O n t h e e x t e n s i o n o f a p a r t i a l m e t r i c t o a t r e e m e t r i c

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1 O n t h x t n s i o n o f p r t i l m t r i t o t r m t r i Alin Guénoh, Bruno Llr 2, Vlimir Mkrnkov 3 Institut Mthémtiqus Luminy, 63 vnu Luminy, F-3009 MARSEILLE, FRANCE, gunoh@iml.univ-mrs.fr 2 Cntr 'Anlys t Mthémtiqu Soils, Éol s Huts Étus n Sins Soils, 4 Rspil, F-7270 PARIS CEDEX 06, FRANCE, llr@hss.fr 3 Déprtmnt Sins Biologiqus, Univrsité Montrél, C.P. 628, su. Cntr-vill, Montrél, Qué H3C 3J7, CANADA, n Institut of Control Sins, 6 Profsoyuzny, Mosow 7806, RUSSIA, mkrnv@mglln.umontrl. Rvis, April 2002 Astrt. Frh, Knnn n Wrnow (99) fin Prolm MCA (mtrix ompltion to itiv) n prov it to NP-omplt: givn prtil issimilrity on finit st X, os thr xist tr mtri xtning to ll pirs of lmnts of X. W us prviously sri simpl mtho of phylognti ronstrution, n its xtnsion to prtil issimilritis, to hrtriz som lsss of polynomil instns of MCA n of rlt prolm. W point out tht ths prolms mit mny othr polynomil instns. W fous prtiulrly on two lsss of gnrliz yls, togthr with th orrsponing mximl yli grphs (2-trs n 2trs). Résumé. Frh, Knnn t Wrnow (99) ont posé l prolèm MCA (mtrix ompltion to itiv) suivnt t ont émontré s NP-omplétu : étnt onné un issimilrité prtill sur un nsml fini X, st-il possil l'étnr n un istn 'rr éfini sur touts ls pirs 'élémnts X. Nous utilisons un métho simpl ronstrution phylogénétiqu, préémmnt érit, t son xtnsion ux issimilrités prtills pour rtérisr s lsss 'instns polynomils MCA t 'un prolèm voisin. Nous montrons qu'n fit uoup 'utrs instns sont ussi polynomils. L'outil prinipl st onstitué pr ux lsss yls générlisés, v ls grphs yliqus mximux (2-rrs t 2-rrs) orrsponnts. Kywors: tr mtri, 2-tr, prtil istn. Introution W onsir prtil mtri on fix finit st X. Prisly, th vlu of is known on sust E of unirt pirs of lmnts of X. Th following ision prolm MCA (Mtrix Compltion to Aitiv) riss in svrl pplition omins,.g. phylognti tr ronstrution: os thr xist vlu X-tr T, suh s th tr

2 mtri T ssoit with it stisfis th following onition: for ny xy E, (x,y) = T (x,y). In othr trms, is it possil to omplt into tr mtri? W lso onsir nonmtri vrsion WMCA (Wk Mtrix Compltion to Aitiv) of MCA, whr ngtiv lngths on som gs of th X-tr T r llow n, s onsqun, T os not nssrily stisfy th mtri tringl inqulity. Frh, Knnn n Wrnow (99) prov tht Prolm MCA is NP-omplt. Hr w hrtriz som polynomil instns of oth Prolms WMCA (Stion 4) n MCA (Stion ). Our pproh is s on phylognti ronstrution mtho rll in Stion 3. This mtho ws prviously sri in Llr n Mkrnkov (998) n rntly xtn to fitting thniqu for infrring tr mtri from prtil mtri (Guénoh n Llr 200). Svrl prviously otin rsults r rll in this ppr, ut ll thos givn with thir proofs r nw. Two typs of gnrliz yliitis will xtnsivly us. On of thm is fin, whrs th othr is rll, in Stion 2. In Stion 6, w point out tht Prolms MCA n WMCA r, in ft, polynomil in wi lss of instns. 2. Nottions n finitions In this stion w giv som si finitions on grphs n trs (Stion 2.), n on issimilritis n mtris (Stion 2.2). This llows us to rll Prolm MCA n stt its nonmtri ountrprt WMCA in Stion Grphs n XLL-trs. W onsir hr only unirt simpl grphs without loops or multipl gs. In suh grph G = (V,E), vrtx v is lf if its gr (v) is qul to. In pth (vv, v v 2,, v k v ) of G twn two vrtis v n v, ll th vrtis r istint xpt, possily, whn v = v n th pth P is yl of G. Th grph G is tr if it is onnt n hs no yls. Th uniqu pth twn two istint vrtis v n v of tr T is not T(v,v ). Th grph G is k-liqu if V = k n uv E for ll u, v V. A tringl of G is sust of V inuing 3-liqu; suh sust is not xyz inst of {x,y,z}. A vlu grph is n orr pir (G,A), whr G is grph n A is rl lngth funtion on th g st E of G. Whn th grph G is onnt n hs no iruits of ngtiv lngth, w st, for ny two istint vrtis v n v of G, G (v,v ) = Min P pth of G twn v n v P A(). In th s of tr T, T (v,v ) = T(v,v ) A(). An XLL tr (lf lll oring to X tr) is tr T stisfying two proprtis: (i) th lf st of T is X; (ii) for ny v V(T)-X, (v) 3. In n XLL-tr, th vrtis in V(T)-X r ll ltnt vrtis. Th mximum numr of ltnt vrtis of 2

