAX), the glol preferentil onsisteny prolem ssoited with ST/TOSORT-VSMAX/MAX. 5 GLOBAL PREFERENTIAL CONSISTENCY AND TOSORT-VSMAX Insted of either listing ll the mximl spnning trees, or finding suh one tree, we will point out the removing of edges elonging to no mximl spnning tree. Espeilly here, we re interested in GPC(ST/TOSORT-VSMAX/MAX). Here is its definition: GPC(ST/TOSORT-VSMAX/MAX): Let G = (V, E) e n undireted grph nd E e P-yli preferene reltion on E representing TOSORT-VSMAX preferene reltion on P(E). Return ll the edges of E elonging to mximl spnning tree for, if suh edges exist. Otherwise return no. Denote S GPC(ST/TV/MAX)(G, E) E, the edges set outputted y n lgorithm solving this prolem. Then, y definition, we hve the following equlity: S GPC(ST/TV/MAX)(G, E) = x x SST/TV/MAX ( G, E ). (2) This equlity is equivlent to the onjuntion of the following ssertions: () for ll e S GPC(ST/TV/MAX)(G, E) E, there exists x S ST/TV/MAX(G, E) P(E), suh tht: e x. () for ll x S ST/TV/MAX(G, E) P(E), x S GPC(ST/TV/MAX)(G, E). The Figure 6 presents n lgorithm solving this preferentil onsisteny prolem. GPCORDINALSTMAX1(G = (V, E): undireted grph, E: P-yli preferene reltion on E): return {edges set, no} egin (1) if ( NBCONNECTEDCOMPONENTS(G) > 1 ) then return no end if (2) A E (3) B E E (4) C(e) E, for every e E (5) while ( B ) do % loop invrints: A B = nd B C(e) = (6) e CHOOSE(M(B, E)) (7) B B \ {e} (8) C(e) C( e') { e'} e' E e (9) if ( NBCONNECTEDCOMPONENTS(V, C(e) {e}) < NBCONNECTEDCOMPONENTS(V, C(e)) ) then A A {e} end if (10) end while (11) return A end GPCORDINALSTMAX1 Figure 6. An lgorithm solving the GPC(ST/TOSORT-VSMAX/MAX) prolem. This lgorithm supposes we know: Another lgorithm NBCONNECTEDCOMPONENTS solving the ounting prolem of the onneted omponents in n undireted grph. This prolem is known solvle in liner time (y depth first serh lgorithm) for ny given undireted grph (see e.g. [22, 6.3 p. 90]). A hoie strtegy CHOOSE outputting n element of the input expliit set in the non-empty se. Otherwise, return no. Exmple 6. By running GPCORDINALSTMAX1 on the instne given in Figure 1 nd Figure 3, we otin s result the respetive edges sets {, } nd E \ {}. We denote (G, E) the size of the instne (G, E) of ST/TOSORT-VSMAX/MAX. This size n e formulted in terms of the verties set rdinlity m = V of grph G, the numer of edges n = E in G, nd the numer of rs p = E in (E, E): Hene, (G, E) is in O(m + n + p). Now, we remrk tht 0 n m² nd 0 p n². Hene, (G, E) is in O(m 4 ). We hve the following results: Property 2. The lgorithm GPCORDINALSTMAX1 hs worst se time omplexity, whih is liner in the size of the input (G, E). Proof: It is simply suffiient to see tht n order of mgnitude for the worst se time omplexity of this lgorithm GPCORDINALSTMAX1 only depends on the seond loop (lines 5 to 10). The lgorithm 35

7 NBCONNECTEDCOMPONENTS, solving the ounting onneted omponents prolem in time liner in the size of its instne ( prtil grph of G), is then in O(m + n). Consequently, the worst se time omplexity of the onditionl instrution if end if (line 9) is out m + n. It is similr for: line 6, where the hoie strtegy neessittes greedy serh of mximl edge, solvle in the worst se in O(n) line 8, where the mximum numer of possile unions is out rdinlity of E, i.e. in O(n). At lst, the ody of the 2 nd loop runs in the worst se in O(m + n) times. Now, the numer of loops is equl to the numer of edges; nd proves tht the omplexity of the lgorithm GPCORDINALSTMAX1 is in O((m + n) n) O(m 4 ), i.e. liner in the input size (G, E). Theorem 3 The lgorithm GPCORDINALSTMAX1 returns the whole MAX-onsistent edges (nd only them) for mximl spnning trees of the ST/TOSORT-VSMAX/MAX prolem, from n instne ((V, E), E), if suh trees exist. Otherwise returns no. The logi underlying this lgorithm onsists in putting n edge e E in est senrio of hoie, in order to elorte liner