Can transitive orientation make sandwich problems easier?

Transcription

1 Disrt Mthmtis 07 (007) Cn trnsitiv orinttion mk snwih prolms sir? Mihl Hi, Dvi Klly, Emmnull Lhr,, Christoph Pul,, CNRS, LIRMM, Univrsité Montpllir II, 6 ru A, 4 9 Montpllir C, Frn Dprtmnt of Mthmtis, Univrsity of Mnito, Winnipg, Mnito, Cn RT N Lortoir l Informtiqu u Prllélism, Éol Norml Supériur Lyon, 46 Allé Itli, 6964 Lyon C 07, Frn Riv 8 Sptmr 00; riv in rvis form 4 Novmr 004; pt Dmr 005 Avill onlin Dmr 006 Astrt A grph G s = (V, E s ) is snwih for pir of grphs G t = (V, E t ) n G = (V, E) if E t E s E. A snwih prolm sks for th istn of snwih grph hving n pt proprty. In sminl ppr, Golumi t l. [Grph snwih prolms, J. Algorithms 9 (995) ] prsnt mny rsults on su-fmilis of prft grphs. W r spilly intrst in omprility (rsp., o-omprility) grphs us ths grphs (rsp., thir omplmnts) mit on or mor trnsitiv orinttions (h orinttion is prtilly orr st or post). Thus, fiing th orinttions of th gs of G t n G rstrits th numr of possil snwihs. W stuy whthr ing n orinttion n rs th omplity of th prolm. Two iffrnt typs of prolms shoul onsir pning on th trnsitivity of th orinttion: th post snwih prolms n th irt snwih prolms. Th orinttions to oth grphs G n G s r trnsitiv in th first typ of prolm ut ritrry for th son typ. 006 Pulish y Elsvir B.V. Kywors: Grph snwih; Comprility grphs; Prtilly orr st. Introution A grph G t = (V, E t ) is spnning su-grph of G = (V, E) if E t E. A grph G s = (V, E s ) is snwih grph for th pir (G t,g)if E t E s E. Golumi t l. [8] introu th following ision prolm: Prolm. GRAPH SANDWICH PROBLEM FOR PROPERTY Π. Instn: Two grphs G t = (V, E t ) n G = (V, E) suh tht E t E. Qustion: Dos thr ist snwih grph G s = (V, E s ) for th pir (G t,g)stisfying proprty Π? In thir ppr, Golumi t l. prsnt s n mpl, th irt Eulrin snwih prolm whr th input grphs r igrphs. Sin igrph is Eulrin iff th in-gr n th out-gr of ny vrt r qul, it is possil to sign polynomil tim lgorithm to i th istn of n Eulrin snwih igrph. Unfortuntly, mny snwih prolms n prov to NP-omplt. Grph snwih prolms n thought of s th gnrliztion Corrsponing uthor. E-mil rss: pul@lirmm.fr (C. Pul). Authors support in prt y th Europn Rsrh Trining Ntwork COMBSTRU (Comintoril Strutur of Intrtl Prolms) X/$ - s front mttr 006 Pulish y Elsvir B.V. oi:0.06/j.is

2 M. Hi t l. / Disrt Mthmtis 07 (007) G t G s G D t D Fig.. For th unirt instn (G t,g)thr ists snwih with n Eulrin yl whil for th irt instn (D t,d)mits no snwih with n Eulrin iruit. of vrious wll-stui grph-thorti prolms. For instn, th s E t =E orrspons to th rognition prolm whil th s E = V is th fill-in prolm. In prti, snwih prolms ris in ivrs rs suh s iology [9,7], ommunition [] n lgr [6]. Rsults onrning grph proprtis n lso foun in [8,,7,4,]. A omplmntry proprty Π n fin s follows: for ny grph G, G stisfis Π iff th omplmnt grph G stisfis Π. Th following ws prov in [8]: Proposition. (Golumi t l. [8]). Thr ists snwih grph G s stisfying proprty Π for th instn (G t,g) iff thr ists snwih grph H s stisfying proprty Π for th instn (G, G t ). It follows tht th Π snwih prolm is polynomilly quivlnt to th Π snwih prolm. Bing omprility grph is omplmntry proprty. In [8], mny rsults onrn su-fmilis of omprility or o-omprility grphs. Thy r of spil intrst sin omprility grph mits on or mor trnsitiv orinttions (h orinttion fins prtilly orr st or post). Lt us fin irt proprty Π s proprty suh tht if igrph G stisfis Π, thn th non-orint grph G stisfis Π. For mpl, th irt Eulrin proprty fits this finition. Similrly, ing post for igrph orrspons, in th