An Introduction to Clique Minimal Separator Decomposition

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1 Alortms 2010, 3, ; o: / Rvw OPEN ACCESS lortms ISSN An Introuton to Clqu Mnml Sprtor Domposton Ann Brry 1,, Romn Poorln 1 n Gnvèv Smont 2 1 LIMOS UMR CNRS 6158, Ensml Sntqu s Cézux, F Auèr, Frn; E-Ml: poorl@sm.r 2 LIRMM, 161 Ru A, F Montpllr, Frn; E-Ml: smont@lrmm.r Autor to wom orrsponn soul rss; E-Ml: rry@sm.r; Tl.: Rv: 7 Aprl 2010 / Apt: 28 Aprl 2010 / Puls: 14 My 2010 Astrt: Ts ppr s rvw w prsnts n xplns t omposton o rps y lqu mnml sprtors. T p s lsurly, w v mny xmpls n urs. Esy lortms r prov to mplmnt ts omposton. T storl n tortl roun s vn, s wll s sts o proos o t struturl rsults nvolv. Kywors: rp omposton; lqu mnml sprtor; mnml trnulton 1. Introuton Ts ppr s sn s prsntton or t not so wll-nown rp omposton w w ll lqu mnml sprtor omposton. In vw o rnt ppltons, ts omposton s promsn tool. A sprtor s st o vrts (o onnt rp), t rmovl o w sonnts t rp, n lqu s st o prws nt vrts, n o ours lqu sprtor s sprtor w s lqu. Clqu mnml sprtor omposton s pross w rs up n unrt rp nto st o surps y opyn t lqu mnml sprtors nto t onnt omponnts ty n, s wll xpln n tl urtr. Hstorlly, lqu sprtor omposton ws stu y rp torsts us som r rp prolms su s vrtx olorn, mxmum lqu n tstn or prton n solv y Dv-n-Conqur ppro y rst solvn tm on t surps n y lqu sprtor omposton, n tn mrn t otn rsults.

2 Alortms 2010, Tr r two mn rws to ts omposton: Frst, not vry rp s lqu sprtor. Son, t surps otn r not sont, w mns tt t ovrlp s lr, lttl s to n y Dv-n-Conqur ppro. Howvr, rnt rsr s molz t y rps (su s txt mnn t [1], op t [2], or ny t rprsnt y symmtr postv mtrx su s stn t). In tos rps, lqu s otn n mportnt lous, sn t rlts mxmum ntrton twn ts lmnts. Ts rps lmost lwys v lqu sprtors. But n ny s, wn ts rps r otn y oosn trsol n stn mtrx, on n oos trsol w ns rp w ontns sunt numr o lqu sprtors. T t tt t surps otn ovrlp n srl proprty n ppltons. For xmpl, wn srn or lustrs o ns, t s n to l to llow n to ntrt wt rnt n roups, w s not t s wt lssl n lustrn. T tory o lqu omposton n lqu mnml sprtor omposton s n stu y svrl utors. T mtmtl nssry to prov t rsults w prsnt r s somwt omplt, n t rsults r strut twn vrous pprs, w s on o t rsons or w ts rp omposton s not s wll-nown s t oul. W wll sr t nown rsults (wl tn t orrsponn rrn pprs wn possl). W lso vot ston to vn t proo (or t lst st o proo) o t mn rsults w us. Our m n ts ppr s to xpln ts omposton s smply s possl; t s pross s not ult to unrstn, n t s not ult to mplmnt. Howvr, t os nvolv som not so sy rp notons su s mnml sprton. W v ornz t rsults so tt rp torsts wll n t tortl roun, n so tt sntsts rom otr rs n us n mplmnt ts omposton, y vn xmpls n xplnn t mplmnttons. T ppr s ornz s ollows: Ston 2 vs som nrl rp notons, som prtulrs on mnml sprton n mnml trnulton. Ston 3 provs prs nton o lqu mnml sprtor omposton, s wll s xmpls; w lso suss ow to n t lqu mnml sprtors o rp ntly, n v som proprts o ts omposton. Ston 4 vs n nt pross or omputn t lqu mnml sprtor omposton o rp. Ston 5 provs r story o lqu omposton, wt t lorp roun s wll s -lvl proos o t rsults w prsnt n ts ppr. W onlu n Ston 6.

