he edge my inerfere wih communicion from he oher se sion dening he edge In mos pplicions inerference occurs whenever he disnce eween he frequencies ss

Transcription

1 The ril Consrin Sisfcion rolem: Fces nd Lifing Theorems Arie MCA Koser 1;2 Sn M vn Hoesel 1;3 Anoon WJ Kolen 1;4 Jnury 30, 1997 Asrc In his pper he pril consrin sisfcion prolem ècsè is inroduced nd formuled s f0; 1g-progrmming prolem We dene he pril consrin sisfcion polyope s he convex hull of fesile soluions for his progrmming prolem As exmples of he clss of prolems sudied we menion he frequency ssignmen prolem nd he mximum sisiliy prolem Lifing heorems re presened nd some clsses of fce-dening vlid inequliies for CS re given Compuionl resuls show h hese vlid inequliies reduce he gp eween L-vlue nd I-vlue susnilly 1 The pril consrin sisfcion prolem A pril consrin sisfcion prolem is dened y èg = èv;eè;d V ; E ;Q V è, where G = èv;eè is conneced grph clled he consrin grph, D V is se of domins D v, v 2 V where ech domin is nie se, E is se of èedge-èpenly funcions fv;wg : ffd v ;d w g j d v 2 D v ;d w 2 D w g! R, fv; wg 2 E, nd Q V is se of èverex-èpenly funcions Q v : D v! R, v 2 V The pril consrin sisfcion prolem is o selec excly one vlue d v in he domin D v for every v 2 V so s o minimize he ol sum of he penlies, ie fv;wg2e fv;wgèfd v ;d w gè+ v2v Q vèd v è The Frequency Assignmen rolem èfaè elongs o he clss of pril consrin sisfcion prolems For he FA verex corresponds o se sion, ie direcionl nenn, in moile elephone nework The domin of verex is he se of frequencies h cn e ssigned o h se sion, nd n edge indices h communicion from one se sion dening 1 Dep of Quniive Economics, Msrich Universiy, OBox 616, 6200 MD Msrich, The Neherlnds 2 e-mil: AKoser@KEUniMsNL; home pge: hp:èèwwwunimsnlè~koserèindexhml 3 e-mil: SvnHoesel@KEUniMsNL 4 e-mil: AKolen@KEUniMsNL 1

2 he edge my inerfere wih communicion from he oher se sion dening he edge In mos pplicions inerference occurs whenever he disnce eween he frequencies ssigned o he sions is less hn given hreshold depending on he wo se sions The penly of n edge reecs he prioriy wih which inerference should e voided, wheres he penly on verex cn e seen s level of preference for he frequencies For noher ype of frequency ssignmen prolems, involving receiver-rnsmier pirs of rdio links, h cn e formuled s pril consrin sisfcion prolem, we refer o Kolen ë2ë The Mximum Sisiliy rolem èmax SATè cn e formuled s pril consrin sisfcion prolem In MAX SAT prolem m cluses c 1 ;:::;c m involving he oolen vriles x 1 ;:::;x n re given Ech cluse conins numer of lierls, where lierl is eiher vrile or he negion of vrile The prolem is o ssign vlue rue or flse o ech vrile so s o mximize he numer of cluses h re sised A cluse is sised if les one lierl in i hs he vlue rue I is no srighforwrd o model MAX SAT s pril consrin sisfcion prolem We inroduce verex v ci for every cluse c i ; i = 1;:::;m nd verex v xj for every vrile x j ; j =1;:::;n The domin of v ci conins vlue for ech lierl in he cluse c i, le us denoe his vlue y he lierl iself The domin of v xj is given y frue; flseg There is n edge eweenverex v ci represening cluse c i, nd verex v xj represening vrile x j if nd only if x j 2 c i or çx j 2 c i èçx j is he negion of x j è If x j 2 c i, hen he penly of he cominion of domin vlues èx j ; flseè is equl o 1 If çx j 2 c i, hen he penly of he cominion of domin vlues èçx j ;rueè is equl o 1 All oher penlies re zero The opiml vlue of his pril consrin sisfcion prolem is k if nd only if he opiml vlue of he corresponding MAX SAT is m, k Furhermore, n opiml soluion of he MAX SAT is given y he domin vlues seleced for he verices corresponding o he vriles in he opiml soluion of he pril consrin sisfcion prolem This shows h he wo prolems re equivlen Since MAX 2 SAT èech cluse conins mos 2 lierlsè is N-hrd ègrey, Johnson nd Sockmeyer ë1ëè inry consrin sisfcion prolem wih jd v j = 2 for ll v 2 V is lredy N-hrd For he MAX 2 SAT prolem more compc formulion is possile We hve verex v xj corresponding o every vrile x j nd he domin is given y frue; flseg There is n edge fv xi ;v xj g if nd only if here exiss cluse conining lierl corresponding o x i nd lierl corresponding o x j The penly corresponding o cominion of vlues for he vriles x i nd x j is equl o he numer of cluses conining lierls corresponding o oh vriles for which he given cominion does no sisfy he cluse The sisiliy prolem èsatè, in which he quesion is wheher here is n ssignmen of he vriles for which ll cluses re sised, cn lso e formuled s pril consrin sisfcion prolem s follows There is one verex for every cluse nd n edge if he wo corresponding cluses conin conicing lierl corresponding o he sme vrile A cominion fx i ; x i g wih x i 2 C j nd x i 2 C k hs penly one All cominions corresponding o nonconicing lierls hve penly zero The prolem is sisle if nd only if he corresponding 2