extension of E (= the miniml numer ssigned to e mong the liner extensions). Suh est senrio onsists in hoosing e s soon s possile, during the topologil sort. For tht, the topologil sorting lgorithm hs to numer every etter edge e thn e for E; next the edge e is numered iff every etter edge thn e is numered, nd so on. In the est se, when e is numered, if the numer of onneted omponents dereses when we dd e to the lredy numered edges, then e n e hosen to elong to mximl spnning tree for TOSORT-VSMAX preferene reltion. Indeed, this est senrio my then e ompleted in mximl spnning tree, y itertively hoosing ny mximl remining edge. Before showing this theorem, here is lemm whih will help us in the demonstrtion. Lemm 2. Given n instne (G, E), with E P-yli, denote C(e) the edges of E for whih there exists pth of strit preferenes towrds e: C(e) = {f E suh tht: f (1),, f (p) E, with p 0, nd f E f (1) E E f (p) E e}. Then, for every A E, C(e) \ A M(C(e) \ A, E) M(E \ A, E) Proof: First of ll, let us lrify the set E \ C(e): E \ C(e) = {f E suh tht: There exists no pth of strit preferenes from f to e in (E, E)} = {f E suh tht: There is no pth of strit preferenes from f to e 1 in (E, E), e 1 C(e)}. Indeed, if suh pth existed from f to e 1, nd y definition of C(e) from e 1 to e, then there would exist pth of strit preferenes from f to e. Show now lemm: Suppose tht C(e) \ A. Then every edge f M(C(e) \ A, E) verifies: f 1 C(e) \ A, not(f 1 E f) There exists no pth of strit preferenes from f 1 to f in (C(e) \ A, E). Hene, if some edges re dded to C(e) \ A in this se E \ (C(e) A) for whih there exists no pth of strit preferenes from f 2 E \ (C(e) A) to e 1 C(e) \ A, then it won t lso exist pth from f 2 to f. At lst, f M(C(e) \ A, E), f 1 E \ A, not(f 1 E f). This shows lemm. Proof (Theorem 3): Firstly, we hve the following equivlene, euse of the Properties 1 () trnslted y the first line of the lgorithm: GPCORDINALSTMAX1(G, E) = no S ST(G) =. Then point out on the first prt of the proposition: Suppose GPCORDINALSTMAX1(G, E) no, nd show y help of Lemm 2 formule (2): GPCORDINALSTMAX1(G, E) = x. x SST/TV/MAX ( G, E) Diret inlusion: For ny e GPCORDINALSTMAX1(G, E) E, there exists x S ST/TV/MAX(G, E) P(E), suh tht: e x. Indeed, suh n x n e designed y using the strtegy desried in the previous remrk with topologil sort of (E, E). So, s long s we re not t n itertion k suh tht e is mximl in the set B k of not yet numered edges, then, during itertions i < k, the hoie strtegy onsists in tking s urrent edge e i mximl edge for (C(e) \ (E \ B i), E), with C(e) \ (E \ B i) B i. Aording to the ove lemm, M(C(e) \ (E \ B i), E) M(B i, E). Therefore, this strtegy is ville, nd the itertion k = C(e). During the itertion k, given e dereses the numer of onneted omponents in C(e) euse e is in GPCORDINALSTMAX1(G, E) nd then verifies the ondition of line 9 in GPCORDINALSTMAX1, then e is hosen to e dded to A k 1, the urrent tree. Next, during itertions i > k, the topologil sort lgorithm tkes s urrent edge, ny edge of M(B i, E). At lst, the elorted liner extension n e ssoited to utility funtion (see sketh of proof of Theorem 1) nd next used s instne of n lgorithm solving ST/Σu/OPT, whih neessrily returns solution x non, ontining e nd then mximl for (G, E). L(f 2 ) Figure 7. Illustrtion for demonstrtion of Theorem 3. f 2 L(e) f 1 L(f 1 ) Legend: Blue : L(e) is n undireted pth in C(e) E etween oth the ends of e. f 1 nd f 2 re two edges of L(e) \ x L(f 1 ) nd L(f 2 ) re 2 undireted pths in x etween oth the ends of respetively f 1 nd f 2. The edges of the old pth is inluded in x. Converse inlusion: For every x S ST/TV/MAX(G, E) P(E), x S GPC(ST/TV/MAX)(G, E). Indeed, reson y ontrdition. Suppose tht: x S ST/TV/MAX(G, E) nd x S GPC(ST/TV/MAX)(G, E). e S GPC(ST/TV/MAX)(G, E) lthough: e x, nd x S ST/TV/MAX(G, E). The numer of onneted omponents does not derese if we dd e in C(e), ording to line 9 of GPCORDINALSTMAX1 Τhere exists in C(e) n undireted pth L(e) etween oth the ends of e. e 36