non-orint vrsion, to ing omprility grph. Proposition.. Lt D t n D ritrry orinttions of th grphs G t n G. If thr ists snwih igrph D s stisfying th irt proprty Π for th instn (D t,d), thn thr ists snwih grph G s stisfying proprty Π for th instn (G t,g). Sin fiing th orinttion of th gs of G t n G rstrits th numr of possil snwihs, it is possil tht whil snwih grph ists in th non-irt instn, no snwih igrph ists in th irt instn. Fig. shows suh n mpl. A proprty Π is strongr thn proprty Π iff ny grph (or post) tht stisfis Π lso stisfis Π. Clrly irt proprty Π is wkr proprty thn Π. Sin, s osrv in [8], thr is no simpl rltionship twn th omplitis of th Π n Π snwih prolms, nithr is thr twn th omplitis of th Π n Π snwih prolms priori. A proprty Π fin on posts is omprility invrint iff whn post P stisfis Π thn ny trnsitiv orinttion of th omprility grph G of P lso stisfis Π. For mpl, ing post of imnsion k is omprility invrint [0]. Comprility invrints r irt proprtis. Evn if w rstrit to omprility invrints, th phnomnon pit in Fig. n still our. To hk Π, w only hv to tst on trnsitiv orinttion of th omprility grph. Howvr, thr is still no simpl omplity rltion (i.., with rspt to th Krp rution, K ). Thr my no trnsitiv orinttion of givn non-orint snwih with th pt proprty (s Fig. ). Thrfor, on must hk ll possil non-orint snwihs, n thr n n ponntil numr of ths.

3 0 M. Hi t l. / Disrt Mthmtis 07 (007) G t G s G P t??? P Fig.. In th ov figur, th pt proprty for th snwih is to hv th gr squn (,,,,, ). Noti tht th gr squn is omprility invrint. Only on solution is possil in th non-orint instn whil thr is no post snwih with tht gr squn. Polynomil NP-Complt Cogrph grph SP Intrvl grph SP Hlf liqu grph SP Prmuttion grph SP Sris prlll irt SP Intrvl irt SP Trnsitiv su hin irt SP imnsionl irt SP Sris prlll post SP Intrvl post SP Trnsitiv su hin post SP imnsionl post SP Fig.. A mp of th rsults. In th ppr, w will fous on vn strongr proprty thn omprility invrint, sin for thos proprtis, if thr ists non-orint snwih stisfying Π, thn thr ists irt snwih stisfying Π. W wonr whthr for suh proprtis ing orinttions to th gs of G t n G hlps to solv th orrsponing snwih igrph prolm. For mpl, th omprility grph snwih prolm hs n prov to NP-omplt [8]. But tsting th istn of trnsitiv snwih igrph D s for pir of igrphs (D t,d)n omplt in polynomil tim. Th lgorithm just hs to tst whthr th trnsitiv losur of D t is inlu in D whih is polynomil prolm. W will fous on proprtis rlt to omprility grphs or o-omprility grphs (whih r quivlnt y Proposition.) tht n nturlly trnslt in trms of post. Two slightly iffrnt prolms r istinguish y th trnsitivity of th input igrphs. Prolm. DIRECTED SANDWICH PROBLEM FOR POSET PROPERTY Π. Instn: Two igrphs D t n D suh tht D t D. Qustion: Dos thr ist snwih igrph D s for th pir (D t,d)stisfying Π? Prolm. POSET SANDWICH PROBLEM FOR POSET PROPERTY Π. Instn: Two posts P t = (V, E t ) n P = (V, E) suh tht E t E. Qustion: Dos thr ist snwih post P s = (V, E s ) for th pir (P t,p)stisfying Π? In th irt snwih prolm, igrphs D t n D of th snwih instn hv n ritrry orinttion. If th rquir proprty Π ls with posts, w n ssum tht D t is trnsitivly orint s its trnsitiv losur is ontin in ny trnsitivly orint snwih grph. Thrfor, th instn will writtn (P t,d). Sin th post snwih prolm is su-prolm of th irt snwih prolm, it follows tht: Post snwih prolm for Π K Dirt snwih prolm for Π. () This inqulity is hlpful in omplity proofs: to show tht irt snwih prolm is NP-omplt, w only n to show th NP-ompltnss of th post snwih prolm, n onvrsly if th prolm is polynomil (Fig. ).