3 pril consrin sisfcion prolem hs opiml vlue zero In Secion 2 of his pper we formule he pril consrin sisfcion prolem s f0; 1gprogrmming prolem, we se he dimension of he prolem nd descrie he rivil fce dening vlid inequliies We prove heorems for lifing fces of suprolem o fces for he originl prolem in Secion 3 In Secion 4 we dene some clsses of fces for he CS Some preliminry compuionl resuls re ddressed in Secion 5, wheres he ls secion conins he concluding remrks 2 Formulion, Dimension nd Trivil Fces To formule he pril consrin sisfcion prolem s f0; 1g-progrmming prolem we inroduce he following f0; 1g-vriles for ll v 2 V nd d v 2 D v n 1 if dv 2 D yèv; d v è = v is seleced 0 oherwise nd for ll fv; wg 2E,d v 2D v nd d w 2 D w n xèv; d v ;w;d w è = 1 if èdv ;d w è2d v D w is seleced 0 oherwise Noe h since he consrin grph is undireced xèv; d v ;w;d w è nd xèw; d w ;v;d w è denoe he sme vrile To e consisen wih he wy we denoe he x nd y-vriles, le pèv; d v ;w;d w è nd qèv; d v è denoe fv;wg èfd v ;d w gè nd Q v èd v è, respecively A f0; 1g-progrmming formulion of he inry consrin sisfcion prolem is given y min s fv;wg2e d w 2D w pèv; d v ;w;d w èxèv; d v ;w;d w è + v2v d v2d v qèv; d v èyèv; d v è è1è d v2d v yèv; d v è=1 8v2V è2è d w2d w xèv; d v ;w;d w è=yèv; d v è 8fv; wg 2E; d v 2 D v è3è d v2d v xèv; d v ;w;d w è2f0;1g 8fv; wg 2E;d v 2 D v ;d w 2D w è4è yèv; d v è 2f0;1g 8v2V; d v 2 D v è5è Consrins è2è model he fc h excly one vlue in he domin of verex should e seleced Consrins è3è model he fc h he cominion of vlues seleced for n edge should e consisen wih he vlues seleced for he verices of h edge 3