3 Alortms 2010, Grp Notons 2.1. Gnrl notons All rps n ts wor r unrt n nt. A rp G = (V, E) s st V o vrts, V = n, n st E o s w r prs o stnt vrts, E = m; n {x, y} s not xy. T omplmnt G = (V, E) o rp G = (V, E) s t rp wr, or ny pr o vrts {x, y}, x y, xy E n only xy E. I X V s st o vrts o G = (V, E), G(X) nots t surp nu y X (w s rp wos vrts r t lmnts o X n su tt xy s n o G(X) x, y X n xy E). Noroos: T noroo o vrtx x n G s N G (x) = {y x xy E}; w omt susrpt G wn t s lr rom t ontxt w rp w wor on. T xtn noroo o vrtx x s N G [x] = N G (x) {x}. W sy tt vrts x n y r nt n only xy E. T noroo o st o vrts A s N G (A) = x A N G (x) A, n ts xtn noroo s N G [A] = N G (A) A. Pts n yls: A pt s squn (x 1, x 2,..., x ) o rnt vrts su tt or <, x s nt to x +1. A or n pt s n x x wr > n +1. A pt s orlss t s no or. A yl s los pt,.., squn x 1, x 2,..., x, x 1 o vrts, wt x 1, x 2,..., x ll rnt, su tt or <, x s nt to x +1 n x s nt to x 1. A orlss yl s yl wt no or. A orlss yl wt 5 or mor vrts s ll ol, t omplmnt o su yl s ll n ntol. Connxty: A rp s onnt tr s pt onntn ny pr o vrts, sonnt otrws ; wn t rp s sonnt, t mxml onnt surps r ll t onnt omponnts o t rp. Clqus: A lqu s st o vrts w r prws nt. An npnnt st s st o vrts wt no s. Prton: A rp s prt t ontns no nu ol or ntol wt n o numr o vrts. Exmpl 2.1 In Fur 1, (,,, ) s orlss yl o lnt 4, {,,, } s lqu, N() = {,,, }, N({, }) = {, }. Fur 1. Grp G, our runnn xmpl.

4 Alortms 2010, Mnml sprton, orl rps, n mnml trnulton Dnton 2.2 A sust S o vrts o onnt rp G s ll sprtor (or somtms utst) G(V S) s not onnt. A sprtor S s ll n -sprtor n l n rnt onnt omponnts o G(V S), mnml -sprtor S s n -sprtor n no propr sust o S s n -sprtor. A sprtor S s mnml sprtor, tr s som pr {, } su tt S s mnml -sprtor. Altrntly, S s mnml sprtor n only G(V S) s t lst 2 onnt omponnts C 1 n C 2 su tt N(C 1 ) = N(C 2 ) = S; su omponnts r ll ull omponnts o S n G. S s mnml xy-sprtor or ny {x, y} wt x C 1 n y C 2. A onnt rp wt no mnml sprtor s lqu. Wn t rp s not onnt, t st o mnml sprtors s t unon o t sts o mnml sprtors o t onnt omponnts. Exmpl 2.3 Lt us llustrt ts notons on tr: tr s two ns o vrts: lvs n rtulton nos (w r ts mnml sprtors). Fur 2 sows ow rmovn rtulton no {} sonnts t rp nto svrl onnt omponnts; {} s -sprtor, s n l n two rnt omponnts. Fur 2. {} s -sprtor. Exmpl 2.4 On t mor omplx rp G o Fur 1: {,,, } s sprtor, ut t s not mnml. {,, } s mnml sprtor, wt ull omponnts {,, } n {,, }. {, } s lso mnml sprtor, wt ull omponnts {, } n {,,,,,, }. Not tt mnml sprtor {, } s nlu n mnml sprtor {,, }. Not tt n n rtrry rp, tr n n xponntl numr o mnml sprtors (O(2 n )). Fur 3 sows n xmpl o ts.

5 Alortms 2010, Fur 3. Ts rp s n xponntl numr o mnml sprtors, w r t omntons rom t Crtsn prout { 1, 1 } { 2, 2 }... { n, n } n n Corl rps: A rp s orl t s no orlss yl o lnt strtly rtr tn 3; orl rps r xtnsons o trs n v mny smlr proprts. Dr [3] ornlly n t onpt o mnml sprtor n orr to rtrz orl rps, y sown tt rp G s orl n only vry mnml sprtor o G s lqu. In orl rp, tr r lss tn n mnml sprtors [4]. Fur 4 sows orl rp H. Fur 4. Corl rp H, wt mnml sprtors: {, }, {, }, {,, }, {, }, {, }, {, }. Prt lmnton orrns: Anotr ntrstn rtrzton o orl rps s tt ty v smpll orrn on t vrts: vrtx s s to smpll ts noroo s lqu. A rp s orl n only on n rptly n smpll vrtx, n rmov t, untl no vrtx s lt. Sn t stp vrtx s rmov, ts ns srs o trnstory rps: t stp + 1, t trnstory rp s nput rp G rom w t rst vrts v n rmov. Ts pross s ll smpll lmnton sm, n ns n orrn on t vrts ll prt lmnton orrn (po). E mnml sprtor o orl rp H s t noroo o vrtx n t trnstory rp n t smpll lmnton sm [4]. W wll sy tt su vrtx nrts mnml sprtor o H (w.r.t. vn po o H):

