European Journal of Operational Research

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1 Eurpea Jural f Operatial Research 232 (2014) 4 4 Ctets lists available at ScieceDirect Eurpea Jural f Operatial Research jural hmepage Discrete Optimizati A aalytical cmparis f the LP relaxatis f iteger mdels fr the k-club prblem Maria Teresa Almeida, Filipa D. Carvalh Istitut Superir de Ecmia e Gestã, Uiversidade Técica de Lisba, Rua d Quelhas 6, Lisba, Prtugal CIO, FC, Uiversidade de Lisba, Blc C6, Pis 4, Lisba, Prtugal article if abstract Article histry Received 21 May 2013 Accepted 2 August 2013 Available lie 12 August 2013 Keywrds Cmbiatrial ptimizati Frmulatis k-clubs Iteger prgrammig Clique relaxatis Give a udirected graph G =(V,E), a k-club is a subset f des that iduces a subgraph with diameter at mst k. The k-club prblem is t fid a maximum cardiality k-club. I this study, we use a liear prgrammig relaxati stadpit t cmpare iteger frmulatis fr the k-club prblem. The cmpariss ivlve frmulatis kw frm the literature ad ew frmulatis, built i differet variable spaces. Fr the case k = 3, we prpse tw ehaced cmpact frmulatis. Frm the LP relaxati stadpit these frmulatis dmiate all ther cmpact frmulatis i the literature ad are equivalet t a frmulati with a -plymial umber f cstraits. Als fr k = 3, we cmpare the relative stregth f LP relaxatis fr all frmulatis examied i the study (ew ad kw frm the literature). Based isights btaied frm the cmparative study, we devise a stregtheed versi f a recursive cmpact frmulati i the literature fr the k-club prblem (k > 1) ad shw hw t mdify e f the ew frmulatis fr the case k = 3 i rder t accmmdate additial cstraits recetly prpsed i the literature. Ó 2013 Elsevier B.V. All rights reserved. 1. Itrducti Give a udirected graph G =(V,E) ad a iteger k, ak-club S is a cluster f des such that ay tw f its elemets ca reach each ther thrugh at mst k 1 itermediate cluster members. If k = 1, the every cluster member is directly liked t all the thers ad S is a clique (Bmze, Budiich, Pardals, & Pelill, 1 Srese, 2004 Alidaee, Glver, Kcheberger, & Wag, 2007 Martis, 2010). If k > 1, a k-club ca be iterpreted as a distacebased relaxati f a clique. The eed fr graph mdels t represet clusters was first ted durig the 170s i the literature f scial etwrks aalysis (Alba, 173Mkke, 17). It was als ted the that the clique mdel is t restrictive fr may real-wrld applicatis fr t all scial relatis require face-t-face iteracti. I fact, scial relatis are frequetly established thrugh itermediaries. This explais the imprtace f clique relaxati mdels fr scial scieces. Recetly, the OR cmmuity has prduced a sigificat umber f studies clique-related mdels with applicatis i bth the scial scieces as well as ther fields such as cmputatial bilgy (Bla_zewicz, Frmawicz, & Kasprak, 2005 Balasudaram, Crrespdig authr. Address Istitut Superir de Ecmia e Gestã, Departamet de Matemática, Rua d Quelhas 6, Lisba, Prtugal. Tel fax addresses talmeida@iseg.utl.pt (M.T. Almeida), filipadc@iseg.utl.pt (F.D. Carvalh). Butek, & Trukhav, 2005 Butek & Wilhelm, 2006) ad data miig (Bgiski, Butek, & Pardals, 2006). Fr ay k > 1, the k-club prblem is t fid a maximum cardiality k-club f a graph. The k-club prblem is NP-hard (Burjlly, Laprte, & Pesat, 2002). The cmputatial perfrmace f all the appraches reprted i the literature depeds the value f k ad the edge desity f the graph. T be able t devise better exact ad apprximate appraches, it is imprtat t gai a deeper uderstadig f the prblem frm a theretical pit f view. A detailed study f the -hereditary ature f k-clubs fr k > 1 is preseted i Mahdavi Pajuh ad Balasudaram (2012). I this study, we ivestigate the relative stregth f the LP relaxatis f iteger prgrammig frmulatis fr the k-club prblem built i differet variable spaces. The rest f the paper is rgaized as fllws. I Secti 2 we prvide defiitis, termilgy, ad tati. Secti 3 ctais the mtivati ad a verview f the study. Secti 4 ctais a review f the literature k-club mdels. Sectis 5 ad 6 are devted t the cmparis f mdels ad the develpmet f ehaced versis, frm the liear prgrammig stadpit. I Secti 7 we derive a frmulati fr a prblem that has recetly bee prpsed i the literature a variat f the 3-club prblem with additial rbustess cstraits. Cclusis are preseted i Secti. 2. Defiitis, termilgy, ad tati Give a udirected graph G =(V, E) ad a de v 2 V, we represet by Nv the set f all des adjacet t v, ad call it the /$ - see frt matter Ó 2013 Elsevier B.V. All rights reserved. http//dx.di.rg/ /j.ejr

2 40 M.T. Almeida, F.D. Carvalh / Eurpea Jural f Operatial Research 232 (2014) 4 4 eighburhd f v. The distace dist G (u,v) betwee tw des u ad v is the miimum umber f edges eeded t lik them, ad the diameter diam(g) is the maximum distace betwee ay u ad v. IfS # V, the subgraph iduced by S is G [S] =(S,E(S)), where E(S) is the set f edges with bth ed des i S. Fr ay k P 1, if diam(g [S] ) 6 k, S is a k-club. A k-club S # V ca be represeted by its icidece vectr =(x 1,...,x jvj ). If C is a subset f des ad s, t 2 VC, we say that C is a s t de cut set if C itersects all chais that lik s ad t. Frmulatis with variables t represet ly des are called atural whereas frmulatis with additial variables t represet edges, chais r paths are called exteded. A frmulati is cmpact if the umber f its variables ad cstraits is buded frm abve by a plymial fucti f the umber f des. If a iteger liear prgrammig frmulati is represeted by [F], the [F] LP represets its liear prgrammig (LP) relaxati, their ptimal values are represeted by Z[F] ad Z[F] LP, ad their feasible sets are represeted by S[F] ad S[F] LP, respectively. Feasible slutis t LP relaxatis f atural (resp. exteded) iteger frmulatis are represeted by, (resp. ð Y ð V, ad ð Z). Iteger frmulatis are cmpared by prjectig their LP relaxati feasible sets t the space f de variables. Give a plyhedr P ¼fðx y 2R q Ax þ By 6 bg, where A, B, a b have m rws, the prjecti f P t the x-space is defied as Prj x ðp ¼fx 2 R y 2 R q ðx y 2Pg. Fr mre plyhedral thery ad prjecti the reader is referred t Nemhauser ad Wlsey (1) ad Balas (2005). Give tw iteger frmulatis [F1] ad [F2], whse bjective fucti is the same ad is t be maximized, if Prj x (S[F1] LP ) is a subset f Prj x (S[F2] LP ), the Z[F1] LP 6 Z[F2] LP, ad we say that [F1] LP is strger tha [F2] LP ad frmulati [F1] dmiates [F2]. If Prj x (S[F1] LP ) = Prj x (S[F2] LP ), the [F1] ad [F2] are LP-equivalet. 