A Polynomial-Space Exact Algorithm for TSP in Degree-8 Graphs

Transcription

1 A Polynomial-Spac Exac Algorihm for TSP in Dgr-8 Graphs Norhazwani Md Yunos Alksandar Shurbski Hiroshi Nagamochi Dparmn of Applid Mahmaics and Physics, Gradua School of Informaics, Kyoo Unirsiy {wani, shurbski, Absrac: Priously, h auhors of his work ha prsnd in a sris of paprs polynomial-spac algorihms for h TSP in graphs wih dgr a mos fi, six and sn. Each of hs algorihms is h firs algorihm spcializd for h TSP in graphs of limid dgr fi, six, and sn rspcily, and h running im bound of hs algorihms ouprforms Gurich and Shlah s O (4 n n log n ) algorihm for h TSP in n-rx graphs (SIAM Journal of Compuaion, 16(3), pp , 1987). Now w ask wha is h highs dgr i unil which a spcializd polynomial-spac algorihm for h TSP in graphs wih maximum dgr i ouprforms Gurich and Shlah s O (4 n n log n ) algorihm? As an answr o his qusion, his papr prsns h firs polynomial-spac xac algorihm spcializd for h TSP in graphs wih dgr a mos igh. W dlop a s of branching ruls o aid h analysis of h branching algorihm, and w us h masur-and-conqur mhod o ffcily analyz our branching algorihm. W obain a running im of O ( n ), and his running im bound dos no gi an adanagous algorihm for h TSP in dgr-8 graphs or Gurich and Shlah s algorihm for h TSP in gnral, bu i gis a limi as o h applicabiliy of our choic of branching ruls and analysis mhod for dsigning a polynomial-spac xac algorihm for h TSP in graphs of limid dgr. Kywords: Traling Salsman Problm, Exac Exponnial Algorihm, Branch-and- Rduc, Masur-and-Conqur. 1 Inroducion Th Traling Salsman Problm, TSP, is on of h mos wll-known combinaorial opimizaion problms. In h muliud of insigad algorihms for h TSP, w confin our xposiion o hos who us polynomial xcuion spac. Gurich and Shlah [5] ha shown ha h TSP in a gnral n-rx graph is solabl in im O (4 n n log n ). This had rmaind h only rsul for narly wo dcads unil Eppsin [2] sard h xploraion ino polynomial-spac TSP algorihms spcializd for graphs of boundd dgr. From his iwpoin, l dgr-i graph sand for a graph in which ach rx has a mos i incidn. 1 Tchnical rpor , Spmbr 6,

2 Eppsin [2] dsignd an algorihm for dgr-3 graphs ha runs in O (1.260 n ) im. Iwama and Nakashima [6] ha claimd an impromn of Eppsin s im bound o O (1.251 n ) im for h TSP in dgr-3 graphs. Lar, Liskiwicz and Schusr [7] ha discord som orsighs mad in Iwama and Nakashima s analysis, and prod ha hir algorihm acually runs in O (1.257 n ) im. Liskiwicz and Schusr hn mad som minor modificaions of Eppsin s algorihm and showd ha his modifid algorihm runs in O ( n ) im. Rcnly, Xiao and Nagamochi [12] ha prsnd an O ( n )-im algorihm for h TSP in dgr-3 graphs, and his improd all prious im bounds for polynomial-spac algorihms. For h TSP in dgr-4 graphs, Eppsin [2] dsignd an algorihm ha runs in O (1.890 n ) im. Lar, Xiao and Nagamochi [13] showd an improd alu for h uppr bound of h running im and showd ha hir algorihm runs in O (1.692 n ) im. Currnly, his is h fass algorihm for h TSP in dgr-4 graphs. To h bs of our knowldg, prsnly h only insigaion on h TSP in graphs of dgr fi and up o sn has bn don by Md Yunos al. [8, 9, 10]. Md Yunos al. [8] ga an O ( n )-im algorihm for h TSP in dgr-5 graphs, followd by an O ( n )-im algorihm for h TSP in dgr-6 graphs [9], and an O ( n )-im algorihm for h TSP in dgr-7 graphs [10]. Abo all, hr xis no rpors in h liraur of xac algorihms spcializd o h TSP in graphs of dgr highr han sn. Furhrmor, h following qusion ariss; unil which alu i of a maximum dgr dos a spcializd polynomial-spac algorihm for dgri graphs ouprform Gurich and Shlah s O (4 n n log n )-im algorihm? Thrfor, in his papr, no only do w prsn h firs polynomial-spac branching algorihm for h TSP in dgr-8 graphs, bu also brach h im bound of O (4 n ). This rsul dos no gi an adanagous algorihm for h TSP in dgr-8 graphs or Gurich and Shlah, bu gis a limi as o h applicabiliy of our choic of branching ruls and analysis mhod for dsigning a polynomial-spac xac algorihm for h TSP in graphs of limid dgr. This mans ha in h qus of dsigning polynomial-spac xac algorihms for h TSP in graphs of limid dgr, possibly diffrn and improd branching ruls and analysis mhod should b sough for in ordr o achi br rsuls. 2 Prliminaris For a graph G, l V (G) dno h s of rics in G, and l E(G) dno h s of in G. A rx u is a nighbor of a rx if u and ar adjacn by an dg u. W dno h s of all nighbors of a rx by N(), and dno by d() h cardinaliy N() of N(), also calld h dgr of. For a subs of rics W V (G), l N(; W ) = N() W. For a subs of E E(G), l N E () = N() {u u E }, and l d E () = N E (). Analogously, l N E (; W ) = N E () W, and d E (, W ) = N E (, W ). Also, for a subs E of E(G), w dno by G E h graph (V, E \ E ) obaind from G by rmoing h in E. W mploy a known gnralizaion of h TSP proposd by Rubin [11], and namd h forcd Traling Salsman Problm by Eppsin [2]. W dfin an insanc I = (G, F ) ha consiss of a simpl, dg wighd, undircd graph G, and a subs F of in G, calld forcd. For briy, hroughou his papr l U dno E(G) \ F. A rx is calld forcd if xacly on of is incidn is forcd. Similarly, i is calld unforcd if no forcd dg is incidn o i. A Hamilonian cycl in G is calld a our if i passs hrough all h forcd 2