6 Alortms 2010, Dnton 2.5 Lt H = (V, E) orl rp, lt α prt lmnton orrn o H. W sy tt vrtx x o numr y α nrts mnml sprtor S w.r.t α S s qul to t st o nors o x o r numr. Tus, to omput t st o mnml sprtors o H, t s sunt to omput po α o H n to slt t vrts tt nrt mnml sprtor o H w.r.t. α. Ts s sy to ompls wt spl n o po n y lortms su s MCS [5], n wll us or our mplmntton n Ston 4. Mnml trnultons: A trnulton s orl omplton o rp: st F o s (ll t ll s)s to t rp n orr to otn orl rp. Dnton 2.6 Lt G = (V, E) rp. A orl rp H = (V, E + F ) s ll trnulton o G, n F s ll t ll. T trnulton s s to : mnml or no propr sust F o F, H = (V, E + F ) s orl. mnmum no otr mnml trnulton s lss ll s. Exmpl 2.7 Grp H rom Fur 4 s mnml trnulton o Grp G rom Fur 1 (s Fur 5). Fur 5. H s mnml trnulton o G. T ll s r rprsnt y ott lns. Computn mnmum trnulton s NP omplt [6], ut omputn mnml trnulton n on n O(nm) tm (or lttl lss or ns rps [7,8]). Ts prolm s vn rs to svrl rnt pprs, so tr r now mny mnml trnulton lortms (s.. [9 12]). W wll us on o tm (MCS-M) n Ston 4 wr w sr n mplmntton. Mnml lmnton orrns: On wy to omput trnulton o rp G = (V, E) s to or t rp nto vn prt lmnton orrn: n n orrn α on V, rptly p t nxt vrtx y α, ny s mssn n ts noroo (n t surp nu y t not yt rmov vrts), n rmov t, untl no vrtx s lt. Ts wll yl trnulton G + α = (V, E + F ) wos ll F s t st o ll s n ts pross.

7 Alortms 2010, An orrn α on t st o vrts o G s mnml lmnton orrn (mo) o G t trnulton G + α o G otn s mnml trnulton o G. Alortm LEX M [13] ws vs to omput n mo. In Ston 4, w wll us smplr vrson o LEX M, MCS-M [9]. Bot ts lortms yl mnml trnulton H o t nput rp G n n orrn α tt s ot mo o G n po o H, n ms t sy to omput t st o vrts nrtn t mnml sprtors o H. 3. Dnn lqu mnml sprtor omposton 3.1. Dntons n xmpls T omposton y lqu mnml sprtors n n y t ollown lortm pross on nput rp G: n ollton o surps (w t t nnn ontns only G) y rptly pplyn t ollown omposton stp on surp w s lqu mnml sprtor, untl non o t surps s lqu mnml sprtor. Domposton Stp 3.1 Gvn rp G, n lqu mnml sprtor S o G n ull omponnt C o S; rpl G wt t ollown 2 surps: G 1 = G (S C), G 2 = G (V C). T omposton s t st o surps otn n t n, w r ll toms. T omposton s unqu, t rsultn st o toms os not pn o t orr n w t omposton stps r xut. Ts s us o t ollown proprty o lqu mnml sprtor omposton: Proprty 3.2 Lt S t st o mnml sprtors o rp G, lt S lqu mnml sprtor o G ; tr pplton o on omposton Stp 3.1, otr mnml sprtor s mnml sprtor o tr G 1 or o G 2. Tus n prtulr, ny lqu mnml sprtor wll lqu mnml sprtor o G 1 or o G 2. Not tt t lqu mnml sprtor w ws us, S, my or my not stll mnml sprtor o G 2, pnn on wtr S mor tn two or only two ull omponnts. T toms otn r rtrz s ollows: Crtrzton 3.3 An tom o rp G s mxml onnt surp ontnn no lqu mnml sprtor. Exmpl 3.4 Fur 6 vs t omposton o tr. On ny orl rp, ts omposton yls t st o mxml lqus o t nput rp. In non-orl rp, som o t toms wll not lqus. Lt us us our rp G rom Fur 1 s n xmpl: T lqu mnml sprtors o G r: {, }, {,}, {,, }. Fur 7 sows omposton tr n t st o toms.

8 Alortms 2010, Fur 6. () A omposton stp usn mnml sprtor {}. T mnml sprtors (xpt or {}) r prtton nto t two sutrs otn. () Totl omposton. () () 3.2. Proprts o t toms T numr o toms s t most n. T ntrston twn toms s lwys lqu (w my mpty), n t s not nssrly mnml sprtor (vn wn t s not mpty). Exmpl 3.5 In our xmpl rom Fur 7, tom {,,, } tom {,,, } = {, }, w s not mnml sprtor; tom {,,, } tom {,,, } = {, }, mnml sprtor An quvlnt pross Anotr wy o on t omposton s usn lqu mnml sprtor S n opyn t n ll t omponnts t ns. For omponnt C w s not ull omponnt, w n to opy only t noroo o C. Ts orrspons to lqu mnml sprtor w s proprly nlu n S. T proprty w us s t ollown: Proprty 3.6 Lt S mnml sprtor, lt (C ) onnt omponnts o G(V S). Tn, N G (C ) s mnml sprtor o G.