3. Mtivati ad verview Like may ther cmbiatrial ptimizati prblems, the k-club-prblem has alterative iteger liear prgrammig frmulatis that are stated with differet sets f variables. The first frmulati i the literature, prpsed i Burjlly et al. (2002), is a cmpact exteded frmulati with de ad chai variables. Fllwig Almeida ad Carvalh (2012), we shall call it the chai frmulati ad dete it by [F_C]. Sice tw-edge chais t =(i,r,j) that lik des i ad j ca be idetified by the cetral de r, there is eed fr chai variables t represet them. Based this bservati, Burjlly et al. (2002) als prpsed a simplified atural frmulati fr the case k = 2. Very recetly, fur ew frmulatis have bee prpsed fr k-clubs. Veremyev ad Bgiski (2012) preset a cmpact exteded recursive frmulati (hecefrth referred t as [F_R]) derived frm a liear mdel by a liearizati prcedure that takes it accut the structure f k- clubs. Fr k = 3, Almeida ad Carvalh (2012) prpse a de cut set atural frmulati [F_S] with a -plymial umber f cstraits ad a cmpact exteded frmulati [F_N] with de ad edge variables. Fr the case k =2, Carvalh ad Almeida (2011) devise a atural frmulati whse cstraits defie facets f the 2-club plytpe P 2c. This frmulati, based Burjlly et al. s mdel, is btaied by liftig redudat cstraits. May authrs have carried ut cmparative studies f alterative frmulatis fr imprtat cmbiatrial ptimizati prblems e.g. tree prblems (Magati & Wlsey, 15) ad travellig salesma-related prblems (Guveia & Vss, 15 Oca, Altiel, & Laprte, 200). T the best f ur kwledge, similar study ca be fud i the literature fr the k-club prblem. Veremyev ad Bgiski (2012, pp ) cmmet the tightess f biary frmulatis based the results btaied with a set f test istaces. They guess that their recursive frmulati is geerally at least as gd (r, at the very least, t substatially wrse) tha the e by Burjlly et al. (2002) ad that their frmulati is rather tight fr larger values f k. Almeida ad Carvalh (2012) have prved that frmulati [F_N] dmiates chai frmulati [F_C] fr k = 3, ad they preset a weakeed versi f [F_N] which is LP-equivalet t [F_C]. The authrs als prve that fr k = 3 de cut set frmulati [F_S] dmiates [F_N], but d t shw hw t stregthe the latter t btai a frmulati LP-equivalet t the frmer. With this study we aim t ivestigate the relative stregth f the LP relaxatis f differet frmulatis fr the maximum k-club prblem. Sice the frmulatis are stated with differet sets f variables, the cmpariss are made by prjectig the LP feasible sets t the space f de variables. The isight gaied it the k-club structure by meas f these cmpariss may be f help t explre pssible ways t frmulate related prblems, as illustrated i Secti 7 with the R-rbust k-club prblem recetly prpsed i the literature by Veremyev ad Bgiski (2012). First, fr the case k = 3, we shw hw t mdify [F_C] i rder t btai a ehaced versi [F_EC], which is LP-equivalet t [F_S]. The, we address the tightess f the recursive frmulati [F_R] i cmparis with the tightess f [F_C] ad shw that the cclusi depeds the value f k fr k = 3 [F_R] dmiates [F_C] but the dmiace des t hld fr k = 4. Fr k = 3, we als shw that [F_R] is dmiated by [F_EC]. Based the isight gaied frm the cmpariss f [F_R], [F_C], ad [F_EC] fr k = 3, we derive a stregtheed versi f [F_R] that is valid fr ay k > 1, deted by [F_ER]. We the shw that fr k = 3 [F_ER] ad [F_EC] are LP-equivalet. We cclude the paper with a frmulati fr the R-rbust 3- club prblem, itrduced i the literature by Veremyev ad Bgiski (2012). T the best f ur kwledge, there is i the literature ther frmulati fr the rbustess cditi whe k > Review f iteger prgrammig frmulatis fr the k-club prblem T set the gruds fr the cmpariss preseted i the fllwig sectis, we shall first review iteger prgrammig frmulatis i the literature. T simplify the tati, we will dete by P k the set f all pairs f des that cat belg simultaeusly t a k-club, because their distace i G =(V,E) is greater tha k, ad by N k the set f all pairs f adjacet des whse distace i G =(V,E) des t exceed k Chai frmulati The frmulati prpsed i Burjlly et al. (2002), with de ad chai variables, ca be preseted as fllws. Fr ay tw adjacet des i, j 2 V, let C k be the set f all chais, f legth at mst k, that lik i ad j, ad let C ¼[ ij2v C k. Let y t, t 2 C, be a biary variable assciated with chai t ad V t be the set f all des i the chai. A maximum k-club is a ptimal sluti fr ½F CŠ max Z ¼ i2v x i st x i þ x j 6 1 fi jg 2P k ð1 y t t2c k fi jg 2N k ð2 y t 6 x r t 2 C r 2 V t ð3 x i 2f0 1g i 2 V ð4 y t 2f0 1g t 2 C ð5

3 M.T. Almeida, F.D. Carvalh / Eurpea Jural f Operatial Research 232 (2014) Cstraits (1) guaratee that if dist G (i,j)>k, the des i ad j are t bth selected fr the k-club. Cstraits (1) will be referred t as packig cstraits. Cstraits (2) impse that a pair f adjacet des i a k-club must be liked by at least e selected chai with at mst k edges. Cstraits (3) impse the iclusi i the k-club f all des i selected chais. Cditis (4) ad (5) defie the variables as biary. Nte that fr every pair fi jg 2N k ad every chai t 2 C k r 2 V t fi jg ca be substituted fr r 2 V t, i cstraits (3), sice i ay sluti such that y t > mifx i x j g the value f variable y t ca be decreased t mifx i x j g, withut alterig the sluti i the x-space. The umber f chai variables y t ad cstraits i [F_C] is O(jVj k+1 ) Recursive frmulati Veremyev ad Bgiski (2012) use a differet apprach t devise a frmulati fr the k-club prblem. First, they develp a liear mdel. The, they trasfrm it it a liear mdel with biary variables x i (i =1,...,jVj) t represet des, ad iteger variables w ðl ði j ¼ 