4 We dene he pril consrin sisfcion polyope XèCSè o e he convex hull of ll f0; 1g-vecors èy; xè sisfying è2è nd è3è Alhough he y-vriles cn e elimined from he formulion, we hve found i more convenien o keep hem in he formulion Noe h once he y-vriles re f0; 1g he x-vriles re forced o e f0; 1g Therefore, he x-vriles cn e relxed o e ë0; 1ë-vriles The dimension of he inry consrin sisfcion polyope is given y Theorem 21 Theorem 21 The dimension of XèCSè, dened y èg =èv;eè;d V è is v2v èjd vj,1è + fv;wg2e èjd vj,1èèjd w j,1è roof We will rs prove h he dimension is less hn or equl o he given vlue y dening v2v jd vj + fv;wg2e jd vjjd w jè= numer of vrilesè, è v2v èjd vj,1è + fv;wg2e èjd vj,1èèjd w j,1è = jv j + fv;wg2e èjd vj + jd w j,1è linerly independen equliies which re sised y ll soluions of CS These liner independen equliies re oined y king ll consrins è2è, nd for every edge fv; wg ll u one of he consrins è3è The consrins è3è for given edge fv; wg cn e viewed s he consrins of rnsporion prolem wih suppliers indiced y èv; d v è wih supply yèv; d v è nd cliens indiced y èw; d w è wih demnd yèw; d w è I is well-known h deleing one of hese consrins resuls in se of liner independen equliies Nex, we will prove h he dimension is greer hn or equl o he given vlue y dening 1+ v2v èjd vj,1è + fv;wg2e èjd vj,1èèjd w j,1è nely independen soluions Noe h once he y-vriles re given, he x-vriles re uniquely deermined y consrins è3è To dene hese soluions we rirrily selec vlue d v 2 D v One soluion is given y yèv; d vè=1 for ll v 2 V For ech v 2 V nd d v 2 D v nfd vg, we dene he soluion yèv; d v è = 1; yèw; d wè = 1 for ll w 6= v Noe h here re v2v èjd vj,1è soluions of his ype For ech fv; wg 2 E, d v 2 D v nfd vg; nd d w 2 D w nfd wg, we dene he soluion yèv; d v è = yèw; d w è = 1 nd yèu; d uè = 1 for ll u 2 V, u 6= v, u 6= w Noe h here re fv;wg2e èjd vj,1èèjd w j,1è soluions of his ype These soluions re nely independen ecuse he v2v èjd vj,1è+ fv;wg2e èjd vj,1èèjd w j,1è vecors oined y surcing he rs soluion from ll oher soluions re linerly independen To see his noe h ech vecor hs one in componen in which ech previously dened vecor hs zero For he soluion dened y èv; d v è ke he componen corresponding o xèv; d v ;w;d wè or xèw; d w ;v;d vè for n edge fw; vg 2 E For he soluion dened y fv; wg 2 E, d v 2 D v, d w 2 D w ke he componen corresponding o xèv; d v ;w;d w è 4

5 I follows srighforwrd h he non-negiviy consrins of he x-vriles dene fces of he polyope if oh domins hve les wo elemens Theorem 22 For every fv; wg 2E,jD v jç2,jd w jç2,d v 2D v,d w 2D w he inequliy xèv; d v ;w;d w èç0 è6è denes fce for XèCSè roof In he proof of Theorem 21 we lised dim XèCSè + 1 nely independen soluions excly one of which hs xèv; d v ;w;d w è = 1 è d v 6= d v, d w 6= d w è nd ll ohers hve xèv; d v ;w;d w è = 0: Hence, we hve dim XèCSè nely independen soluions sisfying xèv; d v ;w;d w èç0 wih equliy 3 Lifing heorems In his secion we will discuss wo dieren ypes of lifing Firsly, we show h fce dening inequliy of pril consrin sisfcion prolem dened y he consrin grph G =èv;eè nd se of domins D v, v 2 V lso denes fce for he pril consrin sisfcion prolem dened y ny consrin grph for which G = èv;eè is n induced sugrph, nd se of domins where he domin for verex v 2 V is unchnged nd ll oher verices hve domin of crdinliy one If XèCSè is dened y èg = èv;eè;d V è, le X u ècsè denoe he CS-polyope dened y he induced sugrph on V nfugwih he sme domins Theorem 31 Le XèCSè e dened yèg=èv;eè;d V è wih jd u j =1, for some u 2 V If çx ç ç 0 is fce dening inequliy for X u ècsè, hen çx ç ç 0 is fce dening inequliy for XèCSè roof The polyopes elonging o oh prolems hve he sme dimension Nex, we show how fce dening inequliy of consrin sisfcion prolem dened y he consrin grph G = èv;eè nd se of domins D v, v 2 V cn e lifed ino fce dening inequliy for he consrin sisfcion prolem y he sme consrin grph nd se of domins D 0 v, v 2 V; where D 0 v = D v, for ll v 2 V, v 6= u, nd D 0 u = D u ëfd 0 ug Theorem 32 ses h if we mke d 0 u copy ofny domin elemen d u 2 D u èie he coecien of xèu; d 0 u ;v;d vè is equl o he coecien of xèu; d u ;v,d v è, for ll v 2 u, d v 2 D v, where u denes he se of neighours of u in he consrin grph; u = fv jfu; vg 2Egè, hen he new inequliy is fce dening for he exended prolem whenever he originl inequliy is fce dening for he originl prolem 5