3 in F. Undr hs circumsancs, h forcd TSP rquss o find a minimum cos our of an insanc (G, F ). Throughou his papr, w assum ha h maximum dgr of a rx in G is a mos igh. W dno a forcd (rsp., unforcd) rx of dgr i as a yp fi rx (rsp., ui rx). W ar inrsd in 12 yps of rics in an insanc of (G, F ), namly, ui and fi for i = 3, 4,..., 8. As shall b sn in Subscion 3.1, forcd and unforcd rics of dgr wo and on ar rad as spcial cass. L V fi (rsp., V ui ), i = 3, 4,..., 8 dno h s of fi-rics (rsp., ui-rics) in (G, F ). 3 A Polynomial-Spac Branching Algorihm Our algorihm consiss of wo major sps which ar rpad iraily. In h firs sp, h algorihm applis rducion ruls unil no furhr rducion is possibl. In h scond sp, h algorihm applis branching ruls in a rducd insanc o sarch for a soluion. 3.1 Rducion Ruls Rducion is a procss of ransforming an insanc o a smallr insanc opimaliy. I aks polynomial im o gnra a soluion of an original insanc from a soluion o a smallr insanc obaind hrough rducion. If an insanc admis no our, w call i infasibl. Obsraion 1 gis wo sufficin condiions for an insanc o b infasibl as obsrd by Rubin [11]. Ths wo sufficin condiions will b chckd whn xcuing h rducion ruls. Obsraion 1 If on of h following condiions holds, hn h insanc (G, F ) is infasibl. (i) d() 1 for somrx V (G). (ii) d F () 3 for somrx V (G). An insanc (G, F ) is calld smi-fasibl if i dos no saisfy any of h condiions in Obsraion 1. If h insanc is smi-fasibl, hn h rducion ruls will b xcud. In his papr, w apply wo rducion ruls as sad in Md Yunos al. [8]. Th rducion ruls as sad in Obsraion 2 prsr h minimum cos our of an insanc, and hy ar applid in ach of h branching opraions. Obsraion 2 Each of h following rducions prsrs h fasibiliy and a minimum cos our of an insanc (G, F ). (i) If d() = 2 for a rx, hn add o F any unforcd dg incidn o hrx ; and (ii) If d() > 2 and d F () = 2 for a rx, hn rmo from G any unforcd dg incidn o rx. Our rducion algorihm is dscribd as Algorihm 1. An insanc (G, F ) is calld rducd if i dos no saisfy any of h condiions in Obsraion 1 and Obsraion 2. 3