1... jvj l ¼ 2... k t represet the umber f paths f legth l that lik adjacet des i ad j. Fially, takig it accut the k-club structure, they devise a recursive biary frmulati with de variables x i (i =1,...,jVj) ad biary variables ðl ði j ¼ 1... jvj l ¼ 2... k such that v ¼ 1 if ad ly if there is at least e path f legth l frm de i t de j i the subgraph iduced by the set S # V whse icidece vectr is =(x 1,...,x jvj ). The recursive frmulati ca be preseted as fllws ½F RŠ max Z ¼ i2v st x i k l¼2 fr i j 2 V j > i i j 2 V ði j R E ð6 6 x i 6 x j x k k2ðn i \N j P x k A þðx i þ x j 2 jvj k2ðn i \N j fr l ¼ 3... k i j 2 V j > i 6 x i 6 v ðl 1 k2n i 0 P jvj v ðl 1 k2n i 1 A þðx i 1 ð7 ð ð ð10 x i 2f0 1g i 2 V ð4 2f0 1g i j 2 V l ¼ 2... k ð11 Cstraits (6) impse that each pair f selected des must be liked by at least a path with at mst k edges. Cstraits (7) ad () defie paths with tw edges. Cstraits () ad (10) defie the recursi fr paths with three r mre edges. Cstraits (11) defie the recursi variables as biary. Recursive frmulati [F_R] has size O(Vj 2 ) Neighburhd ad de cut set frmulatis (k = 3) Fr the case k = 3, Almeida ad Carvalh (2012) prpse tw alterative frmulatis eighburhd frmulati [F_N] ad de cut set frmulati [F_S]. The ratiale fr [F_N] ad [F_S] is summarized ext. Csider a pair f adjacet des i ad j f G =(V,E). They may be icluded i a 3-club S ly if at least e f the fllwig cditis hlds (I) There is a de r i S that is a eighbur f i ad j, i.e., r 2 (N i \ N j ) (II) There are tw adjacet des i S, p ad q, such that p is a eighbur f i, ad q is a eighbur f j, i.e., p 2 N i, q 2 N j, ad (p,q) 2 E. Cditi (I) hlds if ad ly if dist G[S] (i,j)=2. If cditi (II) hlds ad p 2 (N i \ N j ) r q 2 (N i \ N j ), the dist G[S] (i,j) = 2 ad cditi (I) hlds as well. S, i (II), we ca csider ly des p 2 (N i N j ) ad q 2 (N j N i ). Let E dete the set f edges that lik thse des E ¼fðp q 2E p 2ðN i N j q 2ðN j N i g Let us w assciate a variable z with each edge (i,j) 2 E. A maximum 3-club is a ptimal sluti fr ½F NŠ max Z ¼ i2v x i st x i þ x j 6 1 fi jg 2P 3 ð12 x r þ z pq fi jg 2N 3 ð13 r2ðn i \N j ðpq2e z 6 x i z 6 x j z P x i þ x j 1 ði j 2E ð14 x i 2f0 1g i 2 V ð4 z P 0 ði j 2E ð15 Cditis (12) are packig cstraits. Neighburhd cstraits (13) impse that tw adjacet des i ad j cat be bth i a 3-club uless a cmm eighbur is i the 3-club r a pair f eighburs, p ad q, fi ad j respectively, liked by a edge, are i the 3-club. Cstraits (14) guaratee that a edge (i,j) is used if ad ly if bth its ed des belg t the 3-club. Cstraits (15) defie edge variables as egative. Cditis z P x i + x j 1 ca be igred because i ay sluti with z < x i þ x j 1 the value f variable z ca be icreased t mifx i x j g, withut alterig the sluti i the x-space. Nte that, if i the last term f cditis (13) E(N i,n j ) is substituted fr E, the resultig frmulati is LP-equivalet t [F_C] (Almeida & Carvalh, 2012) if i [F_C] fr each pair f des fi jg 2N 3 all variables y t assciated with three-edge chais t =(i,p,q,j) with p 2 (N i \ N j ) r q 2 (N i \ N j ) are remved, the resultig frmulati is LP-equivalet t [F_N] (Almeida & Carvalh, 2012). Neighburhd frmulati [F_N] has jvj + jej variables ad jvj 2 jvj þ 2jEj cstraits. 2 [F_S] is a atural frmulati fr the 3-club prblem based de cut sets f auxiliary graphs built fr pairs fi jg 2N 3. This frmulati is described ext. Csider agai a pair f adjacet des i ad j f G =(V,E) ad E ={(p,q) 2 Ep 2 (N i N j ),q 2 (N j N i )}. E is the set f ier edges f three-edge chais that lik des i ad j, ad whse ier des are t i N i \ N j. Let V represet the set f their ed des. Each de v 2 V is either adjacet t de i r t de j. Therefre V ca be partitied it subsets A ={v2v v 2 (N i N j )} ad B ={v 2 V v 2 (N j N i )}. Let us w assciate with i ad j a subgraph G =(N,H ), where N = V [ {i,j} ad H = E [ {(i,v) 2 Ev 2 A } [ {(v,j) 2 Ev 2 B }.

4 42 M.T. Almeida, F.D. Carvalh / Eurpea Jural f Operatial Research 232 (2014) 4 4 T iclude des i ad j i a 3-club S, it is als ecessary t iclude a de r 2 (N i \ N j ) r a de f each set S # V such that E(V S )=, sice therwise dist G[S] (i,j) > 3. These sets S are i j de cut sets. The set f all miimal S will be deted by S. A maximum 3-club is a ptimal sluti fr ½F SŠ max Z ¼ i2v x i st x i þ x j 6 1 fi jg 2P 3 ð12 r2ðn i \N j x r þ s2s x s fi jg 2N 3 S 2 S ð16 x i 2f0 1g i 2 V ð4 Cditis (12) are packig cstraits. Nde cut set cstraits (16) impse that, if tw adjacet des i ad j are bth selected fr the 3-club, the a cmm eighbur f i ad j r e de frm each S 2 S is als selected fr the 3-club. Nde cut set frmulati [F_S] dmiates eighburhd frmulati [F_N] (Almeida & Carvalh, 2012). 5. Ehaced chai frmulati fr the case k = 3 Fr k = 3, Almeida ad Carvalh (2012) shw that chai frmulati [F_C] is dmiated by frmulati [F_N], which is i tur dmiated by atural frmulati [F_S]. They als shw hw t mdify [F_C] t btai a chai frmulati, LP-equivalet t [F_N]. All these frmulatis are cmpact, except fr [F_S], which has a -plymial umber f de cut set cstraits. The cmpariss i Almeida ad Carvalh (2012) suggest that if we aim t btai a chai frmulati LP-equivalet t [F_S], we have t devise mre striget cstraits t lik de ad chai variables. The ituiti fr the ew likig cstraits is prvided by the example i Almeida ad Carvalh (2012, p. 157) the dmiace f [F_S] LP ver [F_N] LP may be strict ly whe there is at least e pair fi jg 2N 3 ad a de v 2 V such that deg ðv G½V > 1. I this case, de v is shared by tw r mre Š three-edge chais whse cetral edges are represeted by variables z pq, with (p,q) 2 E. I a -iteger feasible sluti ð Z fr [F_N] LP,ifv2A, it may happe that P qðvq2e z vq > x v r, if v 2 B, it may happe that P pðpv2e z pv > x v. I either case, the sluti ca be cut ff by impsig the cditi that the sum f the values f all variables that represet the chais that share de v cat be greater tha x v. T frmulate these ew likig cstraits, we shall rewrite chai variables y t, t =(i,p,q,j), as y pq, i rder t make the idetificati f ier des immediate. The ehaced versi f chai