4 M. Hi t l. / Disrt Mthmtis 07 (007) In this ppr, w stuy post proprtis inspir y th originl rsults of [8]. W r intrst in fining snwihs tht r sris prlll posts, intrvl posts or two-imnsionl posts. Ths fmilis orrspon to o-grphs, intrvl grphs n prmuttion grphs, rsptivly (ll of whih r o-omprility grphs). Morovr, ths thr proprtis r omprility invrints. W prov th following rsults: th sris prlll post snwih prolm is polynomil lik th o-grph snwih prolm; th intrvl post snwih prolm is polynomil whil th intrvl grph snwih prolm in NP-omplt; th two-imnsionl irt snwih prolm is NP-omplt lik th prmuttion grph snwih prolm. W lso introu nw prolm, nmly th trnsitiv su-hin prolm tht sks for th istn in igrph of trnsitiv su-hin of lngth t lst V /. Ruing -SAT to this prolm, w prov its NP-ompltnss. It follows tht th ssoit irt snwih prolm is NP-omplt, whil th post snwih prolm rmins polynomil. Noti tht in trms of non-orint grphs, th trnsitiv su-hin prolm orrspons to th hlfliqu prolm (os thr ist liqu ontining t lst on hlf of th vrtis?). Clrly, th hlf-liqu snwih prolm is NP-omplt. Thrfor, w show tht ny possil onfigurtion with rspt to () ists. Nottion: Lt S P th st of sours of post P, N P () th nighorhoo of vrt in P, P A th post inu y P on vrt st A. Th nottion Su P (), noting th st of out-vrtis (sussors) of in P, is tn to sts: Su P (A) = A SuP ().. Sris prlll posts In rtin shuling prolms, tsks r sujt to prtil orr. Although shuling prolms for n ritrry prtil orr r NP-omplt, thy hv ffiint lgorithms if th prtil orr is sris prlll [4]; ths lgorithms us ivi-n-onqur pproh with th rursiv strutur of ths posts. Thr is linr-tim lgorithm to rogniz sris prlll post u to Vls t l. [9]. A sris prlll post is otin from th singl-vrt post y th pplition of two omposition ruls. Th prlll omposition of posts P n P is th post P + P = (V V,< + ) suh tht u< + v if n only if u, v V n u< v or u, v V n u< v. Th sris omposition of posts P n P is th post P P = (V V,< ) suh tht u< v if n only if u, v V n u< v or u, v V n u< v or u V n v V. Thrfor, sris prlll posts r orgniz in tr strutur (Fig. 4)... Sris prlll post snwih prolm Th fmily of omprility grphs of th sris prlll posts is tly th fmily of o-grphs [8], for whih th snwih prolm hs n prov to polynomil [8]. Th o-grph snwih lgorithm n moifi so tht it pplis to th sris prlll post snwih prolm. By ing n rgumnt out trnsitivity to its proof, w n prov tht th post snwih prolm is lso polynomil lik th o-grph snwih prolm is. Prolm 4. SERIES PARALLEL POSET SANDWICH PROBLEM. Instn: Two posts P t = (V, E t ) n P = (V, E) suh tht E t E. Qustion: Dos thr ist sris prlll snwih post P s for th pir (P t,p)? P S P S f f Fig. 4. A sris prlll post n its nonil omposition tr.

5 04 M. Hi t l. / Disrt Mthmtis 07 (007) Lt us rifly sri th prinipl of this lgorithm. Proofs of Lmms. n. r omitt sin thy n sily u from [8]. Lt us not y P th inomprility grph of th post P. Lmm.. If oth P t n P r onnt, thn thr is no sris prlll snwih post for th instn (P t,p). Lmm.. Lt {C,...,C k } th onnt omponnts of P t. If Ps,...,Pk s r, rsptivly, sris prlll snwih posts for th instns (P t C,P C )... (P t Ck,P Ck ), thn th prlll omposition of Ps,...,Pk s is sris prlll snwih post for (P t,p). Lmm.. Lt {C,...,C k } th onnt omponnts of P. If Ps,...,Pk s r, rsptivly, sris prlll snwih posts for th instns (P t C,P C )... (P t Ck,P Ck ), thn th sris omposition of Ps,...,Pk s is sris prlll snwih post for (P t,p). A similr lmm n foun in [8] for th non-orint s of o-grphs. Th only iffrn is tht w hv to onfirm tht trnsitivity n nsur twn th onnt omponnts {C,...,C k }. Proof. Lt,, thr vrtis of iffrnt onnt omponnts C, C, C of P. By finition of th onnt omponnt, th thr possil rs twn ths thr vrtis long to E. Sin y ssumption P is post, ths thr gs r