4 3.2 Branching Ruls Our algorihm iraily branchs on an unforcd dg in a rducd insanc I = (G, F ) by ihr including ino F, forc(), or xcluding i from G, dl(). By applying a branching opraion, h algorihm gnras wo nw insancs, calld branchs. To dscrib our branching algorihm, l (G, F ) b a rducd insanc. Rcall ha w assum ha an inpu graph has dgr a mos igh. Du o our rducion and branching opraions, h dgr in sub-insanc will nr incras. In (G, F ), an unforcd dg = incidn o a rx of dgr igh is calld opimal, if i saisfis a condiion c-i wih minimum indx i, or all unforcd in (G, F ). W rfr o h following condiions for choosing an opimal dg o branch on, c-1 o c-29, as h branching ruls. Th s of branching ruls for condiions c-1 o c-18 is illusrad in Figur 1, and h s of branching ruls for condiions c-19 o c-29 is illusrad in Figur 2. Dails of our branching algorihm ar dscribd in Algorihm 2. For conninc of h analysis of h algorihm, cass c-5, c-8, c-11, c-14 and c-17 ha bn diidd ino sub-cass according o h cardinaliy of h nighborhood inrscion for rx of dgr igh and rx of dgr four, fi, six, sn and igh, rspcily. Vrx pairs wih inrscions of lowr cardinaliy ak prcdnc or highr ons. Branching Ruls (c-1) V f8 and N U (; V f3 ) such ha N U () N U () = ; (c-2) V f8 and N U (; V f3 ) such ha N U () N U () ; (c-3) V f8 and N U (; V u3 ); (c-4) V f8 and N U (; V f4 ) such ha N U () N U () = ; (c-5) V f8 and N U (; V f4 ) such ha N U () N U () ; (I) N U () N U () = 1; and (II) N U () N U () = 2; (c-6) V f8 and N U (; V u4 ); (c-7) V f8 and N U (; V f5 ) such ha N U () N U () = ; (c-8) V f8 and N U (; V f5 ) such ha N U () N U () ; (I) N U () N U () = 1; (II) N U () N U () = 2; and (III) N U () N U () = 3; (c-9) V f8 and N U (; V u5 ); (c-10) V f8 and N U (; V f6 ) such ha N U () N U () = ; (c-11) V f8 and N U (; V f6 ) such ha N U () N U () ; (I) N U () N U () = 1; (II) N U () N U () = 2; (III) N U () N U () = 3; and (IV) N U () N U () = 4; (c-12) V f8 and N U (; V u6 ); (c-13) V f8 and N U (; V f7 ) such ha N U () N U () = ; (c-14) V f8 and N U (; V f7 ) such ha N U () N U () ; (I) N U () N U () = 1; (II) N U () N U () = 2; (III) N U () N U () = 3; (IV) N U () N U () = 4; and (V) N U () N U () = 5; (c-15) V f8 and N U (; V u7 ); (c-16) V f8 and N U (; V f8 ) such ha N U () N U () = ; (c-17) V f8 and N U (; V f8 ) such ha N U () N U () ; (I) N U () N U () = 1; (II) N U () N U () = 2; (III) N U () N U () = 3; (IV) N U () N U () = 4; (V) N U () N U () = 5; and (VI) N U () N U () = 6; (c-18) V f8 and N U (; V u8 ); (c-19) V u8 and N U (; V f3 ); (c-20) V u8 and N U (; V u3 ); (c-21) V u8 and N U (; V f4 ); (c-22) V u8 and N U (; V u4 ); (c-23) V u8 and N U (; V f5 ); (c-24) V u8 and N U (; V u5 ); (c-25) V u8 and N U (; V f6 ). (c-26) V u8 and N U (; V u6 ); (c-27) V u7 and N U (; V f7 ); (c-28) V u8 and N U (; V u7 ); and (c-29) V u8 and N U (; V u8 ). 4

5 Algorihm 1 Rd(G, F ) Inpu: An insanc (G, F ). Oupu: A rducd insanc (G, F ) of (G, F ); or a mssag for h infasibiliy of (G, F ), which aluas o. 1: Iniializ (G, F ) := (G, F ); 2: whil (G, F ) is no a rducd insanc do 3: if hr is a rx in (G, F ) such ha d() 1 or d F () 3 hn 4: rurn mssag Infasibl 5: ls if hr is a rx in (G, F ) such ha 2 = d() > d F () hn 6: L E b h s of unforcd incidn o all such rics; 7: s F := F E 8: ls if hr is a rx in (G, F ) such ha d() > d F () = 2 hn 9: L E b h s of unforcd incidn o all such rics; 10: s G := G E 11: nd if 12: nd whil; 13: rurn (G, F ). Algorihm 2 sp8(g, F ) Inpu: An insanc (G, F ) such ha h maximum dgr of G is a mos 8. Oupu: Th minimum cos of a our of (G, F ); or a mssag for h infasibiliy of (G, F ), which aluas o. 1: Run Rd(G, F ); 2: if Rd(G, F ) rurns mssag Infasibl hn 3: rurn mssag Infasibl 4: ls 5: L (G, F ) := Rd(G, F ); 6: if V u8 V f8 hn 7: Choos an opimal unforcd dg ; 8: if boh sp8(g, F {}) and sp8(g {}, F ) rurn mssag Infasibl hn 9: rurn mssag Infasibl 10: ls 11: rurn min{sp8(g, F {}), sp8(g {}, F )} 12: nd if 13: ls /* h maximum dgr of any rx in (G, F ) is a mos */ 14: rurn sp7(g, F ) 15: nd if 16: nd if. No: Th inpu and oupu of algorihm sp7(g, F ) ar as follows: Inpu: An insanc (G, F ) such ha h maximum dgr of G is a mos. Oupu: Th minimum cos of a our of (G, F ); or a mssag for h infasibiliy of (G, F ), which aluas o. 4 Analysis 4.1 Analysis Framwork To ffcily analyz h running im of our branching algorihm, w us h masur-andconqur mhod as inroducd by Fomin al. [3]. Gin an insanc I = (G, F ) of h forcd TSP, w assign a nonngai wigh ω() o ach rx V (G) according o is yp. To his ffc, w s a non-ngairx wigh funcion ω : V R + in h graph G, and w 5