9 Alortms 2010, Fur 7. A omposton tr or rp G rom Fur 1 n t st o toms o G. S={,} G({,,,}) G({,,,,,,,,}) G({,,,}) G({,,,,,,}) S={,,} G({,,,}) () Domposton tr G({,,,,,}) () St o toms Ts ls to t ollown omposton stp: Domposton Stp 3.7 Gvn rp G wt lqu mnml sprtor S, lt C 1,..., C t onnt omponnts o G (V S). Rpl G wt t surps G (C N G (C )). Atr su omposton stp usn lqu mnml sprtor S, S n t lqu mnml sprtors susts o S n no lonr mnml sprtor n ny o t surps otn, n ll t otr mnml sprtors o G r prtton nto t surps otn. Exmpl 3.8 Fur 8 sows omposton stp n rp G rom Fur 1 usn Domposton Stp 3.7. An pplton o Domposton Stp 3.7 orrspons to svrl ppltons o Domposton Stp How to omput t lqu mnml sprtors To our nowl, t only nt nown wy to omput t lqu mnml sprtors o rp s to xtrt tm rom mnml trnulton.

10 Alortms 2010, Fur 8. A omposton o G usn Domposton Stp 3.7. S={,,} G({,,,,,}) S={,} G({,,,}) G({,,,,,}) G({,,,}) G({,,,}) Ts mto s s on t ollown proprty: Proprty 3.9 Lt G = (V, E) rp, lt H = (V, E + F ) mnml trnulton o G. T lqu mnml sprtors o G r xtly t mnml sprtors o H tt r lqus n G. Tus n orr to n t lqu mnml sprtors o rp, t nrl pross s s ollows: omput mnml trnulton H o G. omput t mnml sprtors o H. mnml sprtor o H to s wtr t s lqu n G. Computn t mnml sprtors o mnml trnulton o rp n on ntly usn n orrn prov y sr lortm su s MCS-M, n Stps 1 n 2 n mr, s w wll s n Ston 4. Exmpl 3.10 Grp H rom Fur 4 s mnml trnulton o Grp G rom Fur 1. T mnml sprtors o H r: {, }, {, }, {,, }, {, }, {, }, {, }. Tr o tm r lqus n G: {, }, {,, }, {, }, so ty r lqu mnml sprtors o G Som prolms w n solv usn t toms Mnml n mnmum trnulton: mnml trnulton s omput or o t toms o t lqu mnml sprtor omposton o rp, tn t unon o t st o ll s otn ns mnml trnulton or G. Tus or tom mnmum-sz ll s omput, t rsultn ll wll mnmum trnulton o G, sn t sts o ll s n t toms r prws sont. Trwt: t trwt s otn y tn t lrst trwt ovr ll t toms. Prton: Any orlss yl o lnt 4 or mor s prsrv y omposton stp, s wll s ny ntol, so ts omposton prsrvs ols n ntols. Tus rp s prt n only ll ts toms r prt. Colorn: An optml olorn s otn y mrn optml olorns o t toms.

11 Alortms 2010, Mxmum lqu: Any mxml lqu o t rp s prsrv tr omposton stp, so nn t sz o t lrst lqu o tom wll yl t sz o t lrst lqu o t ornl rp. 4. Alortms n mplmnttons Tr r svrl wys o omputn t omposton o rp nto toms. T smplst s to mplmnt Domposton Stp 3.1. W wll us Alortm MCS-M [9], sr lortm w omputs mnml trnulton H = (V, E + F ) o t nput rp G = (V, E), rprsnt y st F o ll s, s wll s n orrn α o t vrts. MCS-M numrs t vrts rom n to 1, n t orrn α otn s mo o t nput rp G n po o t mnml trnulton H w s omput [14]. On ts orrn n ts trnulton r omput, w wll us tm n son lortm, (Alortm Atoms), w wll nrt t toms y snnn t vrts rom 1 to n. W wll strt y vn t nrl MCS-M lortm, tn n xpn vrson, w w ll MCS-M+, s wll xpln trwrs, n nlly v t lortm w yls t toms. G wll t trnstory surp o G nu y t st o stll unnumr vrts. Alortm MCS-M nput : An unrt rp G = (V, E). output: A mnml lmnton orrn α on V n mnml trnulton H = (V, E + F ) o G. nt: F ; G G; Intlz t lls o ll vrts s 0 ; or =n ownto 1 o Coos vrtx x o G o mxml ll ; Y N G (x) ; or vrtx y o G not lonn to N G [x] o tr s pt rom x to y n G su tt vry ntrnl vrtx on t pt s ll strtly smllr tn ll(y) tn Y Y + {y}; or y n Y o F F + {xy}; ll(y) ll(y) + 1 ; α() x ; Rmov x rom G ; H (V, E + F ); W wll now xpn ts lortm n two wys. Frst, w wll mplmnt tr s pt rom x to y n G su tt vry ntrnl vrtx on t pt s ll strtly smllr tn ll(y). W o ts n smlr wy s n t mplmntton o LEX M vn n [13] y snl sr n G. For ll vlu, st r() ontns t r vrts vn ll, s wll s t vrts vn ll strtly smllr tn r rom vrtx vn ll. Son, w wll omput t st X o vrts tt nrt t mnml sprtors o t trnulton H, s sr n [5]. T n ts s tt s lon s t lls o t osn