frmulati [F_C] fr k = 3we prpse i this secti cmbies the ew likig cstraits sketched abve with the ratiale used t remve y t variables assciated with three-edge chais t =(i, p, q, j) with at least e ier de i N i \ N j (see Secti 4.3). Let us csider chai frmulati [F_C] fr k = 3. Fr each pair f des fi jg 2N 3, let C ¼2 (resp. C ¼3 ) represet all chais with tw (resp. three) edges that lik i ad j. Each chai t ¼ði r j 2C ¼2 ca be represeted by its cetral de r. Therefre, all variables y t that represet chais f C ¼2 ca be discarded ad cstraits (2) rewritte as x r þ y t fi jg 2N 3 ð17 r2ðn i \N j t2c ¼3 As discussed i Secti 4.3, fr each fi jg 2N 3, variables y t assciated with chais t ¼ði p q j 2C ¼3 with p 2 (N i \ N j ) r q 2 (N i \ N j ) ca be elimiated. T simplify the tati, fr all ther chais t ¼ði p q j 2C ¼3 cstraits (3) rewritte as y pq 6 x i y pq 6 x p let us write y pq, istead f y t, with y pq 6 x j fi jg 2N 3 ðp q 2E ð1 y pq 6 x q fi jg 2N 3 ðp q 2E ð1 ad cstraits (5) rewritte as y pq 2f0 1g fi jg 2N3 ðp q 2E ð20 Cstraits (2) i [F_C] ca w be lifted t x r þ y pq fi jg 2N 3 ð21 r2ðn i \N j ðpq2e A chai that icludes edge (i,p) (resp. (q,j)) ca ly be used if de p (resp. q) is i the 3-club, ad t iclude a pair f adjacet des i ad j i the 3-club, there is eed t select mre tha e chai with three edges. Therefre, cstraits (1) ca be lifted t y pq 6 x p fi jg 2N 3 p 2 A ð22 qðpq2e pðpq2e y pq 6 x q fi jg 2N 3 q 2 B ð23 A ehaced chai frmulati fr the 3-club prblem is ( ) ½F ECŠ max x i ð12 ð21 ð1 ð22 ð23 ð4 ð20 i2v By cstructi, [F_EC] dmiates [F_C]. Prpsiti 1. Fr k = 3, Z [F_EC] LP 6 Z[F_C] LP. Nte that cstraits (1) ca be igred because i ay sluti with y pq > mi x i x j the value f variable y pq ca be decreased t mi x i x j, withut alterig the sluti i the x-space. Cstraits (1) ca als be writte as P ðpq2e y pq 6 x i ad P ðpq2e y pq 6 x j because the upper budig cstraits xv 6 1, fr all v, dmiate (21) if P ðpq2e y pq > mifx i x j g. This versi f cstraits (1) impses the extra cditi that at mst e chai with three edges ca be selected t lik des i ad j i the iduced subgraph. The cditi is t restrictive fr the 3-club prblem, but it is icmpatible with multiple-chai cditis fr rbustess that have bee recetly prpsed i the i q 1 p q 2 Fig. 1. (a) Csider a feasible sluti fr [F_C] LP i which x i ¼ 2 3 xj ¼ 1 xp ¼ xq ¼ xq ¼ 1, ad yt ¼ yt ¼ 1, where t =(i,p,q 1,j) ad t 2 =(i,p,q 2,j). I the crrespdig sluti fr [F_EC] LP, t cmply with cstrait y pq 1 6 x p, cstrait (21) fr the pair {i,j} is vilated. (b) Csider agai a feasible sluti fr [F_C] LP i which x i ¼ 2 x 3 j ¼ 1 x p ¼ x q1 ¼ x q2 ¼ 1, ad 3 y t1 ¼ y t2 ¼ 1, where t 3 1 =(i,p,q 1,j) ad t 2 =(i,p,q 2,j). I the crrespdig sluti fr [F_R] LP, cstrait (6) fr the pair {i,j} is vilated because þ y pq 2 6 mifx p x j x q1 þ x q2 g¼ 1 ð3 v 3 6 mi x i 6 1 ð2, ad v 3 ¼ 0. (c) Csider a sluti i the x-space such that x i ¼ 2 3 xj ¼ 1 xp ¼ xq ¼ xq ¼ 1. I the crrespdig sluti fr [F_R] LP, cstrait (6) fr the pair {i,j} is vilated (see Secti 4.2). By ctrast, z pq1 ¼ z pq2 ¼ 1 yields a feasible sluti fr [F_N] 3 LP. j

5 M.T. Almeida, F.D. Carvalh / Eurpea Jural f Operatial Research 232 (2014) literature (see Secti 7). By ctrast, the liftig f cstraits (1) t cstraits (22) ad (23) stregthes the LP relaxati f the iteger mdel ad paves the way t the itrducti f rbustess cditis. The dmiace f [F_EC] ver [F_C] may be strict. A illustrati is prvided by the graph depicted i Fig. 1(a). Fr k = 3, the umber f variables i [F_EC] ad [F_C] is O (jvj 4 ). I practice, the umber f variables i [F_EC] is sigificatly smaller tha the umber f variables i [F_C] because all variables that represet chais with tw edges as well as may variables that represet chais with three edges are elimiated. 6. Cmparis f frmulatis fr k = 3 I this secti, we cmpare the LP relaxatis f the iteger mdels described i Sectis 4 ad 5, fr the case k = 3. I these relaxatis, cstraits (4), (5), (11), ad (20) are replaced by 0 6 x i 6 1 i 2 V ð4 0 y t P 0 t 2 C ð5 0 P 0 i j 2 V l ¼ 2... k ð11 0 y pq P 0 fi jg 2N3 ðp q 2E ð20 0 I Secti 6.1, we establish the LP-equivalece f [F_EC] ad [F_S]. I Secti 6.2, we shw that frmulati [F_R] is dmiated by ehaced chai frmulati [F_EC] ad dmiates chai frmulati [F_C]. Based the cmparative study made fr the case k = 3, we prpse a ehaced versi f [F_R] which is valid fr the k-club prblem, fr ay k > 1. We the establish the LP-equivalece betwee [F_EC] ad this ehaced versi f [F_R] fr k = 3. I Secti 6.3, we shw that there is dmiace relati betwee frmulatis [F_N] ad [F_R] Cmparis f the LP relaxatis f [F_EC] ad [F_S] The ratiale used t derive frmulati [F_EC] suggests that with cstraits (22) ad (23) we may btai a chai frmulati as strg as [F_S] frm the LP relaxati stadpit. I this secti we preset a frmal prf that [F_EC] ad [F_S] are, i fact, LP-equivalet. T prve that [F_EC] ad [F_S] are LP-equivalet, we shall shw that the feasible set f [F_S] LP is the prjecti f the feasible set f [F_EC] LP t the x-space. Let ð Y be ay feasible sluti fr [F_EC] LP. Fr ay fi jg 2N 3 ad S 2 S, y pq þ y pq 6 y pq ¼ ðpq2e ðpq2e p2ða \S q2b ðpq2e p2ða S q2ðb \S p2ða \S x p þ q2ðb \S x q ¼ s2s Therefre, satisfies all de cut set cstraits (16), fr the pair fi jg 2N 3. Sice fi jg 2N 3 was chse arbitrarily, is feasible fr [F_S] LP. Cversely, let be ay feasible sluti fr [F_S] LP, ad let us build a feasible sluti ð Y fr [F_EC] LP. Fr every fi jg 2N 3, let us csider subgraph G =(N,H ) (see Secti 4.3), ad a layered digraph D built frm G as fllws. The layers f D are L 1 ={i}, L 2 = A,L 3 = B, ad L 4 ={j}. Fr each edge i H, we create a directed arc i D ad defie the arc capacities as >< x v if u ¼ i ad v 2 A cap ðuv ¼ 1 if u 2 A ad v 2 B > x u if u 2 B ad v ¼ j x s Let f be a maximum i j flw i D. By cstructi, valueðf ¼ P ðpq2e f pq. Let us set y pq ¼ f pq fr all (p,q) 2 E. By the max-flw mi-capacity cut therem, value(f) is als the capacity f a i j cut f miimum capacity i D. Hece, valueðf ¼ P s2c x s, where C is the set f des f V that are icidet t the