6 c-1 c-2 c-3 c c-5(i) c-5(ii) c-6 c c-8(i) c-8(ii) c-8(iii) c-9 c-10 c-11(i) c-11(ii) c-11(iii) c-11(iv) c-12 c c-14(i) c-14(ii) c-14(iii) c-14(iv) c-14(v) c-15 c c-17(i) c-17(ii) c-17(iii) c-17(iv) c-17(iv) : unforcd : forcd c-17(vi) c-18 Figur 1: Illusraion of h branching ruls c-1 o c-18. us h sum of wighs of all rics in h graph as h masur µ(i) of insanc I, ha is, µ(i) = ω(). (1) V (G) I is imporan for h analysis o find a masur which saisfis h following propris (i) µ(i) = 0 if and only if I can b sold in polynomial im; 6

7 c-19 c-20 c-21 c c-23 c c-25 c c-27 c-28 c-29 : unforcd : forcd Figur 2: Illusraion of h branching ruls c-19 o c-29. (ii) If I is a sub-insanc of I obaind hrough a rducion or a branching opraion, hn µ(i ) µ(i). W call a masur µ saisfying condiions (i) and (ii) abo a propr masur. W prform h im analysis of h branching algorihm ia approprialy consrucd rcurrncs or h masur µ = µ(i) of an insanc I = (G, F ), for ach branching rul of h algorihm. L T (µ) dno h numbr of nods in h sarch r gnrad by our algorihm whn inokd on h insanc I wih masur µ. L I and I b insancs obaind from I by a branching opraion, and l a µ(i) µ(i ) and b µ(i) µ(i ) b lowr bounds on h amouns of dcras in h masur. W call (a, b) h branching cor of h branching opraion, and his implis h linar rcurrnc T (µ) T (µ a) + T (µ b). (2) To alua h prformanc of his branching cor, w can us any sandard mhod for linar rcurrnc rlaions. In fac, i is known ha T (µ) is of h form O (τ µ ), whr τ is h uniqu posii ral roo of h funcion f(x) = 1 ( x a + x b). Thalu τ is calld h branching facor of h branching cor (a, b). Th running im of h algorihm is drmind by considring h wors branching facor or all branching cors gnrad by h branching ruls. For furhr dails jusifying his approach, as wll as a solid inroducion o branching algorihms, h radr is rfrrd o h book of Fomin and Krasch [4]. 4.2 Wigh Consrains In ordr o obain a masur which will naurally gi a running im bound as a funcion of h siz of a TSP insanc, w rquir ha h wigh of ach rx o b no grar han on. In wha follows, w xamin som ncssary consrains which hrx wighs should saisfy in ordr for us o obain a propr masur. For ach i = 3, 4,..., 8, w dno by w i h wigh of a ui-rx, and by w i h wigh of an fi-rx. Th condiions for a propr masur rquir ha h masur of an insanc 7

8 obaind hrough a branching or a rducion opraion will no b grar han h masur of h original insanc. Thus, hrx wighs should saisfy h following rlaions: w 8 1, (3) w i w i, 3 i 8 (4) w i w j, 3 i < j 8, and (5) w i w j, 3 i < j 8. (6) Thrx wigh for rics of dgr lss han hr is s o b zro. Lmma 1 sas ha gin Algorihms 1 and 2, sing rx wighs which saisfy h condiions of Eqs. (4) o (6) is sufficin o obain a propr masur. W can pro Lmma 1 in a similar way as Lmma 3 by Md Yunos al. [8, Lmma 3]. Lmma 1 If h wighs of rics ar chosn as in Eqs. (4) o (6), hn h masur µ(i) nr incrass as a rsul of h rducion or h branching opraions of Algorihm 1 and Algorihm 2. To simplify som argumns and h lis of h branching cors w ar abou o dri, w inroduc h following noaion: furhr, i = w i w i, 3 i 8 i,j = w i w j, 3 j < i 8, and i,j = w i w j, 3 j < i 8 m 1 = min{w 3, w 3, 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7 }, (7) m 2 = min{w 3, 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7 }, (8) m 3 = min{w 3, 3, w 4, 4, w 5, 5, w 6, 6, w 7, 7, w 8, 8 }, (9) m 4 = min{ 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7 }, (10) m 5 = min{w 4, w 4, 5,3, 5,3, 6,4, 6,4, 7,5, 7,5, 8,6, 8,6 }, (11) m 6 = min{ 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7 }, (12) m 7 = min{ 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7 }, (13) m 8 = min{ 5,3, 5,3, 6,4, 6,4, 7,5, 7,5, 8,6, 8,6 }, (14) m 9 = min{ 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7 }, (15) m 10 = min{ 6,5, 6,5, 7,6, 7,6, 8,7, 8,7 }, (16) m 11 = min{ 6,4, 6,4, 7,5, 7,5, 8,6, 8,6 }, (17) m 12 = min{ 6,5, 7,6, 7,6, 8,7, 8,7 }, (18) m 13 = min{ 7,6, 7,6, 8,7, 8,7 }, (19) m 14 = min{ 7,5, 7,5, 8,6, 8,6 }, (20) m 15 = min{ 7,6, 8,7, 8,7 }, (21) m 16 = min{ 8,7, 8,7 }, (22) m 17 = min{ 8,6, 8,6 }, (23) m 18 = min{w 3, 3, w 4, 4, w 5, 5, w 6, 6, w 7, 7, 8 }, (24) m 19 = min{w 3, w 3, 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7 }. (25) 8