12 Alortms 2010, vrts p ttn lrr, w r ns lqu o H; wn sunly t ll o t osn vrtx x stops ttn lrr, x nrts mnml sprtor o H. E su vrtx x s to st X. Alortm MCS-M+ nput : An unrt rp G = (V, E). output: A mnml lmnton orrn α on V, mnml trnulton H = (V, E + F ) o G, n t st X o vrts w nrt mnml sprtor o H. nt: F ; G G ; Intlz t lls o ll vrts s 0 ; s 1 ; X ; or =n ownto 1 o Coos vrtx x o G o mxml ll ; Y N G (x) ; ll(x) s tn X X + {x} ; s ll(x) ; Mr x r n mr ll otr vrts o G unr ; or =0 to n 1 o r() or y n N G (x) o Mr y r; A y to r(ll(y)); or =0 to n 1 o wl r() o Rmov vrtx y rom r() ; or z n N G (y) o z s unr tn Mr z r; ll(z) > tn Y Y + {z}; A z to r(ll(z)); ls A z to r(); or y n Y o F F + {xy}; ll(y) ll(y) + 1 ; α() x ; Rmov x rom G ; H (V, E + F ). On w v otn n MCS-M orrn α, s wll s t orrsponn mnml trnulton H o G n t st X o nrtors o t mnml sprtors o H, w run trou t vrts rom 1 to n. W us t trnstory surps G n H o G n H, ntlz s G n H, rsptvly. At stp prossn vrtx x = α(), w wtr x s n t st X. I t s, t noroo N H (x) o x n H s mnml sprtor o H; w wtr t s lqu n G. I t s, tn S s lqu mnml sprtor o G. In tt s w omput t onnt omponnt C o G(V S)

13 Alortms 2010, w ontns x; G (S C) s n tom [15], n s stor s su; C s tn rmov rom G. In ny s, w tn rmov x rom H. Alortm Atoms nput : A rp G = (V, E), mo α o G, t orrsponn mnml trnulton H = (V, E + F ) o G, t st X o vrts tt nrt mnml sprtor o H. output: T st A o toms o G, t st S H o mnml sprtors o H, t st S o lqu mnml sprtors o G. G G ; H H ; A ; S H ; S ; or =1 to n o x α() ; x X tn S N H (x) ; S H S H {S} ; S s lqu n G tn S S {S}; C t onnt omponnt o G S ontnn x ; A A + {G (S C)}; G G C ; Rmov x rom H ; A A + {G }. 5. Tortl n lorpl roun 5.1. A r story o lqu mnml sprtor omposton In 1976, Gvrl [16] sr t lss o lqu sprl rps n t ontxt o solvn n polynoml tm t r prolms o mnmum olorn n mxmum lqu. In 1980, t prolm o nn lqu sprtor n n rtrry rp ws rss y Wtss [17], wo prsnt n O(n 3 ) lortm to n on lqu sprtor. In 1983, Trn [15] rss t prolm o nn omposton o n rtrry rp usn ts lqu sprtors. H not tt usn Wtss lortm, ts woul rqur O(nm 3 ) tm. H propos n O(nm) tm lortm to o ts, y sown tt no ll o mnml trnulton n on two onnt omponnts n y lqu sprtor. Computn n mo rqur O(nm) tm, s sow n [13] (1976), n prov tt usn n mo, lqu sprtor omposton n omput n O(nm) tm. Trn lt opn t quston o nn unqu lqu sprtor omposton. In 1990, Dlus, Krpnsy n Nov [18] propos prlll lortm or lqu sprtor omposton. In 1990, onurrntly, Lmr [19], n ppr snt out or pulton n 1986, sr n vry mtmtl tl ow to otn n optml n unqu omposton usn t lqu mnml sprtors o t rp. Lmr lso sr ow to us LEX M (lso ll t RTL lortm) to n lqu mnml sprtor omposton n O(nm) tm. In 2001, Olsn n Msn [20] ppl lqu mnml sprtor omposton to Bysn ntwors, n prsnt n ppro s on t lqu tr o mnml trnulton to omput