trnsitivly orint. Lt us now prov tht th rs twn two onnt omponnts C,C r orint in th sm irtion. Lt N + () n N (), rsptivly, th out-nighorhoo n in-nighorhoo of in C. Sin P is trnsitiv ll possil rs ist from N + () towrs N (), whih ontrits th ft tht C is onnt in P. Now it is strightforwr to s th lgorithm. If oth P t n P r onnt, thn thr is no sris prlll snwih post. Othrwis if P t is not onnt, rurs on th instns inu y its onnt omponnts C,...,C k ; if P is not onnt, rurs on th instns inu y th onnt omponnts C,...,C k of P. This lgorithm is tly th polynomil tim lgorithm givn in [8]. Thorm.. Th sris prlll post snwih prolm is polynomil... Sris prlll irt snwih prolm Howvr, th ov mtho os not provi n lgorithm for th irt snwih prolm. For this mor gnrl prolm, w provi polynomil-tim lgorithm s on n limintion orring prinipl. Prolm 5. SERIES PARALLEL DIRECTED SANDWICH PROBLEM. Instn: A post P t = (V, E t ) n igrph D = (V, E) suh tht E t E. Qustion: Dos thr ist sris prlll snwih post P s for th pir (P t,d)? Lt us first introu usful proposition. A proprty π is n hritry proprty iff for ny inu su-grph (or inu su-post), th su-grph stisfis th proprty π. Bing tr or sris prlll post is n hritry proprty. Proposition.. Lt π n hritry proprty. If thr ists π-snwih G s for th instn (G t,g)on vrt st V, thn for ny sust V V, th inu su-snwih G s V is π-snwih for th inu instn (G t V,G V ). Proof. First of ll, G s V is snwih for (G t V,G V ). Sin π is hritry G s V lso vrifis proprty π. Lmm.4. If thr is sris prlll snwih post P s for th instn (P t,d)whr P t is onnt, thn thr ists non-mpty st of vrtis A V suh tht A, V\A Su D () n Su P t (V \A) A =.

6 M. Hi t l. / Disrt Mthmtis 07 (007) Fig. 5. A polynomil tim lgorithm for th sris prlll irt snwih prolm. Proof. Suppos thr ists P s, sris prlll snwih post for th instn (P t,d). Osrv tht suh st A must ontin th sour S Ps of P s. Sin P t is onnt, P s is lso onnt. Thus, it is th rsult of sris omposition n its o-omprility grph G s is not onnt. Noti tht S Ps is ontin in onnt omponnt of G s (sin S Ps is n ntihin of P s ). Lt A tht omponnt. A is in sris omposition with V \A. Sin A ontins th sours of P s, th rs twn A n V \A r orint towrs V \A. It follows tht V \A Su D (), A. Morovr, sin P s is post, thr is no r from V \A towrs A. Th inlusion E t E s implis tht Su P t (V \A) A =. Lmm.5. Lt {C,...,C k } th st of onnt omponnts of P t. Th following sttmnts r quivlnt: () Thr is sris prlll snwih post for th instn (P t,d). () For ny i, i k, thr ist: non-mpty st A i suh tht A i, C i \A i Su D C i () n Su P t (C i \A i ) A i = ; sris prlll snwih post PA s i for th instn (P t Ai,D Ai ); sris prlll snwih post PC s i \A i for th instn (P t Ci \A i,d Ci \A i ). Proof. Suppos P s is sris prlll snwih post for th instn (P t,d). By Lmm.4, whnvr i k, thr ists n non-mpty st A i. By finition, A i n ompos in sris with C i \A i in oth D Ci n P s Ci. Sin ing sris prlll post is n hritry proprty, ny inu sust of P s is lso sris prlll post. If follows tht P s Ci \A i n P s Ai r sris prlll snwihs for th instns (P t Ci \A i,d Ci \A i ) n (P t Ai,D Ai ), rsptivly. Assum th onvrs. Sin for ny i, i k, C i \A i Su D C i () n Su P t (C i \A i ) A i =, PC s i \A i n PA s i n ompos in sris to form sris prlll post PC s i for th instn (P t Ci,D Ci ). It follows tht sris prlll snwih post for th whol instn (P t,d)is otin y th prlll omposition of th PC s i s. Assuming tht w r l to omput st A s sri in Lmm.4, w r now l to sign lgorithm s on Lmm.5 for solving th sris prlll irt snwih prolm. Whn solving th prolm rursivly in Fig. 5, Proposition. gurnts tht w n fin st A V suh tht A, V\A Su D () n Su P t (V \A) A =. () To sri how to omput suh st, w shll us thr simpl fts. Ft.6. Lt n y istint vrtis. If A n y/ Su D (), thn y A.