6 In order o prove Theorem 32 we need Lemm 31 nd Lemm 32 The componens corresponding o èv; d v è re given y xèv; d v ;w;d w è for ll w 2 v, d w 2 D w nd yèv; d v è Lemm 31 If v2 u d v 2D v èu; d u ;v;d v èxèu; d u ;v;d v è ç 0, u 2 V, d u 2 D u, is fce dening vlid inequliy for XèCSè, hen he inequliy descries rivil fce roof We rs prove h y dding implici equliies of he CS he vlid inequliy cn e rewrien s v2 u d v2d v èu; d u ;v;d v èxèu; d u ;v;d v èç0 wih èu; d u ;v;d v èç0 Le d v 2 D v, v 2 u, e such h èu; d u ;v;d vè = min dv2d v fèu; d u ;v;d v èg We dd he implici equliies èu; d u ;v;d vèèyèu; d u è, v2d v xèu; d u ;v;d v èè = 0 o he inequliy We oin vlid inequliy of he form v2 u èu; d u ;v;d vèyèu; d u è+ v2 u d v2d v èèu; d u ;v;d v è,èu; d u ;v;d vèèxèu; d u ;v;d v èç0 From he soluion in which we selec d v 2 D v for ll v 2 u nd d u 2 D u for u, nd he vlidiy of x ç 0 i follows h v2 u èu; d u ;v;d vè ç 0 Susiue yèu; d u è= d v2d v xèu; d u ;v;d v è for some v 2 u nd we oin n inequliy where ech coecien is nonnegive Since xèu; d u ;v;d v èç0isvlid for ll fu; vg 2E; d u 2 D u, d v 2 D v i follows h he dimension of he fce of he inequliy is mximl if here is excly one nonzero coecien èu; d u ;v;d v è In h cse he inequliy denes he sme fce s he inequliy xèu; d u ;v;d v èç0 Lemm 32 Le çx ç ç 0 dene non-rivil fce of XèCSè Then for ech u 2 V, d u 2 D u here re excly q = 1+ v2 u èjd u j,1è soluions wih yèu; d u è = 1 nd çx = ç 0 which re nely independen wih respec o he componens corresponding o èu; d u è roof Le èy 1 ;x 1 è;:::;èy p ;x p è e p = dim XèCSè nely independen soluions which sisfy çx ç ç 0 wih equliy Moreover, le èy 1 ;x 1 è;:::;èy q ;x q è e he soluions wih yèu; d u è= 1 which re nely independen wih respec o he componens yèu; d u è nd xèu; d u ;v;d v è for ll v 2 u, d v 2 D v Noe h y yèu; d u è = 1 hese soluions re lso linerly independen We prove h he corresponding mrix A wih 1+ v2 u jd v jrows nd q columns hs rnk 1+ v2 u èjd v j,1è, which implies h here re excly 1 + v2 u èjd v j,1è soluions which re nely independen wih respec o hese componens To prove h he mrix hs rnk 1+ v2 u èjd v j,1è we will prove h here re excly j u j liner independen vecors such h A = 0: Every column of A sises yèu; d u è = d v2d v xèu; d u ;v;d u è for ll v 2 u Therefore here re les j u j liner independen vecors such h A = 0 Assume here exiss noher vecor such h A = 0 which is liner independen from he oher j u j liner independen vecors If he coecien of yèu; d u è is nonzero, hen we use one of he equliies yèu; d u è = d v 2D v xèu; d u ;v;d v è o elimine his coecien 6