3 T is n-2, whr n = X ; whn it is rh, ll th ltnt vrtis hv gr 3 n th tr T is si to rsolv. For mor finitions n proprtis of suh trs, s th ook of Brthélmy n Guénoh (99) Dissimilritis n mtris. A issimilrity on X is rl funtion on X X stisfying (x,y) = (y,x) n (x,y) (x,x) = 0 for ll x, y X: A issimilrity is mtri (or istn) if it stisfis th lssil mtri tringl inqulity: for ll x, y, z X, (x,z) (x,y) + (y,z). It is wll known tht this proprty is stisfi y th minimum pth lngth funtion of ny positivly vlu onnt n unirt grph. So, shortst pth mtri G is ssoit in this wy to th omplt grph on X vlu y issimilrity. A issimilrity is tr mtri if it stisfis th following four-point onition: for ll x, y, z, w X, th inqulity (F) hols: (x,y) + (z,w) mx{ (x,z) + (y,w), (x,w) + (y,z) }. (F) It is now wll-known tht tr mtri is uniquly rprsntl y th lngths of th pths twn th lvs of non-ngtivly vlu XLL-tr T, ll its tr rprsnttion (Bunmn 97). An xtnsion of th prvious rsult (Llr 99) onsists of onsiring th wk four-point onition, whr th inqulity (F) is mt only for ll istint x, y, z, w X. A issimilrity stisfying this onition is not nssrily mtri. Suh issimilrity is ll tr issimilrity. A rl funtion on X X stisfying th wk four-point onition is ll tr funtion. A tr funtion with som ngtiv ntris is sily trnsform into tr issimilrity y ition of onvnint positiv onstnt 2C -min{(x,y)}. Similrly, issimilrity whih is not mtri is trnsform into mtri y ition of positiv onstnt 2C mx{(x,z) - (x,y) - (y,z): x, y, z X}, n this trnsformtion os not fft th four-point onition. Convrsly, ruing y C th lngths of ll trminl gs in positivly vlu XLL-tr T is quivlnt to ruing y 2C th pth lngths twn lvs of T. Thrfor, tr funtion hs gin uniqu XLL-tr rprsnttion, possily with ngtiv lngths on th xtrnl gs (inint to th lvs). Somtims, th issimilrity is prtil, in tht sns tht it is fin only on st E of unorr pirs of lmnts of X. Thus, w hv support grph G = (X,E), vlu y. W sy tht issimilrity xtns, or omplts into if xy E implis (x,y) = (x,y). Without loss of gnrlity, it will ssum in th squl tht G is onnt. W sy tht is prtil mtri if, for ny xy E, (x,y) = G (x,y). For th omplt grph s G, is prtil mtri if n only if it is mtri. Th following proprty is wll-known n sy to otin. Clrly, using minimum pth lngth 3

4 lgorithm, prtil mtri my lwys omplt in polynomil tim into its ssoit minimum pth lngth mtri G. Proposition 2.. A prtil issimilrity omplts into mtri on X if n only if it is prtil mtri Two prolms. Assum tht prtil issimilrity on X with support grph G = (X,E) is givn. Th following "Mtrix Compltion to Aitiv" (MCA) prolm hs n shown to NP-hr y Frh t l. (99): Prolm MCA: givn prtil mtri on X, os it omplt into tr mtri? Aoring to Proposition 2. ov, whn th givn prtil issimilrity is not mtri, th nswr to Prolm MCA is ngtiv. Th following "Wk Mtrix Compltion to Aitiv" (WMCA) prolm rmins of intrst sin suh ompltion still provis tr strutur (ut ngtiv lngths o not fit most of volutionry mols). In suh n xtnsion, it is not importnt to istinguish tr issimilritis from tr funtions in th ompltion ouput. Hr w o not rss th omplxity sttus of WMCA, n just xhiit som polynomil lsss of instns trs n 2-trs In this stion w rll prvious rsults on two lsss of grphs n omplt thm on som points. Ths grphs onstitut th min supporting tools in this stuy yli grphs. Lt G = (X,E) finit unirt simpl grph, n A E st of gs of G. Thn, X A nots th st of ll vrtis inint to t lst on g of A, n G A th sugrph (X A,A) of G. A st C E is si to k-yl ( for gr) of G if ll th vrtis of X C hv gr t lst k+ in G C n C is miniml for inlusion with this proprty. Clrly, -yl is yl. Hr w stuy th s k = 2. Exmpls. If G C is isomorphi to th omplt grph K k+2 or to th omplt iprtit grph K k+,k+, thn C is k-yl. If G C is whl, thn C is 2-yl. A grph with no k-yls is si k-yli. Th mximl k-yli grphs r ll hr k-trs. Thy hv n hrtriz in rursiv wy y To (989): th omplt grph K k with k vrtis is k-tr; if G = (X,E) is k-tr, thn, for ny sust Y X of rinlity k n nw vrtx x X, th grph G = (X {x}, E {xy: y Y}) is k-tr. Thn, grph G = (X,E) is 2-tr if thr xists n orring (x,x 2,,x n ) of X suh tht x x 2 E n, for i = 3,,n, th vrtx x i hs gr 2 in th sugrph G i 4