7 06 M. Hi t l. / Disrt Mthmtis 07 (007) Proof. This irtly follows from th finition of A. Ft.7. Lt n y istint vrtis. If A n Su P t (y), thn y A. Proof. If A n y/ A, thn oth rs y n y woul long to E s : y us Su P t (y) n y us A is ompos in sris with V \A in P s. This is ontrition, sin no pir of symmtri rs n long to post. Ft.8. A S Pt =. Proof. If A S Pt =, thn A, / S Pt n thr ists y V \A suh tht y P t (s thr r no yl in P t ), this ontrits th finition of A. Assum for som sour S Pt, th following lgorithm rturns strit sust A of V. IfA os not rspt (), thn ithr thr ists y V \A suh tht y/ Su D () n it ontrits Ft.6, or thr ists y V \A suh tht for n A, Su P t (y) n it ontrits Ft.7. Thn w just hv to tst Fts.6 n.7 to prov tht A stisfis (). It n thrfor us t lin 4 of th lgorithm pit in Fig. 5 to rurs. SP-R( S Pt,V,P t,d): outputs sust A of vrtis. A ={}. Whil thr ists y/ A n z A suh tht ithr yz E t or zy / E A y to A. Rturn A Thorm.. Th sris prlll irt snwih prolm is polynomil. Proof. Th vliity of th lgoritm sri in Fig. 5 follows from Lmms.4,.5, Proposition. n Fts.6.8. Th numr of rursiv lls in 5 is t most n = V. A rut for nlysis shows tht tsting th istn of st A stisfying () rquirs O(n ) pr sour of P t. It follows tht tsting th istn of sris prlll snwih is polynomil.. Intrvl posts A post P is n intrvl post iff rl intrvl I v =[ v, v ] n ssign to h lmnt v in P, suh tht v w if n only if v w. Prolm 6. INTERVAL DIRECTED SANDWICH PROBLEM. Instn: A post P t = (V, E t ) n igrph D = (V, E) suh tht E t E. Qustion: Dos thr ist n intrvl snwih post P s for th pir (P t,d)? W prov th intrvl irt snwih prolm is polynomil, in ontrst to th unirt s: th intrvl grph snwih prolm is NP-omplt [8]. This omplity gp is proly u to th ft tht n intrvl grph is o-omprility grph tht n hv mny trnsitiv orinttions. Thus, fiing n ritrry trnsitiv orinttion of th intrvl grph rstilly simplifis th prolm. Th sm phnomnon ours for th rognition prolm: lthough oth intrvl grph n intrvl post rognition prolms hv linr tim omplity, th intrvl grph rognition is muh hrr [,,]. Th intrvl post rognition lgorithm [5] is s on th following hrtriztion y Fishurn [5]. Thorm. (Fishurn [5]). A post P is n intrvl post if n only if th st of sussors {Su P (v) ={u V,v u}} v V is linrly orr y inlusion. From th ov hrtriztion, w n u tht non-onnt intrvl post ontins t most on onnt omponnt of mor thn on vrt. In othr wors, non-onnt intrvl post onsists in onnt intrvl post plus som isolt vrtis. This yils to nothr wll-known hrtriztion of intrvl posts in trm of forin su-post: post is n intrvl post iff it os not ontin + s inu post (s Fig. 6).

8 M. Hi t l. / Disrt Mthmtis 07 (007) Fig. 6. Th + post. E t E\E t Fig. 7. In this instn A ={}. To simplify th rwing, w omit th trnsitiv rs of E t. Howvr, E is not nssrily trnsitiv. Lmm. is quit similr to Lmm.4, pt tht smllr prt of th vrtis is this tim sussor of th st A. This rsult is th sis of our lgorithm n n sn s gnrliztion of th rognition lgorithm of [5]. Lmm.. If thr is n intrvl snwih post P s for th instn (P t,d), thn thr ists st A S Pt suh tht V \S Pt Su D (), A. Proof. Lt P s n intrvl snwih post for th instn (P t,d). From Thorm. P s hs its sts of sussors linrly orr y inlusion. Thrfor, thr ists st A of P s sours suh tht vry non-sour vrt of P s is sussor of ny vrt A: Su P s () = V \S Ps. Sin ny sour of P s is sour of P t, A S Pt. An sin A, Su P s () Su D (), it follows tht V \S Pt Su D (), A. Lt us fin th following st of vrtis: A ={ V Su D () V \S Pt }. An mpl of st A in irt snwih instn is givn in Fig. 7. Lmm.. Th following sttmnts r quivlnt: () Thr is n intrvl snwih post for th instn (P t,d). () Thr ists non-mpty st A n n intrvl snwih post for th instn (P t,d) V \A. Proof. First ssum P s is n intrvl snwih post for th instn (P t,d). From Lmm., A is not mpty. Sin ing n intrvl post is n hritry proprty, th post P s V \A is n intrvl post. Morovr, ny r of P t V \A is n r of P s V \A. It follows tht P s V \A is n intrvl snwih post for (P t,d) V \A. Convrsly, ssum P s is n intrvl snwih post for th instn (P t,d) V \A n A =. By Thorm., th vrtis of V \A r linrly orr y inlusion of sussors. Among th st SP s of sours of P s, w n istinguish SP s \S Pt n SP s S Pt. Noti tht ny r y, with A n y \S Pt longs to E (rmrk tht (V \A)\S Pt =V \S Pt sin A S Pt ). Thrfor, ing thos rs to E s fins snwih P s = (V, E s ). Sin for ny A, whv Su P s () = V \S Pt, for ny y V \A, w hv Su P s (y) Su P s (). Sin th st of sussors in P s is linrly orr y inlusion, P s is n intrvl orr. Corollry.. If thr ists n intrvl post snwih for th instn (P t,d), thn thr ists on, sy P s, suh tht S Ps = S Pt. Proof. Th proof follows from th onstrution sri in th proof of Lmm.. Noti tht whn onstruting P s from P s, w nvr rs towrs sours of P t. Applying this rgumnt rursivly omplts th proof.