7 Since ll soluions èy 1 ;x 1 è;èy 2 ;x 2 è;:::;èy p ;x p è sisfy x = 0 i follows h fx 2 X j çx = ç 0 gçfx2xjx =0g If equliy does no hold, hen fx 2 X j x =0g=Xnd x =0 is n implici equliy However, is liner independen from he implici equliies involving xèu; d u ;v;d u è, v 2 u, d v 2 D v Hence fx 2 X j x =0g=fx2Xjçx = ç 0 g I follows h eiher x ç 0orx ç 0isvlid inequliy for XèCSè dening he sme fce s çx ç ç 0 Wihou loss of generliy èmuliply y,1 if necessryè ssume x ç 0 for ll x 2 X I is proved in Lemm 31 h in h cse çx ç ç 0 denes rivil fce Now, we cn prove he min heorem of his pper Theorem 32 Le XèCSè e dened y èg = èv;eè;d V è Le u 2 V, d u 2 D u Dene X 0 ècsè y ègèv;eè;dv 0 è wih D0 v = D v, v 2 V nfug, Du 0 = D u ëfd 0 u g If çx ç ç 0 is non-rivil fce dening inequliy for XèCSè, hen çx + v2èuè d v2d v çèu; d u ;v;d v èxèu; d 0 u ;v;d vèçç 0 è7è is fce dening for X 0 ècsè roof Firs, noe h dim X 0 ècsè = dim XèCSè+1+ v2 u èjd v j,1è Le he soluions èy 1 ;x 1 è;:::;èy p ;x p è, where p = dim XèCSè, e se of nely independen soluions which sisfy çx ç ç 0 wih equliy I follows from Lemm 32 h here exis 1 + v2 u èjd v j,1è soluions which sisfy yèu; d u è = 1 nd re nely independen wih respec o èu; d u è Replce in hese soluions d u y d 0 u Then hese new soluions ogeher wih he old soluions re nely independen In Secion 4 we will dene some fce dening inequliies for pril consrin sisfcion prolem dened y G =èv;eè nd se of domins D v, v 2 V To prove h hey re fce dening we will rs prove h hey re fce dening for consrin sisfcion prolem dened y n induced sugrph G S = ès; E S è of G nd se of domins D v, wih jd v j = 2, v 2 S Nex, Theorems 31 nd 32 re used o exend hese fces dening inequliies o fce dening inequliies for he originl prolem 4 Non-rivil fces of he CS The non-rivil fces we will descrie in his secion re chrcerized y n induced sugrph G S = ès; E S è of he consrin grph G For every v 2 S he domin D v is priioned ino A v nd B v Domin vlues in A v re copies of one noher; likewise he domin vlues in B v Therefore o descrie he fces i is sucien o specify for ech edge fv; wg 2E S he coeciens èv; wè, èv; wè, èv; wè nd èv; wè corresponding o he coecien of xèv; d v ;w;d w è wih respecively fd v ;d w g2a v A w,fd v ;d w g2a v B w,fd v ;d w g2b v A w nd fd v ;d w g2b v B w 7

8 The fce kes he form fv;wg2e S è d v2a v d v2a v d v2b v d w 2A w èv; wèxèv; d v ;w;d w è+ d w 2B w èv; wèxèv; d v ;w;d w è+ d w2a w èv; wèxèv; d v ;w;d w è+ d v2b v d w2b w èv; wèxèv; d v ;w;d w èè ç c è8è where ç 2fç;çg I follows from he lifing heorems in Secion 3 h in order o prove h n inequliy of ype è8è is fce dening i is sucien o prove h è8è is fce dening for XèCSè dened y G S =ès; E S è nd A v = f v g, B v = f v g, for ll v 2 S 41 The cycle-inequliy Firsly, we inroduce he cycle-inequliy A k,cycle inequliy, k ç 3, is dened y S = fv i j i =1;:::;kg E S = ffv i ;v i+1 gji=1; ::; k, 1gg ë ffv k ;v 1 gg èv i ;v i+1 è=èv i ;v i+1 è=1;èv i ;v i+1 è=èv i ;v i+1 è=0 èv k ;v 1 è=èv k ;v 1 è=0;èv k ;v 1 è=èv k ;v 1 è=1 ç = ç c = k,1 i=1;:::;k,1 We will cll domin vlue d v 2 D v n -vlue whenever d v 2 A v ; oherwise i is -vlue Figure 1 shows he 3-cycle nd 4-cycle inequliy The -do represens he A-suse of he domin; he -do represens he B-suse of he domin A line eween wo dos indice h he coecien corresponding o he indiced wo suses is equl o one Theorem 41 The k-cycle inequliy, k ç 3, is vlid for XèCSè roof Consider soluion Ech edge of he cycle conriues mos one o he lef hnd side of è8è If d vk nd d v1 re -vlues, hen he edge fv 1 ;v k g does no conriue o he lef hnd side of è8è nd hence è8è is sised The sme cn e pplied if d vk nd d v1 re -vlues If d v1 is n -vlue nd d vk is -vlue, hen here exiss n i,1 ç i ç k, 1, such h d vi is n -vlue nd d vi+1 is -vlue Hence he edge fv i ;v i+1 g does no conriue o he lef hnd side of è8è, nd hence è8è is sised If d v1 is -vlue nd d vk is n -vlue, hen he sme resoning pplies 8