5 inu y th vrtx st {x,x 2,,x i } (suh n orring is rvrs limintion orr, rvit s RE orr). A 2-tr with n vrtis is 2-onnt n hs 2n-3 gs. It hs t lst on vrtx of gr 2. Both grphs G n G'of Figur r 2trs G G Figur 4 To proposs prour for iing whthr givn st of gs A inlus k-yl. This prour trmins sust Pl(A) of A, ll th kpling of A, s follows: srh vrtx of gr t most k in G A ; if no suh vrtx xists, thn Pl(A) = A; othrwis, lt th vrtx foun with its inint gs, n rpt th oprtion until no vrtis of gr k rmin. Th st of rmining gs is Pl(A). Th st A is k-yli if n only if Pl(A) = k. Suh n lgorithm lrly runs in O(n) tim. A onnt 2-yli grph omplts in mny wys into 2-tr; w giv hr prour tht will usful in Stion 4: If thr xists vrtx x of gr, nw g twn x n n ritrry othr vrtx y, not lry jnt to x. Rpt th oprtion until no vrtx of gr lss thn 2 rmins. List ll th pirs not inlu in E in n ritrry orr n hk thm oring to th list orr. For h suh pir xy, us th 2-pling lgorithm ov to trmin whthr th grph (X, E {xy}) is 2-yli; xy to E if th nswr is positiv, n rjt it othrwis. Stop whn E = 2n-3. Proposition 3.2. If G is 2-yli grph, th ov lgorithm xtns it into 2tr in O(n 3 ) tim. Proof. Clrly, ing n g to vrtx of gr 0 or nnot rt 2-yl. This justifis th first prt of th lgorithm. In th son prt, th finl grph is mximl 2-yli grph, tht is 2-tr; othrwis, furthr pirs woul rtin uring th snning of th list. As fr s th lgorithmi omplxity is onrn, th first prt is in O(n). In th son on, w hv to hk O(n 2 ) pirs, th pling prour ing in O(n) h tim. Th notion of hin n xtn to 2-trs. Lt G = (X,E) 2-tr on X n pir xy E. Th grph G = (X, E {xy}) is no longr 2-yli. It hs uniqu

6 2-yl C xy = Pl(E {xy}). At h stp of th pling lgorithm, vrtx is limint togthr with two gs. So, stting Y = X Pl(E {xy}) n n = Y, th qulity Pl(E {xy}) = 2n -2 hols. Thn, th grph H = (Y,C xy -{xy}) is 2-yli with 2n -3 gs n, so, is 2-tr. This 2-tr hs on or two vrtis of gr two, tkn in th st {x,y}; it is ll th 2-hin G[xy] of G twn x n y. If x n y r two non-jnt vrtis of G[xy], thn th 2-hin G[x y ] is sugrph of G[xy], propr s soon s {x,y } {x,y}. Th lngth of 2-hin, ompris twn 2 n n-2, is th numr of its vrtis minus 2. W thn hv th following proprty: Proposition 3.3. For ny RE orr on X, th lst vrtx of G[xy] is x or y n hs gr 2 in G[xy]. Proof. Assum tht thr xists RE orr L on X n vrtx z of G[xy] suh tht oth x n y r prssors of z in L. Thn, th onstrution ov woul l to 2hin twn x n y without z s vrtx yli grphs. Anothr gnrliztion of yls n trs is mor lssil thn th prvious on, n hs prompt importnt littrtur. Rll tht, givn grph G, ru grph is otin from G y sussiv ontrtions of gs inint to vrtx of gr 2 until no suh possil oprtion rmins. For instn, yl rus to 3-liqu (us yls of lngth two o not xist in simpl grphs). A grph G is homomorphi to grph H without vrtis of gr 2 if its ru grph is isomorphi to H. For k 2, st C E is si to k-yl of G if th grph G C is homomorphi to K k+2. Espilly, -yl is yl. A grph with no k-yls is si k-yli. Th mximl k-yli grphs r th lssil k-trs, whih hv th following wll-known rursiv hrtriztion: th omplt grph K k is k-tr ; if G = (X,E) is k-tr, thn, for ny k-liqu Y X of G n nw vrtx z X, th grph G = (X {z},e {zy: y Y}) is k-tr. So, s it is osrv in To (989): Proposition 3.4. If grph G is k-tr, thn it is k-tr. Th sugrphs of 2-trs r xtly th grphs with no sugrph homomorphi to K 4 ; thy r ll prtil 2-trs or sris-prlll grphs in th litrtur (Wl n Colourn 983). k-trs r lso th mximl tringult (or horl, or rigi iruit) grphs with no (k+2)-liqu (Ros 974), tht is th mximl grphs of trwith k (s.g. Bolnr 997). Suh proprtis mk thm to onstitut n intrsting lss in lgorithmi grph thory. 6