14 Alortms 2010, t toms (w ty ll mxml prm surps) y mrn n t lqu tr ny two nos onnt y n w rprsnts non-lqu mnml sprtor (t rr s rrr to [21] or ull tls on lqu trs) Computn t lqu mnml sprtors o rp W rst prov tt or ny rp G n ny mnml trnulton H o G, t lqu mnml sprtors o G r xtly t mnml sprtors o H tt r lqus n G (proprty 3.9). For ts, t s sunt to sow tt ny lqu o G s t sm numr o ull omponnts n G s n H. Ts ollows rom t ollown proprty, w s n xtnson o Lmm 1 rom [15]. Proprty 5.1 Lt G = (V, E) rp, lt H = (V, E + F ) mnml trnulton o G n lt S lqu o G. Tn G(V S) n H(V S) v t sm onnt omponnts, wt t sm noroos. Proo: Lt H t rp n y H = (V, E + F ), wr F s t st o s n F tt r ontn n N G [C] or som onnt omponnt C o G(V S). By nton o H, G(V S) n H (V S) v t sm onnt omponnts, wt t sm noroos. Lt us sow tt H s orl. Lt µ yl n H o lnt rtr tn 3, n lt us sow tt µ s or n H. As H s orl, µ s or, sy xy, n H. I xy s n H tn t s or o µ n H. Otrws tr s onnt omponnt C o G(V S) su tt x s n C n y s n V N G [C], or onvrsly. Hn tr r two non-onsutv vrts w n z o µ tt r ot n N G (C), n tror ot n lqu S o H. It ollows tt wz s or o µ n H, w omplts t proo tt H s orl. As G H H, wt H orl n H mnml trnulton o G, H = H, n tror G(V S) n H(V S) v t sm onnt omponnts, wt t sm noroos. Corollry 5.2 (Proprty 3.9) Lt G = (V, E) rp, lt H = (V, E+F ) mnml trnulton o G. T lqu mnml sprtors o G r xtly t mnml sprtors o H tt r lqus n G. T t tt mnml sprtor o orl rp H s nrt y vrtx o H w.r.t. po o H [4] s wll-nown. A wy o sltn t xt vrts tt nrt mnml sprtor s suss n [5]. T ollown lortm s s on t xpn vrson o MCS prsnt n [21] to nrt t mxml lqus n lqu tr. Ts pross s ntrt n t mplmntton vn n Ston 4.

15 Alortms 2010, Alortm MCS-Mnsps nput : A orl rp H = (V, E). output: T st S o mnml sprtors o H. nt: V NUM ; H NUM H(V NUM ); H ELIM H; S ; Intlz t lls o ll vrts s 0 ; s 1 ; or =n ownto 1 o Coos vrtx x o H ELIM o mxmum ll ; V NUM V NUM + {x}; H NUM H(V NUM ); ll(x) s tn S S {N HNUM (x)} ; s ll(x) ; or y N HELIM (x) o ll(y) ll(y) + 1; Rmov x rom H ELIM ; 5.3. Domposn rp nto toms Domposton Stp 3.1 n Alortm Atoms o Ston 4 r lrly nspr rom t omposton o rp y lqu sprtors sr y Trn [15]. Trn onsrs lqu sprtors nst o lqu mnml sprtors, wt t rw tt t st o toms otn s not unqu (s Suston 5.4). H lso omputs mo o G n t orrsponn mnml trnulton o G, ut slts t vrts tt nrt lqu sprtor nst o lqu mnml sprtor o G. T proo o t lqu sprtor omposton lortm vn n [15] n sly mo nto proo o Alortm Atoms. W wll v n o ts proo. Torm 5.3 Alortm Atoms omputs lqu mnml sprtor omposton o nput rp G. Proo: ( o t proo) T proo wors y nuton on = X, wr X s t st o vrts o X tt nrt lqu o G w.r.t. α,.., t st o vrts o G tt nrt lqu mnml sprtor o G w.r.t. α. I = 0 tn t omput omposton s {G}, w s orrt sn G s no lqu mnml sprtor. W suppos tt t omposton s orrt X. Lt us sow tt t s stll orrt X = + 1. Lt x t rst vrtx o X n orrn α, lt S t lqu mnml sprtor o G nrt y x w.r.t. α, lt C t onnt omponnt o G(V S) ontnn x, lt G 1 = G(S C) n G 2 = G(V C). For n {1, 2}, lt V t vrtx st o G, lt α t rstrton o α to V, lt H = H(V ) n X = X V. It n sown tt or n {1, 2}, α s mo o G, wt H s orrsponn mnml trnulton o G. Morovr, vrtx o V 2 nrts t sm st n G 2 w.r.t. α 2 s n G w.r.t. α, w s mnml sprtor o H 2 n only t s mnml sprtor o H. Hn X 2 s xtly t st o vrts o G 2 tt nrt lqu mnml sprtor o G 2 w.r.t. α 2, wt X 2 sn X 2 X {x}. It ollows y nuton ypotss tt G 2 s orrtly ompos. It rmns to sow tt G 1 s n tom o G. It n sown tt vrtx o C {x} prs x n orrn α, n nrts t sm st n G 1 w.r.t. α 1 s n G w.r.t. α, w s mnml sprtor o H 1 n only t s mnml sprtor o H. As no vrtx o C {x} nrts