9 4.3 Main Rsul L a rx wigh funcion ω() b chosn as follows: w 8 = 1 for a u8-rx w 8 = w 7 = w 7 = w 6 = w 6 = ω() = w 5 = w 5 = w 4 = w 4 = w 3 = for an f8-rx for a u7-rx for an f7-rx for a u6-rx for an f6-rx for a u5-rx for an f5-rx for a u4-rx for an f4-rx for a u3-rx w 3 = for an f3-rx 0 ohrwis (26) Thrx wigh funcion ω() gin in Eq. (26) is obaind as a soluion o a quasiconx program, according o h mhod inroducd by Eppsin [1]. All h branching cors ar in fac consrains in h quasiconx program. Lmma 2 If hrx wigh funcion ω() is s as in Eq. (26), hn ach of h branching opraions in Algorihm 2 has a branching facor no grar han A proof of Lmma 2 can b drid analyically by analyzing h branching cors which rsul by applying h branching and rducion opraions. From Lmma 2, w g our main rsul as sad in Thorm 1. Thorm 1 Th TSP in an n-rx graph G wih maximum dgr igh can b sold in O ( n ) im and polynomial spac. In h rmaindr of h analysis, for an opimal dg =, w rfr o N U () by {, 2,..., a }, a = d U (), and o N U ( ) \ {} by { a+1, a+2,..., a+b }, b = d U ( ) 1. W assum wihou loss of gnraliy ha +i = a+i for i = 1, 2,..., c, whr c = N U () N U ( ) is h numbr of common nighbors of and. If hr xiss an f3-rx a+i in N U ( ) \ {}, l x N U ( a+i ) \ {, }. W s ha h choic of rx x is uniqu, bcaus a+i is of yp f3 and N U ( a+i ) \ {, } = 1. This rx x will plays a ky rol in our analysis, as shown in Fig Branching on Edgs around f8-rics (c-1 o c-18) This subscion will show how w dri h branching cors for h branching opraions on an opimal dg =, incidn o a forcd rx of dgr igh, disinguishing h 18 cass for condiions c-1 o c-18. W analyz h branching cors in a similar mannr wih h analysis of h algorihm for h TSP in dgr-5 graphs by Md Yunos al. [8]. Cas c-1. Thr xis rics V f8 and N U (; V f3 ) such ha N U () N U ( ) = (s Figur 4): W branch on h dg. No ha N U ( ) \ {} = { 8 }. 9

10 a+i x (a) forc() : forcd by rducion ruls : nwly dld by rducion ruls x a+i (b) dl() by h branching opraion : nwly dld by h branching opraion Figur 3: Illusraion of (a) nwly forcd and (b) dld dg by a branching opraion and rducion ruls for an f3 rx a+i (a) forc( ) in c-1 (b) dl( ) in c-1 : unforcd : nwly dld : forcd Figur 4: Illusraion of branching rul c-1, whrrx V f8 and N U (; V f3 ). In h branch of forc( ), h dg will b addd o F by h branching opraion, and 2, 3, 4, 5,,, and 8 will b dld from G by h rducion ruls. Boh and will bcomrics of dgr wo. From Eq. (26), h wigh of rics of dgr wo is zro. So h wigh of rx dcrass by w 8 and h wigh of rx dcrass by w 3. Each of hrics 2, 3, 4, 5, and can b any of h possiblrx yps f3, u3, f4, u4, f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 1 = min{w 3, w 3, 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7 }. In h branch of dl( ), h dg will b dld from G by h branching opraion, and h dg 8 will b addd o F by h rducion ruls. Th wigh of rx dcrass by 8,7 and h wigh of rx dcrass by w 3. Thr ar wo cass for rx 8 ; 1) rx 8 is of yp f3, and 2) ohrwis. W will analyz hs wo cass sparaly for ach of branchs forc( ) and dl( ). Firs, w will analyz h cas whrrx 8 is an f3-rx (s Figur 3). Rcall ha in his cas, w dno by x h uniqurx in N U ( 8 ) \ { }. In h branch of forc( ), dg x 8 will b addd o F by h rducion ruls. Hnc h wigh of rx 8 dcrass by w 3. If rx x is an f3-rx (rsp., u3, f4, u4, f5, u5, f6, u6, f7, u7, f8, or a u8-rx), hn h wigh dcras α 1 of rx x will b w 3 (rsp., 3, w 4, 4, w 5, 5, w 6, 6, w 7, 7, w 8, and 8). Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 + w 3 + w 3 + 6m 1 + α 1. In h branch of dl( ), dg x 8 will b dld from G b h rducion ruls. Hnc h wigh of rx 8 dcrass by w 3. If rx x is an f3-rx (rsp., u3, f4, u4, 10