9 08 M. Hi t l. / Disrt Mthmtis 07 (007) Fig. 8. A polynomil tim lgorithm for th intrvl irt snwih prolm. Thorm.. Th intrvl irt snwih prolm is polynomil. Proof. Th orrtnss of th lgorithm pit in Fig. 8 is impli y Lmms. n.. Tsting for th istn of th st A n lrly on in O(n ) whr n = V. Sin thr r t most n rursiv lls, th whol omplity is polynomil. Corollry.4. Th intrvl post snwih prolm is polynomil. 4. Th trnsitiv su-hin prolm In irt grph D = (V, E), trnsitiv su-hin is hin [v,...,v k ] suh tht ny rs v i,v j (with i<j k) longs to E. This stion introus th trnsitiv su-hin prolm tht sks for th istn of trnsitiv su-hin ontining t lst hlf of th vrtis. Using rution from -SAT inspir from [6], w first prov tht this prolm is NP-omplt. Thn w show tht th orrsponing post snwih prolm is polynomil whil th irt snwih prolm is NP-omplt. Prolm 7. TRANSITIVE SUB-CHAIN PROBLEM. Instn: A igrph D = (V, E). Qustion: Dos D ontins trnsitiv su-hin of lngth t lst V /? Thorm 4.. Th trnsitiv su-hin prolm is NP-omplt. Proof. W ru -SAT to th trnsitiv su-hin prolm. Lt I n instn of k luss i = (i i i ) ( i k). W trnsform I into igrph D = (V, E) with V ={ 0 } i k { i,i, i, i }. Th thr litrls of lus i r inpnnt. For ny i k n ny j, j i i is n r of E. Morovr for i<j k n h, h, i hh j longs to E iff i h = h j. Finlly for ny v = 0, w th rs 0 v. Fig. 9 givs n mpl. Clrly D is polynomil tim onstrutil. W now prov tht -SAT instn I is stisfil iff th ssoit igrph ontins trnsitiv hin of lngth qul to V /. Assum th instn I is stisfil. Thn t lst on litrl, not y l i, pr lus hs n stisfi n for ny i = j, l i = l j. Thn thr ists hin in D from 0 to k through th l i s suh tht th sugrph inu y th vrtis of th hin is trnsitiv. Suh hin ontins k + vrtis; thrfor, its lngth is k, whih is qul to V / sin V =4k +. Convrsly, if trnsitiv hin of lngth V / ists in D, thn it hs to ontin ny i (0 i k) plus on litrl l i pr lus. Th trnsitivity nsurs tht for no i = j,whvl i = l j. It thrfor fins n ssignmnt tht stisfis th instn.