8 So tht e should e hosen during the design of x, euse e x, it is neessry tht e e mximl t n itertion k E, if we use the Kruskl s lgorithm to solve ST/TO-SORT-VSMAX/MAX. At this itertion, e A k, the tree t itertion k, nd e dereses the numer of onneted omponents in A k 1. If e M(E, E), then e S GPC(ST/TV/MAX)(G, E), this is ontrdition with the initil ssumption. Aordingly, C(e). Moreover, every edge of C(e) hs lredy een hosen in the senrio of the topologil sort lgorithm during itertions i < k, in order tht e e mximl during the itertion k. It is sure tht C(e) x euse it would exist n undireted pth L(e) {e} in x; tht is ontrditory with x is tree. Hene C(e) \ x. And for every f C(e) \ x, f hs not een dded to x euse during the itertion i < k where it hs een hosen, there lredy exists n undireted pth L(f) in A i x etween the ends of f. Now, the edge set (C(e) x) L( f ) is n undireted f C( e)\ x pth (or ontins suh pth) in x etween the ends of e, mking up with e n undireted pth in x (see Figure 7). This ontrdits the ssumption x is tree. Tht demonstrtes the onverse inlusion. 6 CONCLUSION AND PERSPECTIVES One of the limits, devolved upon deision proesses sed on listing of preferred solutions suggested y Perny & Spnjrd [18] to solve ordinl omintoril prolems, ws the intrtility of lrge size inputs. We introdued nother kind of omputle prolems, preferentil onsisteny ones. Their outputs n e proessed in rel-time y humn eing (i.e. liner in the input size). These omputle prolems re sed oth on the notion of onsisteny pointing out y onstrint progrmming (CP), nd on the notion of hoie investigted in deision iding (DA). In the se of mximl spnning trees prolems stisfying the TOSORT-VSMAX ondition, we proposed n lgorithm solving the glol onsisteny prolem, with liner worst se time omplexity in its input size. One of the ims of this rtile is to ring together the CP nd OR-DA ommunities, to proess more effiiently omintoril prolems exploiting omplex preferenes. Their mutul ontriutions open new wy of intertive solving of semi-strutured omintoril prolems. Consequently, the perspetives re numerous: At first, with preferentil onsisteny: Glol preferentil onsisteny n e used in n intertive deision proess, where the user mkes some lol deisions (hoie), nd where the DSS is restrited to remove preferentil inonsistent domin-vlues. However, suh support systems my not lwys wrrnt preferred solution for the initil instne. Consequently, we hve explored this wy, for exmple y identifying domin-vlues whih re in ll preferred solutions or, y investigting rtionl hoie theory [23] to identify some suffiient properties so tht the deision proess lwys returns preferred solution for the initil instne, if suh solution exists. Next, with effiient spnning trees prolems nd the prtiulr ompt representtion of preferenes used in this rtile: We hve een srutinizing the onept of expressive power of ompt representtion. Any kind of ompt representtions models only suset of preferene reltions. For exmple, utility funtions model only totl preorders. In order to etter understnd the type of ompt representtion used in this rtile, we fous our reserhes on its expressive power for spnning trees prolems. At lst, with pplitions: Wht mkes good theory, it is its ppliility to rel world prolems. The possile pplitions re numerous. And t this time, we work on n utonomous eletril network designing prolem llowing severl not neessry rdinl riteri. Shortly, these prolems rise in isolted regions s some Pifi islnds or in remote villges in rinforest. The isoltion of these popultions implies tht the ontinuous supply of fossil fuels is very expensive to the ommunity, nd exoritntly expensive if you wnted to onnet to n existing eletriity grid. 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