16 Alortms 2010, lqu mnml sprtor o G w.r.t. α (sn x s t rst vrtx tt os), no vrtx o C {x} nrts lqu mnml sprtor o G 1 w.r.t. α 1 tr. Now vrtx o S {x} nrts sust o S, w s not sprtor o G 1, w.r.t. α 1. Hn G 1 s no lqu mnml sprtor, n s tror n tom o G T unty o lqu mnml sprtor omposton Trn [15] ornlly n t omposton stp o rp y lqu sprtor s ollows: Domposton Stp 5.4 Lt S lqu sprtor o G = (V, E ), lt (S, A, B) prtton o V su tt no vrtx n A s nt to vrtx n B. Rpl G y G 1 = G (S A) n G 2 = G (S B). H rmr tt t st o toms otn y rptly pplyn ts omposton stp s not unqu, vn ountrxmpl tt s rll n Fur 9, n lt t prolm o t unty o t omposton opn. Lmr [19] solv ts prolm, sown tt usn lqu mnml sprtors nst o lqu sprtors nsurs unty o t omposton (wt t tonl onton tt on o A n B ontns ull omponnt o S n G n t s o Domposton Stp 5.4). W prsnt som lmnts o t proo o ts rsult, w s ssot n [19] wt proos o mor omplx rsults w m t ult or non-splsts to ollow. W rst rll som ntons vn n [19]. Fur 9. A rp wt rnt ompostons y lqu sprtors. () Grp. () On omposton, w s t unqu omposton y lqu mnml sprtors. () Anotr omposton. {,} {,,} {,,,} {,,} {,,,} {,} () () () {,,} {,,} Dnton 5.5 A rp s prm t s onnt n s no lqu sprtor (or quvlntly, no lqu mnml sprtor). A mp-surp o rp G s mxml prm surp o G. Lmr prov tt t lqu mnml sprtor omposton o rp G s t st o mp-surps o G (Crtrzton 3.3). T proo o ts rtrzton rls on t ollown proprty. Proprty 5.6 Lt G = (V, E) rp, lt S lqu o G vn t lst on ull omponnt n G. Tn tr s sust A o V su tt S A n G(A) s prm.

17 Alortms 2010, Proo: ( o t proo) T proprty ols G s orl sn n tt s, s S s lqu o G vn ull omponnt, sy C, n G, tr s vrtx x o C su tt S N G (x) (ts ollows rom t rsults n [4]), n tror S {x} s lqu n nus prm rp. Lt us sow tt t ols or ny rp G. Lt G t rp otn rom G y mn mp-surp o G nto lqu. G s orl n s t sm mp-surps s G [19]. Tr s sust A o V su tt S A n G (A) s prm. Lt B sust o V su tt A B n G (B) s mp-surp o G. Tn S B n G(B) s prm. H lso sow t ollown rtrzton, w s Crtrzton 3.3: Crtrzton 5.7 T lqu mnml sprtor omposton o rp G s t st o mp-surps o G. Proo: W suppos tt t omposton stp w s us s Domposton Stp 5.4 wt t tonl onton tt on o A n B ontns ull omponnt o S n G, w s mor nrl tn Domposton Stp 3.1. T proo os y nuton on t numr n o vrts o t rp. T proprty trvlly ols n = 1. Suppos tt t ols or ny rp wt t most n vrts. Lt us sow tt t ols or rp G = (V, E) wt n + 1 vrts. T proprty trvlly ols G s no lqu mnml sprtor. W suppos tt G s lqu mnml sprtor S. Lt (S, A, B) prtton o V su tt no vrtx n A s nt to vrtx n B n on o A n B ontns ull omponnt o S n G. Lt G 1 = G(S A) n G 2 = G(S B). Lt us sow tt t st toms(g) o toms otn y rpln G y G 1 n G 2 n urtr omposn G 1 n G 2 s qul to t st mp-surps(g) o mp-surps o G. Clrly tom(g) = toms(g 1 ) toms(g 2 ). As y nuton ypotss toms(g 1 ) = mp-surps(g 1 ) n toms(g 2 ) = mp-surps(g 2 ), t s sunt to sow tt mp-surps(g) = mp-surps(g 1 ) mp-surps(g 2 ). Lt us rst sow tt mp-surps(g) mp-surps(g 1 ) mp-surps(g 2 ). Lt G(M) n mp-surps(g). Tn M S A or M S B (otrws M S woul lqu sprtor o G(M)) n tror G(M) mp-surps(g 1 ) mp-surps(g 2 ). Lt us sow now tt mp-surps(g 1 ) mp-surps(g 2 ) mp-surps(g). Lt G(M) mp-surps(g 1 ) mp-surps(g 2 ), sy G(M) mp-surps(g 1 ). By Proprty 5.6, s A ontns ull omponnt o S n G, no sust o S nus mp-surp o G 1. Hn M A. Lt G(M ) mp-surp o G wt M M. As mp-surps(g) mp-surps(g 1 ) mp-surps(g 2 ), G(M ) mp-surps(g 1 ) mp-surps(g 2 ), n s M A (sn M A n M M ), G(M ) mp-surps(g 1 ). As G(M) n G(M ) r ot mp-surps o G 1 wt M M, M = M. Hn G(M) mp-surps(g). It ollows tt mp-surps(g) = mp-surps(g 1 ) mp-surps(g 2 ), w omplts t proo y nuton. Corollry 5.8 T omposton y lqu mnml sprtors s unqu.