11 f5, u5, f6, u6, f7, u7, f8, or a u8-rx), hn h wigh dcras β 1 of rx x will b w 3 (rsp., w 3, 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, and 8,7). Thus h oal wigh dcras for his cas in h branch of dl( ) is a las w 8 w 7 + w 3 + w 3 + β 1. As a rsul, for h ordrd pair (α 1, β 1 ) aking alus in {(w 3, w 3 ), ( 3, w 3 ), (w 4, 4,3 ), ( 4, 4,3 ), (w 5, 5,4 ), ( 5, 5,4 ), (w 6, 6,5 ), ( 6, 6,5 ), (w 7, 7,6 ), ( 7, 7,6 ), (w 8, 8,7 ), ( 8, 8,7 )}, w g h following 12 branching cors: (w 8 + 2w 3 + 6m 1 + α 1, w 8 w 7 + 2w 3 + β 1 ). (27) Nx, w xamin h cas whrrx 8 is no an f3-rx. In h branch of forc( ), if rx 8 is a u3-rx (rsp., f4, u4, f5, u5, f6, u6, f7, u7, f8, or a u8-rx), hn h wigh dcras α 2 of rx 8 will b w 3 (rsp., 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, and 8,7). Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 + w 3 + 6m 1 + α 2. In h branch of dl( ), if rx 8 is a u3-rx (rsp., f4, u4, f5, u5, f6, u6, f7, u7, f8, or a u8-rx), hn h wigh dcras β 2 of rx 8 will b 3 (rsp., w 4, 4, w 5, 5, w 6, 6, w 7, 7, w 8, and 8). Thus h oal wigh dcras for his cas in h branch of dl( ) is a las w 8 w 7 + w 3 + β 2. As a rsul, for h ordrd pair (α 2, β 2 ) aking alus in {(w 3, 3 ), ( 4,3, w 4 ), ( 4,3, 4 ), ( 5,4, w 5 ), ( 5,4, 5 ), ( 6,5, w 6 ), ( 6,5, 6 ), ( 7,6, w 7 ), ( 7,6, 7 ), ( 8,7, w 8 ), ( 8,7, 8 )}, w g h following 11 branching cors: (w 8 + w 3 + 6m 1 + α 2, w 8 w 7 + w 3 + β 2 ). (28) Cas c-2. Cas c-1 is no applicabl, and hr xis rics V f8 and N U (; V f3 ) such ha N U () N U ( ) : Wihou loss of gnraliy, assum ha N U () N U ( ) = { 2 } (s Figur 5). W branch on h dg (a) forc( ) in c (b) dl( ) in c-2 : unforcd : nwly dld : forcd Figur 5: Illusraion of branching rul c-2, whrrx V f8 and N U (; V f3 ). In h branch of forc( ), h dg will b addd o F by h branching opraion, and 2, 3, 4, 5,, and 2 will b dld from G by h rducion ruls. So h wigh of rx dcrass by w 8, and h wigh of rx dcrass by w 3. Each of hrics 3, 4, 5, and can b any of h possiblrx yps f3, u3, f4, u4, f5, u5, f6, u6, f7, u7, f8 and a u8-rx, and ach of hir wighs dcrass by a las m 1 = min{w 3, w 3, 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. If rx 2 is an f3 or a u3-rx, afr applying h branching opraion, 2 would bcom a rx of dgr on. From Obsraion 1, cas (i), his is infasibl, and h algorihm will rurn a mssag of infasibiliy and rmina. Ohrwis, if rx 2 is an f4-rx (rsp., u4, f5, u5, f6, u6, f7, u7, f8, or a u8-rx), hn h wigh dcras α 3 of 11