arcs i a miimum capacity i j cut i D. Therefre, valueðf ¼mi P s2s x s S 2 S, ad cstraits (16) fr {i,j} are satisfied. Sice the pair {i,j} was chse arbitrarily, we ca cclude that ð Y is feasible fr [F_EC] LP. Prpsiti 2. Z½F ECŠ LP ¼ Z½F SŠ LP 6.2. Cmparis f the LP relaxatis f [F_R], [F_C], ad [F_EC] As pited ut by Veremyev ad Bgiski (2012), a theretical cmparis f the tightess f the LP relaxati f their frmulati [F_R] with that f the LP relaxati f [F_C] is a very challegig task, due t the recursive ature f [F_R]. As we shall shw ext, the result f the cmparis is t idepedet f the value f k [F_R] dmiates [F_C] fr k = 3, but the dmiace des t hld fr k =4. We shall cclude this secti by shwig that [F_EC] dmiates [F_R] fr k =3. We shall first shw that [F_R] dmiates [F_C] fr the case k =3. Recallig the ratiale preseted i the prf f Prpsiti 2, itis quite ituitive that cstraits (7) play a imprtat rle i ay cmparis f [F_R] LP ad [F_C] LP fr k = 3. Give a pair fi jg 2N 3 ad a de p 2 A, cstrait 6 x p may cut ff feasible slutis f [F_C] LP i the x-space, i a way that bears a clear resemblace with the effect f cstraits (22) i [F_EC]. Prpsiti 3. Fr k = 3, Z [F_R] LP 6 Z[F_C] LP. Prf. Let ð V be feasible fr [F_R] LP. We shall assume, withut lss f geerality, that ¼ mi x ab a x b P x r2ðna\n b r ad, fr all a, b 2 V. v ð3 ab ¼ mi x a P k2na v ð2 kb Let us w build a feasible sluti ð Y fr [F_C] LP. All packig cstraits are satisfied by because fr all pairs fi jg 2P 3. ¼ v ð3 ¼ 0 Let fi jg 2N 3 be chse arbitrarily. If N i \ N j =, the ¼ 0 ad P r2ðn i \N j x r ¼ 0. Otherwise, let D be a cmplete layered digraph with five layers L 1 ¼fig L 2 ¼fi 0 g L 3 ¼ N i \ N j L 4 ¼fj 0 g ad L 5 ¼fjg (where i 0 ad j 0 are cpies f i ad j, respectively) ad arc capacities defied by x i if u ¼ i ad v ¼ i 0 >< x r if u ¼ i 0 ad v ¼ r 2ðN i \ N j cap ðuv ¼ 1 if u 2ðN i \ N j ad v ¼ j > 0 x j if u ¼ j 0 ad v ¼ j Suppse that f is a maximum i j flw i D. By cstructi, valueðf ¼ P r2ðn i \N j f i 0 r. If we set y t ¼ f i 0 r, fr every chai t ¼ði r j 2C ¼2 with variables v ð3, it fllws that P t2c ¼2 y t ¼. We are w left ad chais t 2 C ¼3, fr all fi jg 2N 3. Fr every pair fi jg 2N 3, let us build a layered digraph D, with layers

6 44 M.T. Almeida, F.D. Carvalh / Eurpea Jural f Operatial Research 232 (2014) 4 4 L 1 ={i}, L 2 ={i 0 }, L 3 ={pp 2 N i ad N p \ N j g L 4 ¼ ad L 5 ={j} (where i 0 is a cpy f i) ad whse set f arcs is t t 2 C ¼3 A ¼fði i 0 g [ fði 0 p p 2 N i ad N p \ N j g[ [ ðp t t 2 C ¼3 ad edge ði p belgs t chai t [ ðt j t 2 C ¼3 with arc capacities defied by capðu v x i if u ¼ i ad v ¼ i 0 >< if u ¼ i 0 ad v ¼ p 2 L 3 ¼ 1 if u 2 L 3 ad v ¼ t 2 L 4 > mifx w w is a de f tg if u ¼ t 2 L 4 ad v ¼ j Suppse that f is a maximum i j flw i D. By cstructi, valueðf ¼ P t2c f ¼3 tj ¼ v ð3. By settig y t ¼ f tj, fr all t 2 C ¼3, it fllws that P t2c y ¼3 t ¼ valueðf. We ca w cclude that P t2c 3 y t ¼ þ v ð3 ad csequetly, sice fi jg 2N 3 was chse arbitrarily, that ð Y is feasible fr [F_C] LP. The dmiace f [F_R] ver [F_C] fr k = 3 may be strict. A illustrati is prvided by the graph depicted i Fig. 1(b). The dmiace des t hld fr k = 4, as illustrated by the example preseted i Appedix A. Let us w csider [F_EC] ad [F_R] fr k =3. Let fi jg 2N 3 be chse arbitrarily. I [F_EC] cstraits (23) impse that P pðpq2e h, y pq 6 x q fr every q 2 B. I [F_R] cstraits () impse ly that v ð3 6 x i ad v ð3 6 P k2n i. Due t cstraits (23), feasible slutis fr [F_R] LP the x-space, may be ifeasible fr [F_EC] LP. A illustrati is prvided by the graph depicted i Fig. 2(a). We shall ext shw that fr k = 3 [F_EC] dmiates [F_R]. Prpsiti 4. Fr k = 3, Z [F_EC] LP 6 Z[F_R] LP. Prf. We shall shw that the prjecti t the x-space f the feasible set f [F_EC] LP is icluded i the prjecti t the x-space f the feasible set f [F_R] LP by shwig that if ð Y is feasible fr [F_EC] LP, the ð V is feasible fr [F_R] LP, where ( ) ( ) x r ab ¼ mi x a x b fr all a b 2 V i r2ðn a\n b ad v ð3 ¼ mi x ab a kb k2n a p 2 p 1 Fig. 2. (a) Csider a feasible sluti fr [F_R] LP i which x i ¼ 2 x 3 j ¼ 1 x p1 ¼ x p2 ¼ x q ¼ 1 ð2 v 3 ¼ 0 ð2 p 1 j ¼ v ¼ 1 ð3 p 2 j, ad v 3 ¼ 2. I the crrespdig sluti fr [F_EC] LP, cstrait (21) fr the pair {i,j} is vilated because 3 y p 1 q þ y p 2 q 6 x q must hld. (b) Csider agai k = 3 ad a feasible sluti fr [F_R] LP i which x i ¼ 2 3 xj ¼ 1 xp ¼ xp ¼ xq ¼ 1 ð2 v ¼ 0 ð2 p 1 j ¼ v ¼ 1, ad p 2 j 3 v ð3 ¼ 2. T cmply with 3 (00 ), cstrait (6) is vilated. q j We shall assume, withut lss f geerality, that fr all pairs fi jg 2N 3 P ðpq2e y pq 6 x i ad P ðpq2e y pq 6 x j. Nte that, t satisfy cstraits (7) ad (), ad v ð3 cat assume values greater tha the es they are beig assiged, which makes cstraits () ad (10) irrelevat. If fi jg 2P 3, the ¼ v ð3 ¼ 0, ad csequetly, þ v ð3 if ad ly if x i þ x j 6 1. Let fi jg 2N 3 be chse arbitrarily. If ¼ x i r ¼ x j, cstrait (6) is satisfied, regardless f the value f v ð3. Otherwise, ¼ P r2ðn i \N j x r. If v ð3 ¼ x i, cstrait (6) is satisfied, regardless f the value f < x i ad v ð3. Otherwise, P k2n i ¼ P k2n i. Let us assume that v ð3 ¼ P k2n i. By cstructi, P ðpq2e y pq ¼ P P p2a qðpq2e y pq. Let p be chse arbitrarily i A ad remember we are assumig, withut lss f geerality, that P ðpq2e y pq 6 x j. Sice P qðp q2e y p q 6 P ðpq2e y pq, we ca cclude that Pqðp q2e y p q 6 x j. By (22), Pqðp q2e y p q 6 x p, ad by (1), P qðp q2e x q ¼ P q2n p x q 6 P q2ðn p \N j x q. q2n j N i Pqðp q2e y p q 6 It fllws that P qðp q2e y p q 6 mi x p x j P r2ðn p \N j x r p j ad, therefre P ðpq2e y pq 6 mi x i P k2n i ¼ v ð3. ¼ Sice fi jg 2N 3 was chse arbitrarily, we ca cclude that if ð V is feasible fr [F_EC] LP, the ð Y is feasible fr [F_R] LP. h Ehaced versi f [F_R] The study f the relatis betwee [F_EC] LP ad [F_R] LP fr k =3 gives hits hw t stregthe the recursive frmulati frm the LP relaxati stadpit fr ay iteger k P 3. Nte that fr k = 2, [F_C] ad [F_R] are essetially the same (Veremyev & Bgiski, 2012). Fr k = 3, the dmiace f [F_EC] LP ver [F_R] LP stems frm tw reass the elimiati f all variables that represet chais t ¼ði p q j 2C ¼3 with p 2 (N i \ N j )rq2(n i \ N j ) i [F_EC] ad the liftig effect prvided by cstraits (23). Fr ay k P 3, a similar elimiati is btaied by replacig () with 6 x i 6 v ðl 1 k2ðn i N j ð 0 Due t their recursive ature, variables represet paths directed frm de i t de j. Give ay udirected graph, there is at least e path frm i t j if ad ly if there is at least e path frm j t i. Therefre, fr ay k P 3, we ca add t [F_R] the fllwig cditis fr l =3,..., k i, j 2 V, j > i, 6 x j 6 0 P jvj v ðl 1 ki k2ðn j N i v ðl 1 ki k2n j 1 A þðx j 1 ð 00 ð10 0 The resultig ehaced recursive frmulati fr the k-club prblem is ( ) ½F ERŠ max x i ð6 ð ð 0 ð 00 ð10 ð10 0 ð4 ð11 i2v