9 ç 2 ç 3 Figure 1: Cycle Inequliies The proof of Theorem 41 lso indices he srucure of he soluions which sisfy è8è wih equliy If d v1 nd d vk re oh -vlues, hen ll oher domin vlues in he cycle mus e -vlues s well If d v1 nd d vk re oh -vlues, hen ll oher domin vlues in he cycle mus e -vlues s well If d v1 is n -vlue nd d vk -vlue, hen here exiss n i, 1 ç i ç k, 1, such h d vj, 1 ç j ç i, isn-vlue, nd d vj ;i+1çjçk is -vlue If d v1 is -vlue nd d vk is n -vlue, hen here exiss n i, 1 ç i ç k, 1, such h d vj ; 1 ç j ç i is -vlue, nd d vj, i +1çjçk is n -vlue Theorem 42 The k-cycle inequliy, k ç 3, is fce dening for XèCSè roof By he resuls of Secion 3 i is sucien o prove h he k-cycle inequliy is fce dening for XèCSè dened y he k-cycle consrin grph nd A vi = f i g, B vi = f i g, i = 1;:::;k The dimension of Xè CS è is 2k The 2k nely independen soluions sisfying he k-cycle inequliy wih equliy re given elow Afer ech soluion we hve indiced componen for which his soluion is he unique soluion hving one in his componen This proves h hese soluions re nely independen è 1 ;:::; k è èxèv 1 ; 1 ;v k ; k èè è 1 ;:::; i ; i+1 ;:::; k è i=1;:::;k,1 èxèv i ; i ;v i+1 ; i+1 èè è 1 ;::: ; k è èxèv 1 ; 1 ;v k ; k èè è 1 ;:::; i ; i+1 ;:::; k è i=1;:::;k,1 èxèv i ; i ;v i+1 ; i+1 èè 9

10 42 The clique-cycle inequliy A second clss of fce dening vlid inequliies re he clique-cycle inequliies A k,cliquecycle inequliy, k ç 3, is dened y S = fv i j i =1;:::;kg E S = ffv i ;v j gji; j =1;:::;k;i éjg E C = ffv i ;v i+1 gji=1;:::;k,1gëffv k ;v 1 gg suse forming k-cycle èv; wè =èv; wè =èv; wè =0;èv; wè =1 fv; wg 2E S ne C èv i ;v i+1 è=èv i ;v i+1 è=èv i ;v i+1 è=1;èv i ;v i+1 è=0 i=1;:::;k,1 èv k ;v 1 è=èv k ;v 1 è=èv k ;v 1 è=1;èv k ;v 1 è=0 ç = ç c = k,1 Figure 2 shows clique-cycle inequliy for k = 3 nd k =4 For k = 3 he fce descried y clique-cycle inequliy is he sme s fce descried y @@,,,,, ç 2 ç 3 Figure 2: Clique-Cycle Inequliy Theorem 43 The k-clique-cycle inequliy, k ç 3, is vlid for XèCSè roof Consider n rirry soluion Whenever n -node is seleced in D vi, hen he edge in he k-cycle fv i ;v i+1 g èor fv k ;v 1 g whenever i = kè conriues excly one o he lef hnd side of è8è Hence è8è is vlid whenever les k, 1 -nodes re seleced Therefore, le us ssume h k, p -nodes nd hence p -nodes re seleced, p ç 2 If v vn w re oh -nodes, hen he edge fv; wg conriues one o he lef hnd side of è8è The ol conriuion of ll edges eween -nodes is pèp, 1è=2 10