12 rx 2 will b w 4 (rsp., w 4, 5,3, 5,3, 6,4, 6,4, 7,5, 7,5, 8,6, and 8,6). Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 + w 3 + 5m 1 + α 3. In h branch of dl( ), h dg will b dld from G by h branching opraion, and h dg 2 will b addd o F by h rducion ruls. So h wighs of rics and dcras by 8,7 and w 3, rspcily. If rx 2 is an f4-rx (rsp., u4, f5, u5, f6, u6, f7, u7, f8, or a u8-rx), hn h wigh dcras β 3 of rx 2 will b w 4 (rsp., 4, w 5, 5, w 6, 6, w 7, 7, w 8, and 8). Thus h oal wigh dcras for his cas in h branch of dl( ) is a las w 8 w 7 + w 3 + β 3. As a rsul, for h ordrd pair (α 3, β 3 ) aking alus in {(w 4, w 4 ), (w 4, 4 ), ( 5,3, w 5 ), ( 5,3, 5 ), ( 6,4, w 6 ), ( 6,4, 6 ), ( 7,5, w 7 ), ( 7,5, 7 ), ( 8,6, w 8 ), ( 8,6, 8 )}, w g h following 10 branching cors: (w 8 + w 3 + 5m 1 + α 3, w 8 w 7 + w 3 + β 3 ). (29) Cas c-3. Cas c-1 and cas c-2 ar no applicabl, and hr xis rics V f8 and N U (; V u3 ) (s Figur 6): W branch on h dg. No ha N U ( )\{} = { 8, 9 } (a) forc( ) in c (b) dl( ) in c-3 : unforcd : nwly dld : forcd Figur 6: Illusraion of branching rul c-3, whrrx V f8 and N U (; V u3 ). In h branch of forc( ), h dg will b addd o F by h branching opraion, and 2, 3, 4, 5, and will b dld from G by h rducion ruls. So h wigh of rx dcrass by w 8, and h wigh of rx dcrass by 3. Non of hrics 2, 3, 4, 5, and can b an f3-rx bcaus i would ha bn chosn as an opimal dg in som prious cas. Hnc, ach of hrics 2, 3, 4, 5, and can only b on of yps u3, f4, u4, f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 2 = min{w 3, 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 + w 3 w 3 + 6m 2. In h branch of dl( ), h dg will b dld from G by h branching opraion, and 8 and 9 will b addd o F by h rducion ruls. So h wigh of rx dcrass by 8,7, and h wigh of rx dcrass by w 3. Each of rics 8 and 9 can b any of h possiblrx yps f3, u3, f4, u4, f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 3 = min{w 3, 3, w 4, 4, w 5, 5, w 6, 6, w 7, 7, w 8, 8}. Thus h oal wigh dcras for his cas in h branch of dl( ) is a las w 8 w 7 + w 3 + 2m 3. As a rsul, w g h following branching cor: (w 8 + w 3 w 3 + 6m 2, w 8 w 7 + w 3 + 2m 3 ). (30) 12

13 Cas c-4. Non of h prious cass ar applicabl, and hr xis rics V f8 and N U (; V f4 ) (s Figur 7): W branch on h dg. No ha N U ( ) \ {} = { 8, 9 } (a) forc( ) in c (b) dl( ) in c-4 : unforcd : nwly dld : forcd Figur 7: Illusraion of branching rul c-4, whrrx V f8 and N U (; V f4 ). In h branch of forc( ), h dg will b addd o F by h branching opraion, and 2, 3, 4, 5,,, 8, and 9 will b dld from G by h rducion ruls. So h wigh of rx dcrass by w 8, and h wigh of rx dcrass by w 4. Each of hrics 2, 3, 4, 5, and can only b on of yps f4, u4, f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 4 = min{ 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Each of hrics 8 and 9 can b any of h possiblrx yps f3, u3, f4, u4, f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 1 = min{w 3, w 3, 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 + w 4 + 6m 4 + 2m 1. In h branch of dl( ), h dg will b dld from G by h branching opraion. So h wigh of rx dcrass by 8,7, and h wigh of rx dcrass by 4,3. Thus h oal wigh dcras for his cas in h branch of dl() is a las w 8 w 7 + w 4 w 3. As a rsul, w g h following branching cor: (w 8 + w 4 + 6m 4 + 2m 1, w 8 w 7 + w 4 w 3). (31) Cas c-5. Non of h prious cass ar applicabl, and hr xis rics V f8 and N U (; V f4 ) such ha N U () N U ( ) : W disinguish wo sub-cass, according o h cardinaliy of h inrscion N U () N U ( ), (c-5(i)) N U () N U ( ) = 1; and (c-5(ii)) N U () N U ( ) = 2. Cas c-5(i). Wihou loss of gnraliy, assum ha N U () N U ( ) = { 2 } (s Figur 8): W branch on h dg. No ha N U ( ) \ {} = { 8 }. In h branch of forc( ), h dg will b addd o F by h branching opraion, and 2, 3, 4, 5,,, 2 and 8 will b dld from G by h rducion ruls. So h wigh of rx dcrass by w 8, and h wigh of rx dcrass by w 4. Vrx 2 can only b on of yps f4, u4, f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and is wigh dcrass by a las m 5 = min{w 4, w 4, 5,3, 5,3, 6,4, 6,4, 7,5, 7,5, 8,6, 8,6}. Each of hrics 3, 4, 5,, and can only b on of yps f4, u4, f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 4 = min{ 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Vrx 8 can b any of h possiblrx yps f3, u3, f4, u4, f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and is wigh dcrass by a las 13