7 W r gin intrst in th s whr k = 2. A grph G = (X,E) is 2-tr if thr xists RE orr (x,x 2,,x n ) of X suh tht x x 2 E n, for i = 3,,n, th vrtx x i hs gr 2 n longs to uniqu tringl in th sugrph G i inu y th vrtx st {x,x 2,,x i }. A 2-tr with n vrtis hs t lst two vrtis of gr 2. 2-trs r th mximl tringult grphs without 4-liqu. Th grph G of Figur is 2-tr whil G is not (not tht it is 2-yl). An O(mx(m,n)) lgorithm to i whthr givn grph G is 2-yli ws vis y Liu n Glmhr (980). Th nlogous prolm is NP-hr for k 3 (Arnorg t l. 987). As onsqun of th ov rursiv hrtriztions of 2-trs n 2- trs, on otins th following proprty whih, for k = 2, is vrint of wll-known rsult of Dir (92; s,.g., Wlsh 976, p. 238, or Aignr 979, p. 387): Proposition 3.. A k-yl inlus t lst on k-yl. Proof. Othrwis, lt C k-yl suh tht G C hs no k-yl. So, G C is sugrph of mximl grph G with no k-yl, tht is k-tr. But G is lso k-tr, ontrition with th hypothsis tht C is k-yl. 4 Th tringl mtho Th tringl mtho llows on to xtn ny issimilrity with 2-tr s support grph to tr funtion. It ws introu in Llr (99) in prtiulr s, thn formliz n stui in mor tils in Mkrnkov (997), Llr n Mkrnkov (998), n xtn in Guénoh n Llr (200). It is rll hr in Stion 4. n us to show our first min rsult: Prolm WMCA is polynomil in 2-yli grph. Thrfor, n tking into ount nw uniity rsult otin in Stion 4.2, Prolm MCA is solv for gnrl instns of 2-tr support grphs. 4.. A solution of WMCA in 2-yli grphs. W first ssum tht th support grph G of th givn prtil issimilrity is 2-tr. In this s, th tringl mtho uils vlu XLL-tr T suh tht T (x,y) = (x,y) for ny xy E. Th si osrvtion is tht tringl {x,y,z}, wight oring to, fins vlu {x,y,z}ll-tr T of th 3-str typ, tht is, with uniqu ltnt vrtx u. Th vlus (x,y), (x,z) n (y,z) r uniquly otin s pth lngths in T ftr rsolving th following systm of linr qutions 2 T (x,u) = (x,y) + (x,z) - (y,z), 2 T (y,u) = (y,x) + (y,z) - (x,z) n 2 T (z,u) = (z,x) + (z,y) - (x,y). Th orr of vrtis in G is n ritrry RE orr x, x 2,, x n. So, for vry x i, thr xist xtly two lmnts y, z {x,, x i- } suh tht oth x i y n x i z long to E. Th tringl {x i,y,z} will hng into n {x i,y,z}ll-tr of th 3-str typ, n th otin 3-strs will sussivly glu togthr to finlly otin n XLL-tr. 7

8 First, th tringl {x,x 2,x 3 } is rprsnt s 3-str T 3. Thn, th sm oprtion is m on th tringl {x 4,y,z}, whr y, z {x,x 2,x 3 }. A son 3-str T 4 is otin with th pth T 4 (yz) ommon with T 3 (yz), with th sm lngth (y,z). Th trs T 3 n T 4 r glu on this pth to otin n {x,x 2,x 3,x 4 }LL-tr. A nw tringl with th vrtis x i, y, z suh tht y, z {x,x 2,,x i- } is onsir t h stp; th xistn of suh tringl is gurnt y th proprtis of RE orrs. If yz E, thn its vlu is fix s T i-(y,z), th lngth of th pth twn y n z in th urrnt {x,x 2,,x i- }LL-tr T i-. So, th 3-str orrsponing to th tringl {x i,y,z} provis grfting of th nw vrtx x i onto this tr. Finlly, n XLL-tr T =T n is otin, prsrving ll th issimilrity vlus in. Applying th tringl mtho is polynomil with O(n 2 ) omplxity. Thus: Proposition 4.. If G = (X,E) is 2-tr, thn thr xists vlu XLL-tr T suh tht th tr funtion T xtns. Thorm 4.2. If G = (X,E) is 2-yli, thn thr xists vlu XLL-tr T suh tht th tr funtion T xtns. Proof. Assum G is 2-yli n us th lgorithm givn in Stion 3. to otin 2-tr G = (X, E ) with E E. Giv ritrry positiv lngths to ll th pirs in E -E. Now xtns to prtil issimilrity with 2-yli support grph n th rsult follows from Proposition 4.. Sin this lgorithm runs in O(n 3 ) n th tringl mtho runs in O(n 2 ), w r l to onlu: Corollry 4.3. Prtil issimilritis with 2-yli support grphs onstitut polynomil lss of WMCA Prolm. In th prtiulr s whr G is 2-tr, Llr n Mkrnkov (998) show tht th finl X-tr os not pn on th orr on tringls. Thir rgumnts r xtn hr to 2-trs. Thorm 4.4. Th tringl mtho uniquly xtns prtil issimilrity with 2tr support grph G = (X,E) to tr funtion T, inpnntly of th us RE orr. Proof. Lt x, y X suh tht xy E. W pro y inution on th lngth k of th 2hin G[xy]. Th rsult is ovious for k 4. Assum tht it is tru for ll 2-trs of lngth t most k- n, without lost of gnrlity, tht x hs gr 2 in G[xy]. Lt z n z th vrtis jnt to x in this grph. As osrv in Stion 3., ithr th pir zz longs to E, or G[zz ] G[xy]. In oth ss, T (z,z ) is givn or, y th inution 8