11 The ol conriuion of he -nodes is equl o he numer of -nodes k, p To prove h he ol conriuion is les k, 1wehve o prove h pèp, 1è=2+èk,pèçk,1, ie p 2, 3p +2ç0orèp,1èèp, 2è ç 0 This holds since p ç 2 The proof of Theorem 43 lso indices he srucure of he soluions which sisfy è8è wih equliy A soluion sises è8è wih equliy if nd only if he numer of -nodes seleced is eiher one or wo Theorem 44 The k-clique-cycle inequliy, k ç 3, is fce dening for XèCSè roof By he resuls of Secion 3 i is sucien o prove h he k-clique-cycle inequliy is fce dening for XèCSè dened y he k-clique consrin grph nd A vi = f i g, B vi = f i g, i = 1;:::;k The dimension of Xè CS è is k+kèk,1è=2 The k+kèk,1è=2 nely independen soluions sisfying è8è wih equliy re given elow For ech soluion we will lso specify componen for which his soluion hs vlue one nd ll previously dened soluions hve vlue zero his componen This proves h he soluions re nely independen For ech p =1;:::;k we dene v p v i is -node i 6= p is n -node wih componen xèv p ; p ;v p+1 ; p+1 èèxèv k ; k ;v 1 ; 1 èifp=kè For ech p; q =1;:::;k, péq,we dene v p v q v i is -node is -node i 6= p; q is n -node wih componen xèv p ; p ;v q ; q è 5 Compuionl Resuls A rs es of he quliy of he vlid inequliies descried ove is done on 11 insnces wih jd v j = 2 for v 2 V These insnces re suprolems of he Frequency Assignmen CALMAinsnce celr8, which hve o e solved in he crossover of he geneic lgorihm descried y Kolen ë2ë We used he cllle lirry of CLEX 40 o solve he liner progrmming relxion èz L è, he è0; 1è-progrmming prolem èz I èswell s he liner progrmming relxion wih 3- cycle vlid inequliies èz 3 è The selecion of violed vlid inequliies ws done y enumrion of ll vlid inequliies wih k = 3 èie 4 vlid inequliies for ech 3-cycle were villeè For 11

12 ll insnces we hve jv j = 458 nd jej = 1655 The resuls re presened in Tle 1 The progrm wrien in C++ ws running on DEC 2100 A500M worksion wih 128M inernl memory The le shows h for ll insnces he L-relxion wih 3-cycle vlid inequliies gives n ineger soluion The numer of violed inequliies which hd o e dded is given in he ls column The compuion imes were in verge reduced y 764è insnce z L z 3 z I CU z L CU z 3 CU z 3+I CU z I èvi c c c c c c c c c c c p Tle 1: Compuionl resuls An insnce wih lrge gp eween L nd I ws given y suprolem of Frequency Assignmen rolem of lrge elecommunicion compny This insnce hs 708 verices nd 1677 edges ègin ll domins conins 2 vluesè The 3-cycle inequliies close he gp eween L nd I wih 926è Wih hese vlid inequliies CLEX needed 113 nodes rnch-ndound nodes o oin nd prove he opiml vlue CLEX ws no le o solve his insnce o opimliy wihou dding vlid inequliies 6 Concluding Remrks In he cse jd v j = 2 for ll v 2 V he numer of k-cycle inequliies which descrie dieren fces of he polyope is 2 k,1, which give us he possiiliy he enumere ll vlid inequliies for smll k èwhich is done for he insnces menioned in he previous secion However, if he numer of domin elemens grows he numer of ville cycle nd clique-cycle inequliies which dene dieren fces increses enormously Therefore, in fuure pper he seprion prolems for ech clss of vlid inequliies will e discussed Heurisics for hese seprion prolems hve o e developed, nd hve o e implemened in Brnch-nd-Cu frmework o solve lrge-size rel-life prolems èlike he CALMA-insncesè Moreover, due o he size of hese insnces, prolems will rise in solving he L relxion 12

13 References ë1ë M Grey, D Johnson, nd L Sockmeyer ësome simplied N-complee grph prolems" Theoreicl Compuer Science, 1:237í267, 1976 ë2ë AWJ Kolen ëa geneic lgorihm for frequency ssignmen prolems" Technicl repor, Msrich Universiy,