7 M.T. Almeida, F.D. Carvalh / Eurpea Jural f Operatial Research 232 (2014) By cstructi, [F_ER] dmiates [F_R]. Prpsiti 5. Z [F_ER] LP 6 Z[F_R] LP. The dmiace f [F_ER] ver [F_R] may be strict. A illustrati is prvided by the graph depicted i Fig. 2(b). It is quite ituitive that if k = 3, [F_EC] ad [F_ER] are LPequivalet. The prf f Prpsiti 6 fllws the ratiale used i the prf f Prpsiti 4. A detailed prf is preseted i Appedix B. Prpsiti 6. Fr k = 3, Z [F_EC] LP = Z[F_ER] LP Cmparis f the LP relaxatis f [F_N] ad [F_R] Fr k = 3, frmulati [F_C] is dmiated bth by [F_N] ad [F_R], but the dmiaces stem frm differet reass. The dmiace f [F_N] ver [F_C] is due t the substituti f E fr E(N i,n j ) i the last term f iequalities (13) (Almeida & Carvalh, 2012). The dmiace f [F_R] ver [F_C] is, rughly speakig, a csequece f (7). It is the atural t suspect that i the x-space the feasible sets f [F_N] LP ad [F_R] LP are differet ad iclusi relati hlds fr them. The graph used t illustrate the dmiace f [F_R] ver [F_C] (see Fig. 1(c)) ca als be used t shw that i the x-space a feasible sluti fr [F_N] LP may t be feasible fr [F_R] LP. O the ther had, i the x-space a feasible sluti fr [F_R] LP may t be feasible fr [F_N] LP. A illustrati is prvided by the graph depicted i Fig. 3. Sice there is iclusi relati i the x-space betwee the feasible sets f [F_R] LP ad [F_N] LP, there is dmiati relati betwee [F_R] ad [F_N] Neighburhd frmulati revisited Fr the case k = 3, eighburhd frmulati [F_N] is mre cmpact tha ay ther frmulati i the literature. Sice, als fr k = 3, [F_EC] ad [F_S] are LP-equivalet, ad [F_S] dmiates [F_N], we ca cclude that [F_N] is als dmiated by [F_EC]. Nte that there is way f impsig cstraits the set f three-edge chais that lik each pair f des fi jg 2N 3 usig ly de ad edge variables. Hwever, it is pssible t derive [F_EC] frm [F_N] by itrducig chai variables ad addig valid iequalities fr the 3-club plytpe. Let us csider agai variables y pq (see Secti 5), ad let us add cstraits (1), (20), (22), ad (23) t [F_N]. After addig these cstraits, cstraits (13) ca be replaced with r2ðn i \N j x r þ ðpq2e y pq fi jg 2N 3 ð21 Nw, edge variables z play rle i the frmulati ad ca be drpped, tgether with cstraits (14). T summarize, (13) ad (14) i [F_N] ca be replaced with 1 ad (20) (23). Thus, by itrducig chai variables y pq ad liftig variable likig cstraits, [F_N] is trasfrmed it [F_EC]. This prcedure prvides a alterative prf f the dmiace f [F_S] ver [F_N] preseted i Almeida ad Carvalh (2012). The large umber f variables added t [F_N] t trasfrm it it [F_EC] raises the questi f the magitude f the differece Z [F_N] LP Z [F_EC] LP. A upper bud this differece is easy t derive by csiderig a variat f frmulati [F_S], deted by [F_S c ], which results frm replacig cstraits (16) with ðc s 1x s fi jg 2N 3 S 2 S ð24 r2ðn i \N j x r þ s2s where c s is the degree f de s i subgraph G. Sice P s2s ðc s 1x s P 1 if ad ly if P s2s x s P 1, frmulati [F_S c ] is valid fr the 3-club prblem. Sice c s P 2 fr all s 2 S ad S 2 S Z½F SŠ LP 6 Z½F S c Š LP. We shall ext shw that [F_N] dmiates [F_S c ]. Prpsiti 7. Z[F_N] LP 6 Z[F_S c ] LP. Prf. Let ð Z be feasible fr [F_N] LP. Fr all fi jg 2N 3 S 2 S, z pq ¼ z pq þ z pq ðpq2e ðpq2e p2ða \S q2b 6 s2ða \S ðpq2e p2ða S q2ðb \S ðc s 1x s þ ad is feasible fr [F_S c ] LP. s2ðb \S ðc s 1x s ¼ s2s h ðc s 1x s ad The dmiace f [F_N] ver [F_S c ] may be strict. A illustrati is prvided by the graph depicted i Fig. 4. The graphs depicted i Figs. 3 ad 4 prvide a illustrati that there is dmiace relati betwee [F_S c ] ad [F_C]. 7. k-clubs with additial cstraits A k-club is by defiiti a subset f des f a graph which iduces a subgraph with diameter at mst equal t k. Ifk is small, a k-club represets a cluster with gd cectivity betwee each pair f its members every member eeds at mst k 1 itermediaries t reach ay ther member. If the uderlyig graph represets a cmmuicati etwrk, the diameter has bee csidered a atural way f describig the reliability f the etwrk (Besch, Harary, & Kabell, 11). Hwever, a k-club is a fragile structure i that if e f its elemets breaks dw by accidet, the cmmuicatis amg members may be severely affected. A k-club is als quite vulerable t exteral attacks the destructi f a selected sigle elemet (de r lik) may reder the cmmuicati amg members ttally impssible. T idetify etwrk clusters that cmbie gd cectivity amg members with better p 1 i j i q 1 p Fig. 3. Csider a sluti i the x-space such that x i ¼ 2 3 xj ¼ 1 xp ¼ xq ¼ 1.By 3 settig ¼ ¼ v ð3 ¼ 1 we get a feasible sluti fr [F_R] 3 LP. Sice P r2ðn i \N j xr ¼ 1 ad E 3 =, cstrait (13) i [F_N] LP fr the pair {i,j} is vilated. q p 2 q 2 Fig. 4. Csider a sluti i the x-space i which x i ¼ 5 xj ¼ 1 xp 1 ¼ xq 2 ¼ 1 4, ad x p2 ¼ x q1 ¼ 1. I [F_S c] LP,S = {A,{p 1,q 2 },B }, thus de cut set cstraits (24) are satisfied fr all S 2 S, but i [F_ N] LP, t cmply with (14), eighburhd cstrait (13) fr {i,j} is vilated. j