9 hypothsis, uniquly trmin y th tringl mtho ppli to G[zz ]; th sm for oth pirs yz n yz. Finlly, on hs T (x,y) = mx{ (x,z) + T (y,z ), T (x,z ) + T (y,z) } - T (z,z ). Exmpl 4.. Consir Figur 2, whih shows 2-tr now with prtil mtri. Figur 3 shows th 3-strs ssoit to its tringls n thir sussiv inorportion, until th finl X-tr is otin. In ll our xmpls, th lphti orr on th vrtis will n RE orr f Figur 2: 2-tr now with prtil mtri f 20 f f Figur 3: onstrution of n X-tr from th vlu 2-tr of Figur 2 9

10 4.2. Rsolv 2-trs n th MCA prolm. Th tringl mtho provis uniqu tr funtion xtnsion of ny prtil issimilrity fin on 2-tr. But it my xist othr xtnsions, not givn y this mtho Figur 4 Figur Exmpl 4.6. Consir th vlu 2-tr of Figur 4, with th RE orr (,,,, ). Th tr of Figur shows four possil grftings of on th initil {,,}-tr, mong n infinity; hr, is th grfting provi y th tringl mtho. Th rson of suh n miguity is th qulity of th sums (,)+(,) n (,)+(,). So, th thir sum T (,)+(,) n tk ny vlu infrior to 2 (orrsponing to th tringl mtho), n suprior to 4 if mtri is rquir. Thn, th grfting of provis tr mtri only for 2 T (,) 6: 2 n 4 r th xtrm plmnts of omptil with this onition. With slight hng on th givn prtil issimilrity, sy (,) = 7., th tringl mtho xtnsion oms th uniqu possil on sin, oring to th fourpoint onition, on thn otins T (,)+(,) = 2. n, so, T (,) = 0.. Although n hv no longr th sm grfting point on th pth T(), th otin XLL-tr is vry los to th prvious on with th plmnt for. In tht sns, th tringl mtho xtnsion givs prtiulrly stl tr. Not lso tht, lthough th t onstitut prtil mtri, th uniqu possil xtnsion is not tr mtri. Dfinition 4.7. A vlu 2-tr G on X is si to rsolv if it ls, y pplying th tringl mtho, to rsolv XLL-tr. Thorm 4.8. Th tr funtion xtnsion of prtil issimilrity with 2-tr support grph G is uniqu if n only if G is rsolv. 0

11 Proof. If G is rsolv, h of th n-2 ltnt vrtis is pl in turn t n intrior point of n g of th urrnt tr. Thr is no hoi for this plmnt n th proof is sily otin y inution on n. For th onvrs, ssum tht, t som stp of th tringl mtho, nw vrtx x i = x is grft on th pth T(yz) t point u whih is lry ltnt vrtx. So, two gs xu n uv, oth not longing to th pth T(yz), r otin, with rsptiv lngths A(xu) n A(uv). Dtrmin nw tr T y rpling xu n uv with thr gs uu, u x n u v, n giv thm lngths rsptivly qul to A (uu ) = ε, A (u x) = A(ux)-ε, A (u v) = A(uv)-ε, whr ε is smll nough stritly positiv onstnt. Th mtri in th otin vlu tr T is n ltrntiv xtnsion of th vlus of twn ll th pirs of prssors of x in th onsir RE orr. Strting from T to ontinu th tringl mtho pross ls to n xtnsion of tht iffrs from th tringl mtho on. Exmpl 4.9. St X = {,,,, } n onsir th vlu 2-tr G (hr, 2-tr) in Figur 6. Th tringl mtho givs th XLL-tr of Figur 7, with th vrtx u of gr 4. With ε =, th oprtion sri in th ov proof provis th ltrntiv tr T of Figur 8, whr ll th lngths of gs of G r still prsrv s pth lngths. On th ontrry, th vlu 2-tr of Exmpl 4. is rsolv. Consquntly, it ls to uniqu XLL-tr, whih is pit in Figur u u u Figur 6 Figur 7 Figur 8 Givn prtil issimilrity with 2-tr support grph G, on my us th tringl mtho (in O(n 2 )) to trmin th orrsponing XLL-tr T. If T is rsolv, thn th xtnsion of is uniqu. This xtn msur will mt th mtri onition if n only if non of xtrnl gs in T hs strongly ngtiv lngth. So: Corollry 4.0. Prtil issimilritis with rsolv 2-tr support grphs onstitut polynomil lss of Prolm MCA. Thorm 4.8 will usful in Stion 6 to fin othr polynomil lsss. W n this stion with thr rmrks:

12 Rmrk 4.. As Exmpl 4.6 shows, Prolm MCA rmins iffiult in n unrsolv 2-tr. In tht s, mny tr funtion xtnsions r possil, n th prolm is to trmin whthr som of thm r mtri. Exmpl 4.9 shows n sy s, whr th tringl mtho xtnsion is lry mtri. Rmrk 4.2. For similr rsons, Corollry 4.0 nnot xtn to 2-yli grphs. In this s gin, mny tr funtion xtnsions xist. Although it is lwys possil to xtn prtil mtri from 2-yli to 2-tr support grph (st th lngth of nw g s th minimum pth lngth twn its xtrmitis whnvr this quntity is fin), suh n xtnsion is not gurnt to prtil tr mtri. Rmrk 4.3. Th s of rsolv 2-trs my onsir s th gnrl on, sin unrsolv ons orrspon to itionl linr pnnis twn th vlus of. For instn, in Exmpl 4.9, w hv 2A(,u) = (,)+(,)-(,) = (,)+(,)- (,). Prtil 2-trs n Prolm MCA Compr to 2-trs, 2-trs provi, s support grphs, itionl informtion out th orrsponing vlu XLL-tr. In this stion, it is shown tht Prolm MCA is polynomil for ny prtil mtri with ny 2-yli support grph G, through n qut ompltion of G into 2-tr. Th strtpoint is th following rsult: Proposition. (Llr n Mkrnkov 998). Lt prtil issimilrity with 2-tr support grph G, n T th tr funtion xtning otin y th tringl mtho. Thn, T is tr mtri if n only if is prtil mtri. Sin 2-trs r horl grphs, it is sy to vrify tht, with 2-tr support grph G = (X, E), is prtil mtri if n only if ll th tringls of G r mtri. Assum tht G is just 2-yli. W thn n nw pirs to E until 2-tr is otin. To gt tr mtri xtnsion y th tringl mtho, w hv to giv to h nw g xy lngth prsrving th proprty of ing gnrlly prtil gin - mtri. For tht purpos, simpl solution onsists of tking th minimum lngth G (x,y) of pth of G twn x n y s (x,y). As rll in Stion 3.2, fst lgorithms xist to rogniz prtil 2-trs, tht r grphs without sugrphs homomorphi to K 4 (tht is, 2-yls). Hr w giv simpl lgorithm tht omins this rognition with 2-tr xtnsion of givn prtil mtri. Th lgorithm is s on th onstrution of RE orr, togthr with 2

13 mrking som nw gs. Lt x vrtx with minimum gr (x) in th urrnt grph: if (x) =, lt y th vrtx jnt to x n z vrtx, iffrnt from x, jnt to y; onvnint lngth is ssign to th pir yz, whih is mrk, n x is limint togthr with th g xy; if (x) = 2, lt y n z th vrtis jnt to x; if th pir xy is not lry n g of th urrnt grph, thn this g is mrk n, onvnint lngth is ssign to it, n x is limint togthr with th gs xy n xz; if (x) > 2, th lgorithm stops; G is not prtil 2-tr. Th irruil prt of G, not Irr(G) is th grph otin t th n of this prour; ithr Irr(G) = K 3 or ll its vrtis hv gr t lst 3. Lt E th st E ugmnt with ll th mrk pirs. Thorm.3. Th ov prour xtns givn prtil mtri with th support grph G to prtil mtri with 2-tr support grph if n only if G is 2-yli. Proof. Th ompltion-limintion prour trmins RE orr on X. W first show tht th limintion of x nnot hng th vntul 2-yliity of G. This is ovious whn (x) =. For (x) = 2, whn lting x, 2-yl C of G inluing th gs xy n xz oul rmov. Th xistn of suh 2-yl implis nothr on C otin y sustituting yz to ths two gs; if yz E, thn th 2-yl C xists in G n is not fft y th ltion of x; if yz E, thn C is sustitut to C for lting x. In ll ss, th nw grph is 2-yli if n only if G is. So, if Irr(G) = K 3, ll th sussivly onsir grphs r 2-yli. Othrwis, Irr(G) hs 2-yl, s fin in Stion 3. n, y Proposition 3.2, is not 2-yli. Assum Irr(G) = K 3 n onsir th grph G = (X,E ): on this grph, th ov vlution n limintion prour onsists of sussiv limintions of vrtx longing to uniqu tringl, whih is mtri. So, G is mtri 2-tr n th rsult follows. Corollry.4. Prtil mtris with 2-yli support grphs onstitut polynomil lss of Prolm MCA. Exmpl.. Consir th prtil mtri of Figur 9 with yl support grph. Whil liminting vrtis n, gs n r with rsptiv lngths 0 n 7. Th rsulting 2-tr orrspons to th XLL-tr of Figur 0, with null lngths for th gs jnt to n, whih givs tr mtri xtnsion of. 3

14 Figur 9 Figur 0 Rmrk.6. Whn th ompltion-limintion prour ls to omplt grph Irr(G) with vrtx st Y of rinlity t lst 4, it oul xpt tht w just hv to trmin whthr Y is tr mtri. In ft, this is only th s if ll of th gs of Irr(G) hv not riv thir lngths uring th prour. Othrwis, mny possil lngths wr onvnint, provi thy r omptil with th tringl mtri onition. So, it is not possil to giv gnrl onlusion. Exmpl.7. Applying th ompltion-limintion prour to th vlu grph of Figur ls to K 4. In tht xmpl, no possil systm of lngths on th nw gs n giv tr mtri f 3 Figur Th limintion of vrtis n implis ition of gs f n. Ths oprtions l to th omplt grph on Y = {,,, f}. On my hv (,f) 0 n (,) ; so, (,) + (,f) = 9, 0 (,f) + (,) n (,) + (,f) 9. Th four-point onition nnot stisfi. In th xmpl ov, th nswr to Prolm MCA ws otin with polynomil numr of lmntry oprtions. In mor gnrl s, th grph Irr(G) is now with g lngths, som of thm ompris into intrvls. On hs thn n instn of th "snwih prolm", lso prov NP-omplt y Frh t l. (99). 6 Furthr polynomil ss Clrly, 2-yli grphs r sprs, whr th g nsity rss s th numr of vrtis inrss. In this stion, mor ns grphs r invstigt. It is first osrv in Stion 6. tht suffiint numr of mningful yls of lngth four l to n 4