10 M. Hi t l. / Disrt Mthmtis 07 (007) missing trnsitiv rs 0 Fig. 9. Th igrph ssoit to th -SAT instn ( ) ( ) ( ) whr =, = n =. Th trnsitiv rs r omitt n ott rs o not long to th igrphs. 4.. Trnsitiv su-hin post snwih prolm In post ny hin is trnsitiv. Th hight h(p ) of post P is th lngth of th longst hin of P. It turns out th trnsitiv su-hin prolm for post is polynomilly quivlnt to th omputtion of th hight, whih is polynomil (it n on using DFS). Prolm 8. TRANSITIVE SUB-CHAIN POSET SANDWICH PROBLEM. Instn: A post P t = (V, E t ) n post P = (V, E) suh tht E t E. Qustion: Dos thr ist post P s = (V, E s ) suh tht E t E s E n h(p s ) V /? It is wll known tht th hight of post is monoton inrsing funtion of th st of gs. Th following lmm stts tht proprty. Lmm 4.. Lt Q = (V, E) n Q = (V, E ) two posts suh tht E E. Thn h(q) h(q ). Thorm 4.. Th trnsitiv su-hin post snwih prolm is polynomil. Proof. Lmm 4. shows tht omputing h(p ) suffis to tst th istn of snwih post P s suh tht h(p s ) V /. 4.. Trnsitiv su-hin irt snwih prolm Prolm 9. TRANSITIVE SUB-CHAIN DIRECTED SANDWICH PROBLEM. Instn: A post P t = (V, E t ) n igrph D = (V, E) suh tht E t E. Qustion: Dos thr ist post P s = (V, E s ) suh tht E t E s E n hight(p s ) n/? As for th trnsitiv su-hin prolm, -SAT rus to th ov prolm. Stting P t = (V, ) n D s fin in Thorm 4. is nough to prov its NP-ompltnss. Thorm 4.. Th trnsitiv su-hin irt snwih prolm is NP-omplt. 5. Two-imnsionl posts Lt L n P, rsptivly, totl orr n post on th sm vrt st. If < P y implis < L y, thn L is linr tnsion of P. Th imnsion of post P is th minimum numr k of linr tnsions suh tht < P y if n only if < Li y for ny i, i k. Th omprility grphs of two-imnsionl posts r th prmuttion grphs.

11 040 M. Hi t l. / Disrt Mthmtis 07 (007) i = δ( ) i i = δ( i ) i i i Fig. 0. Th ggt ssoit to tripl T i = ( i, i, i ). W not i = δ( i ) n i = δ( i ). Th prmuttion grph snwih prolm is known to NP-omplt [8]. Unlik th intrvl s whr th irt prolm is polynomil whil th non-orint prolm is NP-omplt, this stion shows tht for two-imnsionl posts, th post snwih prolm is lso NP-omplt. In this s, fiing th orinttion of th gs os not hlp. Prolm 0. TWO-DIMENSIONAL POSET SANDWICH PROBLEM. Instn: P t = (V, E t ) n P = (V, E) two posts on th sm groun vrt st suh tht E t E. Qustion: Dos thr ist snwih post P s = (V, E s ) of imnsion? Thorm 5.. Th two-imnsionl post snwih prolm is NP-omplt. Proof. W ru to th sm BETWEENNESS prolm tht ws us in [8] for prmuttion snwih grphs. Sin w r ling with posts, w n som itionl rgumnts. Prolm. BETWEENNESS PROBLEM. Instn: A groun st S n st T ={T,...,T k } of tripls of S. Qustion: Dos thr ist linr orring λ suh tht for ny tripl T i = ( i, i, i ), ithr i < λ i < λ i or i < λ i < λ i? Lt T ={T,...,T k } st of tripls on groun st S n instn of BETWEENNESS. W ssoit 5-vrt ggt to ny tripl T i = ( i, i, i ) (s Fig. 0): { i, i, i,i, i } whr i = δ(i ) n i = δ(i ). To ny instn of BETWEENNESS w ssoit pir of posts P t = (V, E t ) n P = (V, E) with E t E, s on th ggt of Fig. 0 s follows: V = S X whr X ={i i k} { i i k}, E t = {i i,i i,i i,i i} n E = X S\{uv v = δ(u)}. i k Clrly, P t n P r polynomil tim onstrutil. First, suppos thr is two-imnsionl snwih post P s for (P t,p). W writ u<v if uv is n r of P s, n u v if u n v r inomprl. Lt L s = (L,L ) rlizr of P s. Thn P s is su-orr of th plnr ltti L = L L sin prout-imnsion n intrstion-imnsion r qul. W n fin n orring λ on L y u< λ v if n only if L (u)>l (v) n L (u)<l (v). Sin S is n ntihin in P s, th rstrition of λ to S is linr orring (s ny two lmnts i, j of S stisfy y finition ithr L (j)>l (i) n L (j)<l (i),orl (i)>l (j) n L (i)<l (j)). Lt us onsir tripl T i =( i, i, i ). Without loss of gnrlity, w n ssum tht i < λ i. Sin i < i n i i, thn i < λ i. In if w suppos tht i λ i, this mns tht i n i r omprl in L L, whih is ontrition. An if w suppos i > λ i, thn y trnsitivity of λ w hv i < λ i, n sin i< λ i, y trnsitivity i < λ i, whih is lso ontrition. Sin i < i n i i, it follows tht i < λ i. Similrly, onsiring i w n prov tht i < λ i. Thrfor, i is twn i n i in λ. Thus, solution to th two-imnsionl post snwih prolm (P t,p)implis solution to th BETWEENNESS prolm on S with tripls T.