8 46 M.T. Almeida, F.D. Carvalh / Eurpea Jural f Operatial Research 232 (2014) 4 4 reliability prperties, Veremyev ad Bgiski (2012) itrduced the ew ccept f R-rbust k-club, which exteds the rigial k-club defiiti by impsig the additial cditi that there must be at least R iterally de-disjit paths betwee every pair f cluster members. A key feature f this ew ccept is that the elimiati f up t R 1 elemets will t destry the k-club structure, which meas that the R-rbust k-clubs have much better errr ad attack tlerace characteristics tha k-clubs (Veremyev & Bgiski, 2012). As pited ut by the authrs, develpig mathematical prgrammig appraches fr fidig large etwrk clusters with gd errr ad attack tlerace characteristics is t a easy task. Fr the case k = 2, all distict paths that lik a give pair f des i ad j are iterally de-disjit, sice each path is either edge (i,j) r a tw-edge path f the frm (i,r,j) with de r i N i \ N j. The R-rbust 2-club prblem is frmulated i Veremyev ad Bgiski (2012) as fllws max Z ¼ x i i2v st a þ k2ðn i \N j x k P Rðx i þ x j 1 i j 2 V i < j ð25 x i 2f0 1g i 2 V ð4 where a = 1 if (i,j) 2 E ad a = 0 therwise. Fr k > 2, t guaratee that there are at least R iterally dedisjit paths betwee ay pair f des it is ecessary t idetify the ier des f each path with mre tha tw edges. I Veremyev ad Bgiski (2012), the w ðl variables that represet the umber f distict paths with l edges (l =2,...,k) that lik each pair f des i ad j (i,j =1,...,jVj) are defied by recursi, ad by its very ature a defiiti by recursi is icmpatible with the idetificati f ier des path by path. Fr that reas, fr k > 2Veremyev ad Bgiski (2012) csidered a relaxati f the R-rbust k-club ccept, btaied by replacig the rigial cditi R iterally de-disjit paths by the relaxed cditi R distict paths. The relaxed R-rbust k-club prblem is frmulated i Veremyev ad Bgiski (2012) with biary de variables x i (i =1,...,jVj) ad iteger path variables w ðl ði j ¼ 1... j V j l ¼ 2... k as fllws max st Z ¼ i2v x i a þ k w ðl P Rðx i þ x j 1 ð26 l¼2 w ð2 6 x k þjvjð2 x i x j ð27 k2ðn i \N j w ð2 P x k jvjð2 x i x j ð2 k2ðn i \N j w ð2 6 jvjx i w ð2 P jvjx i ð2 w ð2 6 jvjx j w ð2 P jvjx j ad fr l ¼ 3... k ð30 w ðl w ðl x i 2f0 1g 6 k2n i w ðl 1 þjvj l 1 ð1 x i ð31 P k2n i w ðl 1 jvj l 1 ð1 x i ð32 w ðl 6 jvj l 1 x i w ðl P jvj l 1 x i ð33 w ðl 2 Z þ ð34 where a = 1 if (i,j) 2 E, a = 0 therwise, ad i, j =1,..., jvj. ð4 Cstraits (26) impse that each pair f selected des must be liked by at least R paths with at mst k edges. Cstraits (27) (30) defie the umber f tw-edge paths that lik each pair f des, ad cstraits (31) (33) defie the umber f paths with three r mre edges that lik each pair f des. Cstraits (34) defie the path variables as egative ad iteger. While fr k = 2 the cditis R iterally de-disjit paths ad R distict paths are equivalet, fr k > 2 the latter is csiderably weaker tha the frmer, sice distict paths may share eve all ier des. This meas that the errr ad attack tlerace prperties f R-rbust k-clubs are t guarateed by cditis (26) (34) ad (4). Fr k = 3, the errr ad attack tlerace prperties f R-rbust k-clubs ca be guarateed by adaptig frmulati [F_EC], as described ext. Give a pair f des i ad j t liked by a edge i G, cstraits (22) ad (23) impse that at mst e chai that icludes de p ad at mst e chai that icludes de q ca be selected t lik them i the iduced subgraph. Furthermre, i frmulati [F_EC], variables y pq are defied ly fr (p,q) 2 E such that p 2 (N i N j ) ad q 2 (N j N i ). Thus, chais represeted by variables y pq are iterally de-disjit with tw-edge chais that lik i ad j (which are represeted by their cetral des). I shrt, all that is missig t mdel R-rbust 3-clubs is a set f variables t idetify each chai with three edges that liks pairs f des i ad j, liked by a edge i G. Fr each pair f des i ad j adjacet i G, let us defie variables y pq, with p 2 N i (N j [ {j}) ad q 2 N j (N i [ {i}). Each variable is assciated with e edge i E 1 ¼fðp q 2E p 2 N i ðn j [fjg q 2 N j ðn i [figg. T btai a valid frmulati fr the R-rbust 3-club prblem, we substitute cstraits r2ðn i \N j x r þ ðpq2e y pq P Rðx i þ x j 1 fi jg 2N 3 ð37 fr cstraits (21) ad elarge frmulati [F_EC] with the ew cstraits x r þ y pq P ðr 1ðx i þ x j 1 ði j 2E ð3 r2ðn i \N j y qðpq2e 1 ðpq2e 1 pq 6 x i y pq 6 x j ði j 2E ðp q 2E 1 ð3 y pq 6 x p ði j 2E p 2 N i ðn j [fjg ð40 pðpq2e 1 y pq 6 x q ði j 2E q 2 N j ðn i [fig ð41 y pq 2f0 1g ði j 2E ðp q 2E1 ð42 Cditis (37) ad (3) impse that each pair f des must be liked by at least R selected chais with at mst three edges. Cditis (3) (41) impse that all des i selected chais that lik adjacet des are icluded i the sluti ad that these chais are iterally de-disjit. T summarize, a ptimal sluti fr ( ½F EC RŠ max ) x i ð12ð1ð22ð23ð4ð20ð37 ð42 i2v is a maximum 3-club that satisfies the additial cditi that every pair f its des is liked by at least R iterally de-disjit chais.. Cclusis I this paper, we have preseted a cmparative study f the LP relaxatis f iteger frmulatis fr the k-club prblem. The

9 M.T. Almeida, F.D. Carvalh / Eurpea Jural f Operatial Research 232 (2014) [F_S] [F_EC] [F_ER] [F_N] [F_R] [F_S γ ] [F_C] Fig. 5. Cectis amg frmulatis fr the 3-club prblem. cmpariss ivlved mdels kw frm the literature ad three ew mdels. Sme mdels are valid fr ay k > 1 whereas sme ther were desiged fr the case k =3. Fr the case k = 3, the cectis amg all these frmulatis are sythesised i Fig. 5. Fr geeric frmulatis [F_ ] ad [F_], [F_ ]? [F_] meas that the LP relaxati f [F_ ] is strger tha the LP relaxati f [F_], ad [F_ ] M [F_] meas that [F_ ] ad [F_] are LP-equivalet. Thick arrws represet the relatis derived i this study ad thi arrws represet relatis derived i Almeida ad Carvalh (2012). A dashed lie betwee [F_ ] ad [F_] meas that there is relati betwee their LP relaxatis. Fr k = 4, we shwed that [F_R] LP is t strger tha [F_C] LP. Based the isight btaied by cmparig [F_EC] ad [F_R] fr k = 3, we devised [F_ER] fr the k-club prblem ad shwed that it dmiates [F_R], fr ay k > 1. I additi, we have derived a frmulati fr the maximum R-rbust 3-club prblem. T the best f ur kwledge, this is the first frmulati i the literature fr the maximum R-rbust 3-club prblem. Ackwledgemets This wrk is supprted by Natial Fudig frm FCT-Fudaçã para a Ciêcia e Teclgia, uder the prject PEst-OE/MAT/ UI0152. The authrs