15 lgri solution. Stion 6.2 prsnts n pproh irtly rlt with th rsults of Stions Skw C 4 s. A C 4 of G is yl of lngth 4 (th usul trm 4-yl hs nothr mning in this ppr). A C 4 xyzw is si skw if (x,y)+(z,w) (x,w)+(y,z). Thn, if, sy, xz E, w hv (y,w) = mx{(x,y)+(z,w), (x,w)+(y,z)} - (x,y). If xyzw os not mit hor, w hv th linr qution (y,w) + (x,z) = mx{ (x,y) + (z,w), (x,w) + (y,z) } with two vrils. A grph G with nough skw 4-yls ls in this wy to systm of linr qutions, whos rsolution my giv n nswr to prolms MCA n WMCA in polynomil tim. Exmpl 6.. Th support grph of Figur 2 hs 9 gs; so, it rmins 6 untrmin vlus for n xtnsion of f Figur 2 Figur 3 G hs 9 skw C 4 s ling to th following systm of qutions: : (,)+(,) = 2 f : (,)+(,f) = 3 : (,)+(,) = 2 f : (,)+(,f) = 3 f : (,)+(,f) = 2 f : (,)+(,f) = 2 : (,)+(,) = f : (,f)+(,) = 2 f : (,)+(,f) = This systm hs th solution: (,) = (,) = 7, (,) =(,f) =, (,f) = (,) = 6. Th orrsponing tr mtri is rprsnt y th XLL-tr of Figur G inlus 2-tr. 2-yli grphs r sprs, us thy hv t most 2n-3 gs. Whn th numr of gs inrss, it my xpt tht th support grph G mits 2-tr H = (Y,F) s sugrph (w ll H 2-sutr of G). Th tringl mtho thn provis n YLL-tr T n, if H is rsolv, uniqu tr funtion T xtning th rstrition F. On my thn, in first stp, ompr th otin vlus of T to th vlus of on pirs of E not in F, ut with xtrmitis in Y. If th vlus r not intil, Prolm WMCA (n so MCA) hs ngtiv nswr for th t. Othrwis, th positiv nswr to WMCA otin for prt of th t n lso, oring to Stion 4.2, ngtiv on for MCA. In oth prolms, on my, with positiv nswr on Y, try to xtn th otin solution to th rmining pirs of E, or to sk nothr 2-tr in G. 2 f

16 This pproh llows on to solv Prolms WMCA n MCA in mny ss. W prsnt hr mor formliz prour tht works s soon s G mits sugrph H whih is rsolv 2-tr on X. Dfinition 6.2. A imon D of G is qurupl xyzw of lmnts of X suh tht {xy, yz, zw, wx, xz} E n yw E (so, D is C 4 with uniqu hor). Th imon D is rsolv if th C 4 xyzw is skw. It ws osrv in Stion 6. ov tht Conition (F) uniquly trmins th vlu of th son hor of rsolv imon. Th min stp of th prour is s follows: sn ll th qurupls of lmnts of X. Whn qurupl is 4-liqu of G, hk whthr it stisfis th four-point onition (F); if not, stop: is not tr funtion. Whn qurupl xyzw is imon, hk whthr it is rsolv; if it is th s, st (y,w) = mx{ (x,y) + (z,w), (x,w) + (y,z) } - (x,y) n th pir yw to E. This stp is itrt until ithr ll th pirs hv riv tr funtion vlu, or 4- liqu not stisfying Conition (F) is foun, or no nw pirs n vlu; in th lst s, th prolms rmin untrmin. Assum tht G inlus rsolv 2-tr H n onsir th lmnts of X in RE orr. Th first four vrtis of rsolv 2-tr onstitut rsolv imon in H; t th nxt stp, th fith vrtx onstituts rsolv imon with som tripls in th prvious vrtis, n so on. Aoring to Thorm 4.8, th xtnsion is uniqu or ls to ontritory Conition (F); in oth ss, prolms WMCA n MCA r solv. Th snning of qurupls is in O(n 4 ) n th numr of itrtions is oun y th numr of pirs not in E, tht is O(n 2 ). Finlly, th prour ns t worst O(n 6 ) tim. Thorm 6.3. Prtil issimilritis with support grphs inluing rsolv 2-tr onstitut polynomil instns of Prolms WMCA n MCA. In phylognti pplitions, for instn, thr r gnrlly svrl, ut fw, missing ntris in onsir volutionry istn mtrix. Thrfor, n lgorithm s on th ov-isuss prour n usfull in this sitution. In Guénoh n Llr (200), th tringl mtho ws ppli, in tr mtri pproximtion purpos, to mor thn 00 prtil mtri tls orrsponing to vrtrt homologous gns issu from th HOVERGEN ts (Durt t l. 994). Among thm, th nswr to MCA rmin untrmin for only ozn tls, with unonnt support grphs. Exmpl 6.4. Consir th vlu grph in Figur 4; it hs n = 6 vrtis n m = 0 gs. Th ltion of th g provis rsolv 2-tr (whil th ltion of f givs n unrsolv on). In th ov prour, first snning of qurupls givs 6