12 M. Hi t l. / Disrt Mthmtis 07 (007) s s k s n i i Fig.. Compltion of P t into two-imnsionl snwih post: i X k implis tht ny r s j with j k longs to E s ; i X k implis tht ny r s j with k j longs to E s. Th rs rwn in ol long to E t. For th onvrs, lt λ linr orring on S tht solvs th BETWEENNESS prolm for th tripls T. By rvrsing som of th tripls, w n ssum tht i < λ i < λ i for vry i. W shll fin two-imnsionl post P s tht is snwih for th pir (P t,p). Lt S ={s k k n}, whr s k < λ s l if n only if k<l. For m n, lt Xm ={ i i = s m } n fin Xm similrly. Th post P s hs th gs s k, whr ithr Xm with m k, or X m with m k (Fig. ). Consir th following two listings of V. L : X,X,...,X n,s,x,s,x,s,...,s n,xn,s n,s n, L : Xn,...,X,X,s n,xn,s n,...,s 4,X,s,X,s,s. Not tht th listing us for h st X j m in L is rvrs in L. If L n L r onsir to linr orrs, thn thir intrstion is P s, proving tht P s is two-imnsionl. Rfrns [] K.S. Booth, G.S. Lukr, Tsting for th onsutiv ons proprtis, intrvl grphs n grph plnrity using PQ-tr lgorithm, J. Comput. Systms Si. (976) [] M. Crioli, H. Evrtt, C.H.M. Figuiro, S. Klin, Th homognous st snwih prolm, Inform. Pross. Ltt. 67 () (998) 5. [] D.G. Cornil, S. Olriu, L. Stwrt, Th ultimt intrvl grph rognition lgorithm? in: Proings of th Ninth Annul ACM-SIAM Symposium on Disrt Algorithms (SODA), 998, pp [4] C.M.H. Figuiro, S. Klin, K. Vusković, Th grph snwih prolm for -join omposition is NP-omplt, Disrt Appl. Mth. (00) 7 8. [5] P.C. Fishurn, Intrnsitiv iniffrn in prfrn thory: survy, Opr. Rs. 8 (970) [6] H.N. Gow, S.N. Mhshwri, L.J. Ostrwil, On two prolms in th gnrtion of progrm tst pths, IEEE Trns. Softwr Eng. () (976) 7. [7] M.C. Golumi, H. Kpln, R. Shmir, On th omplity of DNA physil mpping, Av. in Appl. Mth. 5 (994) 5 6. [8] M.C. Golumi, H. Kpln, R. Shmir, Grph snwih prolms, J. Algorithms 9 (995) [9] M.C. Golumi, R. Shmir, Complity n lgorithms for rsoning out tim: grph-thorti pproh, J. ACM 40 (99) 08. [0] M. Hi, Comprility invrints, Ann. Disrt Mth. (984) [] M. Hi, E. Lhr, C. Pul, A not on fining ll homognous st snwihs, Inform. Pross. Ltt. 87 (00) [] M. Hi, R. MConnll, C. Pul, L. Vinnot, L-BFS n prtition rfinmnt, with pplitions to trnsitiv orinttion, intrvl grph rognition n onsutiv ons tsting, Thort. Comput. Si. 4 (000) [] P.B. Hnrson, Y. Zlstin, A grph-thorti hrtriztion of th pv hunk lss of synhronizing primitivs, SIAM J. Comput. 6 (977) [4] E.L. Lwlr, Squning jos to minimiz totl wight ompltion tim sujt to prn onstrints, Ann. Disrt Mth. (978) [5] C.H. Ppimitriou, M. Ynnkkis, Shuling intrvl-orr tsks, SIAM J. Comput. 8 (979) [6] J.D. Ros, A grph-thorti stuy of th numril solution of sprs positiv finit systms of linr qutions, in: R.C. R (E.), Grph Thory n Computing, Ami Prss, NY, 97, pp [7] S.-M. Tng, F.-L. Yh, Y.-L. Wng, An ffiint lgorithm for solving th homognous st snwih prolm, Inform. Pross. Ltt. 77 (00) 7. [8] J. Vls, Prsing flowhrts n sris prlll grphs, Thnil Rport STAN-CS-78-68, Computr sin Dprtmnt, Stnfor Univrsity, Stnfor, CA, 978. [9] J. Vls, R.E. Trjn, E.L. Lwlr, Th rognition of sris prlll igrphs, SIAM J. Comput. () (98) 98.