thak the aymus referees fr their cmmets ad suggestis. Thaks are als due t A Heshall fr her assistace i editig this versi f the paper. Appedix A Csider graph G =(V,E) depicted i Fig. A.1 ad k =4. Suppse that x i ¼ x j ¼ 1 x 1 ¼ x 2 ¼ x 3 ¼ 1, ad x 5 4 ¼ I [F_C] LP there are six variables y t, assciated with chais (i,1,j), (i,1,2,j), (i,3,2,j), (i,3,2,4,j), (i,1,2,4,j), ad (i,3,2,1,j). Csider ay ð Y. T be a sluti fr [F_C] LP, it must cmply i 3 2 Fig. A.1. Graph t illustrate that [F R ] des t dmiate [F C ] fr k= j with P 6 t¼1 y t 6, but the cstrait (2) fr the pair {i,j}, is 10 vilated. Csider w ð V, with, ab ¼ mi x a x b ¼ mi x a ab v ðl 1 kb k2n a x k fr all a b 2 V fr all a b 2 V l ¼ 3 4 k2ðn a\n b P The, 4 l¼2 vðl ¼ 1 þ 2 þ 2 ¼ 1, which satisfies cstrait (6) i [F_R] LP, fr the pair {i,j}. Appedix B. prf f Prpsiti 6 Prpsiti 6. Fr k = 3, Z[F_EC] LP = Z[F_ER] LP. Prf. Suppse that ð V is feasible fr [F_ER] LP. Let us shw that it is pssible t build frm it a feasible sluti ð Y fr [F_EC] LP. We shall assume, withut lss f geerality, that < ¼ mi x = ab a x b x k fr all a b 2 V k2ðn a\n b < ¼ mi x ab a x b v ð3 k2ðn an b kb ka k2ðn b N a = fr all a b 2 V Fr packig cstraits ad fr chais with tw edges, the ratial f the prf is the same ratiale used i Secti 6.2. Let us w address chais with three edges that lik des i ad j. Let C e ¼3 ¼ t ¼ði p q j 2C ¼3 p R N j q R N i be the set f chais assciated with variables y pqðfi jg 2N 3 p 2 A q 2 B i [F_EC] LP. Csider a pair fi jg 2N 3, chse arbitrarily. Let D be a layered digraph with seve layers L 1 ¼fig L 2 ¼fi 0 g L 3 ¼ A L 4 ¼ t t 2 C e ¼3 L 5 ¼ B L 6 ¼fj 0 g ad L 7 ¼fjg (where i 0 ad j 0 are cpies f i ad j, respectively) whse set f arcs is A ¼fði i 0 g [ fði 0 p p 2 A g[ [ ðp t p 2 A t 2 C e ¼3 ad p is a de f t [ [ ðt q q 2 B t 2 C e ¼3 ad q is a de f t [ [fðq j 0 q 2 B g[fðj 0 jg ad arc capacities are defied by

10 4 M.T. Almeida, F.D. Carvalh / Eurpea Jural f Operatial Research 232 (2014) 4 4 >< cap ðuv¼ x i if u ¼ i ad v ¼ i 0 if u ¼ i 0 ad v ¼ p 2 A x p if u ¼ p 2 A ad v ¼ t 2 e C ¼3 x q if u ¼ t 2 e C ¼3 qi if u ¼ q 2 B ad v ¼ j 0 > x j if u ¼ j 0 ad v ¼ j ad p is a de f t ad v ¼ q 2 B ad q is a de f t Nte that each edge (p,q) 2 E is represeted i D by exactly e de t 2 L 4. This de will be deted by u(p,q). Suppse that f is a maximum i j flw i D. By cstructi, value ðf ¼ P p2a f pt ¼ P q2b f tq. Als by cstructi, t2e C ¼3 ðuv P u2l3 v2l4cap x p P < mi x p x j x k p2a p2a k2b ad ðuv P u2l4 v2l5cap x q P < mi x q x i x k q2b q2b k2a Therefre, value ðf ¼mi x i Pp2A = t2e C ¼3 ¼ p2a = ¼ q2b Pq2B qi x j qi ¼ v ð3. If we set y pq ¼ f puðpq fr all (p,q) 2 E, the value ðf ¼ Pðpq2E y pq. Fr the pair {i,j}, all cstraits (22) ad (23) are satisfied, due t de flw cservati i L 3 ad L 5. Sice cstraits (21) fr {i,j} are als satisfied, ad the pair fi jg 2N 3 was chse arbitrarily, sluti ð Y is feasible fr [F_EC] LP. Cversely, suppse that ð Y is feasible fr [F_EC] LP. By repeatig the argumets used i the prf f Prpsiti 4, we cclude that ð V is feasible fr [F_ER] LP, where < ¼ mi x = ab a x b x k fr all a b 2 V k2ðn a\n b < v ð3 ¼ mi x ab a x b = kb ka fr all a b 2 V k2ðn an b k2ðn b N a Refereces Alba, R. D. (173). A graph-theretic defiiti f a scimetric clique. Jural f Mathematical Scilgy, 3, Alidaee, B., Glver, F., Kcheberger, G., & Wag, H. (2007). Slvig the maximum weight clique prblem via ucstraied quadratic prgrammig. Eurpea Jural f Operatial Research, 11, Almeida, M. T., & Carvalh, F. D. (2012). Iteger mdels ad upper buds fr the 3- club prblem. Netwrks, 60, Balas, E. (2005). Prjecti, liftig ad exteded frmulati i iteger ad cmbiatrial ptimizati. Aals f Operatis Research, 140, Balasudaram, B., Butek, S., & Trukhav, S. (2005). Nvel appraches fr aalyzig bilgical etwrks. Jural f Cmbiatrial Optimizati, 10, Bla_zewicz, J., Frmawicz, P., & Kasprak, M. (2005). Selected cmbiatrial prblems f cmputatial bilgy. Eurpea Jural f Operatial Research, 161, Besch, F. T., Harary, F., & Kabell, J. A. (11). Graphs as mdels f cmmuicati etwrk vulerability Cectivity ad persistece. Netwrks, 11, Bgiski, V., Butek, S., & Pardals, P. M. (2006). Miig market data A etwrk apprach. Cmputers & Operatis Research, 33, Bmze, I. M., Budiich, M., Pardals, P. M., & Pelill, M. (1). The maximum clique prblem. I D.-Z. Du & P. M. Pardals (Eds.), Hadbk f cmbiatrial ptimizati (pp. 1 74). Drdrecht, The Netherlads Kluwer Academic Publishers. Burjlly, J.-M., Laprte, G., & Pesat, G. (2002). A exact algrithm fr the maximum k-club prblem i a udirected graph. Eurpea Jural f Operatial Research, 13, Butek, S., & Wilhelm, W. E. (2006). Clique detecti mdels i cmputatial bichemistry ad gemics. Eurpea Jural f Operatial Research, 173, 1 7. Carvalh, F. D., & Almeida, M. T. (2011). Upper buds ad heuristics fr the 2-club prblem. Eurpea Jural f Operatial Research, 210, Guveia, L., & Vss, S. (15). A classificati f frmulatis fr the (timedepedet) travelig salesma prblem. Eurpea Jural f Operatial Research, 3, 6 2. Magati, T. L., & Wlsey, L. A. (15). Optimal trees. I M. O. Ball, T. L. Magati, C. L. Mma, & G. L. Nemhauser (Eds.). Hadbks i peratis research ad maagemet sciece (Vl. 7, pp ). Elsevier. Mahdavi Pajuh, F., & Balasudaram, B. (2012). O iclusiwise maximal ad maximum cardiality k-clubs i graphs. Discrete Optimizati,, 4 7. Martis, P. (2010). Exteded ad discretized frmulatis fr the maximum clique prblem. Cmputers & Operatis Research, 37, Mkke, R. J. (17). Cliques, clubs ad clas. Quality ad Quatity, 13, Nemhauser, G. L., & Wlsey, L. A. (1). Iteger ad cmbiatrial ptimizati. New Yrk Jh Wiley. Oca, T., Altiel, I. K., & Laprte, G. (200). A cmparative aalysis f several asymmetric travelig salesma prblem frmulatis. Cmputers & Operatis Research, 36, Srese, M. (2004). New facets ad brach ad cut algrithm fr the weighted clique prblem. Eurpea Jural f Operatial Research, 154, Veremyev, A., & Bgiski, V. (2012). Idetifyig large rbust etwrk clusters via ew cmpact frmulatis f maximum k-club prblems. Eurpea Jural f Operatial Research, 21,