17 th imons (unrsolv), ((,) = 7) n f ((,f) = 6). Th gs n f r to E n son snning is prform. From th imons f, f n f, w fin, rsptivly, (,) = 6, (,) = 6 n (,f) = 0. Consquntly, os not stisfy Conition (F) on th 4-liqu f. Th onlusion is tht this prtil mtri nnot omplt into tr funtion f Figur 4 7. Conlusion W sri n stlish nw proprtis of two lsss of yli-lik grphs (Stion 3), whih l to th following min rsults. For prtil issimilrity with support grph G: If G is 2-yli, thn xtns to tr funtion in polynomil tim (Stion 4.); If, morovr, G is rsolv 2-tr, thn iing whthr xtns or not to tr mtri n solv in polynomil tim (Stion 4.2); If G is 2-yli, thn th prolm whthr is not prtil mtri n os not xtn to mtri of ny typ (Stion 2.2) or is prtil mtri n xtns to tr mtri n solv in polynomil tim (Stion ); If G inlus rsolv 2-tr G, thn th prolm whthr xtns or not to tr funtion or to tr mtri n solv in polynomil tim. Morovr th prour propos in th rtil os not rquir th rovry of G (Stion 6.2). So, mny prtil instns of prolms WMCA n MCA r rsolvl in polynomil tim. Hr is our lst xmpl, prsnting smingly iffiult instn. Th Ptrsn grph of Figur is 2-yl n hs no 2-sutrs. It is now with prtil tr mtri, xpt th vlu of on g (in ol), inrs y. It my xpt tht suh prtil mtri no longr xtns to tr mtri. How to prov (or isprov) tht? 7

18 Figur Rfrns M. Aignr (979), Comintoril Thory, Springr-Vrlg, Brlin. S. Arnorg, D.G. Cornil, A. Proskurowski (987), Complxity of fining mings in k-tr, SIAM J. Alg. Dis. Mth. 8, J.P. Brthélmy, A. Guénoh (99), Trs n Proximity Rprsnttions, Lonon, J. Wily. H.L. Bolnr (997), Trwith: lgorithmi thniqus n rsults, in Proings 22n Intrntionl Symposium on Mthmtil Fountions of Computr Sins, MFCS 97, I. Privr, P. Ruzik (s), Ltur Nots in Computr Sins 29, Brlin, Springr-Vrlg, pp P. Bunmn (97), Th rovry of trs from msurs of issimilrity in Mthmtis in Arhologil n Historil Sins, F.H. Hoson, D.G. Knll, P. Tutu (Es.), Eimurg, Eimurg Univrsity Prss, pp G.A. Dir (92), A proprty of 4-hromti grphs n som rmrks on ritil grphs, J. Lonon Mth. So. 27, L. Durt, D. Mouhirou, M. Gouy (994), HOVERGEN: ts of homologous vrtrt gns, Nuli Ais Rs. 22, M. Frh, S. Knnn n T. Wrnow (99), A roust mol for fining optiml volutionry trs, Algorithmi, 3, -79. A. Guénoh, S. Grnols (999), Approximtion pr rr 'un istn prtill, Mthémtiqus, Informtiqu t Sins humins 46, -64. A. Guénoh, B. Llr (200), Th tringls mtho to uil X-trs from inomplt istn mtris, RAIRO Oprtions Rsrh 3, B. Llr (99), Minimum spnning trs for tr mtris: rigmnts n justmnts, J. of Clssifition 2 (99)

19 B. Llr, V. Mkrnkov (998), On som rltions twn 2-trs n tr mtris, Disrt Mth. 92, P.C. Liu, R.C. Glmhr (980), An O(mx(m,n)) lgorithm for fining sugrph homomorphi to K 4, in Proings th Southstrn Confrn on Comintoris, Grph Thory n Computing, pp V. Mkrnkov (997), Propriétés omintoirs s istns 'rr : Algorithms t pplitions, Ph.. thsis, EHESS, Pris. R.E. Pipprt, L.W. Bink (969), Chrtristion of 2-imntionl trs, in Th Mny Fts of Grph Thory, G. Chtrn, SF. Kpoor (Es.), Ltur Nots in Mthmtis 0, Brlin, Springr-Vrlg, pp A. Proskurowski (984), Sprting sugrphs in k-trs: ls n trpillrs, Disrt Mth. 49, D.J. Ros (974), On simpl hrtriztions of k-trs, Disrt Mth. 7, P. To (989), A k-tr gnrliztion tht hrtrizs onsistny of imnsion nginring rwings, SIAM J. Dis. Mth. 2 (2), A. Wl, C.J. Colourn (983), Stinr trs, prtil 2-trs n minimum IFI ntworks, Ntworks 3, D.J.A. Wlsh (976) Mtroi Thory, Lonon, Ami Prss. 9

20 Psuoos Algorithm 3.. Compltion of 2-yli grph into 2-tr. Whil thr xists x X suh tht (x) = Slt y X suh tht y x n xy E E := E {xy} En Whil If E = 2n-3 thn stop Algorithm Els Mk list L of ll pir xy not inlu in E Whil E < 2n-3 Slt ny pir xy from L Apply 2-pling lgorithm to hk whthr th grph G = (X, E {xy}) is 2-yli L: = L \ {xy } If G is 2-yli E := E {xy} En Whil En Els Algorithm.2. Compltion-limintion prour. Whil thr xists x X suh tht (x) = Slt y suh tht xy E n z x suh tht yz E Mrk xz (x,z) := G (x,z) X := X \ {x} n E := E \ {xy} En Whil Whil thr xists x X suh tht (x) = 2 Slt y n z suh tht xy E n xz E If yz E Mrk yz E := E {yz} (y,z) := G (y,z) X := X \ {x} E := E \ {xy,xz} En Whil 20