18 Alortms 2010, Conlusons Clqu mnml sprtor omposton s smpl to mplmnt n prtulrly wll-sut to ppltons nvolvn rps osn rom stn mtrx. Tou t tortl roun s somwt omplt, t s not nssry to mstr t n orr to us ts omposton. Clqu mnmum sprtor omposton s lso n us rntly s tool to solv t mxmum wt stl st prolm or spl rps lsss [22], n s n ppl to trwt y usn sprtors w r lqus or lmost lqus [23]. On o t promsn spts o ts omposton s tt on trsol s osn, t lustrs w r orm s toms o t orrsponn rp r unquly n, pnn solly on t strutur o t t. Rrn rp vsulzton, t st o toms n ornz nto mt-rp wr t vrts r t toms. For nstn [2], tr n n twn two toms wn tr ntrston s lqu mnml sprtor. Ts yls mor lol vw o t rp, wl splyn som o ts struturl proprts. Rrns 1. D B, M.; K, B.; Murs, M.J.; SnJun, E. Grp omposton ppros or trmnoloy rps. Pro. MICAI 2007, K, B.; Pnt, N.; Llns, G.; Syrt, A.; Brry, A. Clustrn n xprsson t usn rp sprtors. In Slo Bol. 2007, 7, Dr, G.A. On r rut rps. A. Mt. Sm. Unv. Hmur 1961, 25, Ros, D.J. Trnult rps n t lmnton pross. J. Mt. Anl. Appl. 1970, 32, Brry A.; Poorln, R. A smpl lortm to nrt t mnml sprtors o orl rp. Rsr Rport LIMOS RR LIMOS UMR CNRS: Auèr, Frn, Ynns, M. Computn t mnmum ll-n s NP-omplt. SIAM J. Alr. Dsrt Mto 1981, 2, Krts, D.; Spnr, J. Mnml ll n O(n 2.69 ) tm. Dsrt Mt. 2006, 306, Hrns, P.; Tll, J.A.; Vllnr, Y. Computn mnml trnultons n tm O(n α lon) = o(n ). SIAM J. Dsrt Mt. 2005, 19, Brry, A.; Blr, J.R.S.; Hrns, P. Mxmum rnlty sr or omputn mnml trnultons o rps. Alortm 2004, 39, Brry, A.; Bort, J.P.; Hrns, P.; Smont, G.; Vllnr, Y. A w-rn lortm or mnml trnulton rom n rtrry orrn. J. Alor. 2006, 58, Brry, A.; Hrns, P.; Vllnr, Y. A vrtx nrmntl ppro or mntnn orlty. Dsrt Mt. 2006, 306, Hrns, P. Mnml trnultons o rps: A survy. Dsrt Mt. 2006, 306, Ros, D.J.; Trn, R.E.; Lur, G.S. Alortm spts o vrtx lmnton on rps. SIAM J. Comput. 1976, 5,

19 Alortms 2010, Brry, A.; Krur, R.; Smont, G. Mxml ll sr lortms to omput prt n mnml lmnton orrns. SIAM J. Dsrt Mt., , Trn, R.E. Domposton y lqu sprtors. Dsrt Mt. 1985, 55, Gvrl, F. Alortms on lqu sprl rps. Dsrt Mt. 1977, 19, Wtss, S. An lortm or nn lqu utsts. In. Pross. Ltt. 1981, 12, Dlus, E.; Krpns, M.; Nov, M.B. Fst prlll lortms or t lqu sprtor omposton. In Prons o t Frst Annul ACM-SIAM Symposum on Dsrt Alortms: SODA 90, Pllp, PA, USA, Jnury 1990; pp Lmr, H.G. Optml omposton y lqu sprtors. Dsrt Mt. 1993, 113, Olsn, K.G.; Msn, A.L. Mxml prm surp omposton o Bysn ntwors. IEEE Trns. Syst. Mn Cyrnt. B 2002, 32, Blr, J.R.S.; Pyton, B.W. An ntrouton to orl rps n lqu trs. Grp Tory Sprs Mtrx Comput. 1993, 84, Brnstät, A.; Hoàn, C.T. On lqu sprtors, nrly orl rps, n t Mxmum Wt Stl St Prolm. Tor. Comput. S. 2007, 389, Bolnr, H.L.; Kostr, A.M.C.A. S sprtors or trwt. Dsrt Mt. 2006, 306, y t utors; lns MDPI, Bsl, Swtzrln. Ts rtl s n Opn Ass rtl strut unr t trms n ontons o t Crtv Commons Attruton lns ttp://rtvommons.or/lnss/y/3.0/.