14 (a) forc( ) in c-5(i) (b) dl( ) in c-5(i) : unforcd : nwly dld : forcd Figur 8: Illusraion of branching rul c-5(i), whrrx V f8 and N U (; V f4 ). m 1 = min{w 3, w 3, 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 + w 4 + m 5 + 5m 4 + m 1. In h branch of dl( ), h dg will b dld from G by h branching opraion. So h wigh of rx dcrass by 8,7, and h wigh of rx dcrass by 4,3. Thus h oal wigh dcras for his cas in h branch of dl() is a las w 8 w 7 + w 4 w 3. As a rsul, w g h following branching cor: (w 8 + w 4 + m 5 + 5m 4 + m 1, w 8 w 7 + w 4 w 3). (32) Cas c-5(ii). Wihou loss of gnraliy, assum ha N U () N U ( ) = { 2, 3 } (s Figur 9): W branch on h dg (a) forc( ) in c-5(ii) (b) dl( ) in c-5(ii) : unforcd : nwly dld : forcd Figur 9: Illusraion of branching rul c-5(ii), whrrx V f8 and N U (; V f4 ). In h branch of forc( ), h dg will b addd o F by h branching opraion, and 2, 3, 4, 5,,, 2 and 3 will b dld from G by h rducion ruls. So h wigh of rx dcrass by w 8, and h wigh of rx dcrass by w 4. Each of hrics 2 and 3 can only b on of yps f4, u4, f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 5 = min{w 4, w 4, 5,3, 5,3, 6,4, 6,4, 7,5, 7,5, 8,6, 8,6}. Each of hrics 4, 5,, and can only b on of yps f4, u4, f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 4 = min{ 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 + w 4 + 2m 5 + 4m 4. In h branch of dl( ), h dg will b dld from G by h branching opraion. So h wigh of rx dcrass by 8,7, and h wigh of rx dcrass by 4,3. Thus h oal wigh dcras for his cas in h branch of dl() is a las w 8 w 7 + w 4 w 3. 14

15 As a rsul, w g h following branching cor: (w 8 + w 4 + 2m 5 + 4m 4, w 8 w 7 + w 4 w 3). (33) Cas c-6. Non of h prious cass ar applicabl, and hr xis rics V f8 and N U (; V u4 ) (s Figur 10): W branch on h dg (a) forc( ) in c (b) dl( ) in c-6 : unforcd : nwly dld : forcd Figur 10: Illusraion of branching rul c-6, whrrx V f8 and N U (; V u4 ). In h branch of forc( ), h dg will b addd o F by h branching opraion, and 2, 3, 4, 5, and will b dld from G by h rducion ruls. So h wigh of rx dcrass by w 8, and h wigh of rx dcrass by 4. Each of hrics 2, 3, 4, 5, and can only b on of yps u4, f5, u5, f6, u6, f7, u7 f8, and a u8-rx, and ach of hir wighs dcrass by a las m 6 = min{ 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 + w 4 w 4 + 6m 6. In h branch of dl( ), h dg will b dld from G by h branching opraion. So h wigh of rx dcrass by 8,7, and h wigh of rx dcrass by 4,3. Thus h oal wigh dcras for his cas in h branch of dl( ) is a las w 8 w 7 + w 4 w 3. As a rsul, w g h following branching cor: (w 8 + w 4 w 4 + 6m 6, w 8 w 7 + w 4 w 3 ). (34) Cas c-7. Non of h prious cass ar applicabl, and hr xis rics V f8 and N U (; V f5 ) (s Figur 11): W branch on h dg. No ha N U ( )\{} = { 8, 9, 0 } (a) forc( ) in c (b) dl( ) in c-7 : unforcd : nwly dld : forcd Figur 11: Illusraion of branching rul c-7, whrrx V f8 and N U (; V f5 ). In h branch of forc( ), h dg will b addd o F by h branching opraion, and 2, 3, 4, 5,,, 8, 9, and 0 will b dld from G by h 15

16 rducion ruls. So h wigh of rx dcrass by w 8, and h wigh of rx dcrass by w 5. Each of hrics 2, 3, 4, 5, and can only b on of yps f5, u5, f6, u6, f7, u7 f8, and a u8-rx, and ach of hir wighs dcrass by a las m 7 = min{ 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Each of hrics 8, 9 and 0 can b any of h yps f3, u3, f4, u4, f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 1 = min{w 3, w 3, 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 + w 5 + 6m 7 + 3m 1. In h branch of dl( ), h dg will b dld from G by h branching opraion. So h wigh of rx dcrass by 8,7, and h wigh of rx dcrass by 5,4. Thus h oal wigh dcras for his cas in h branch of dl() is a las w 8 w 7 + w 5 w 4. As a rsul, w g h following branching cor: (w 8 + w 5 + 6m 7 + 3m 1, w 8 w 7 + w 5 w 4). (35) Cas c-8. Non of h prious cass ar applicabl, and hr xis rics V f8 and N U (; V f5 ) such ha N U () N U ( ) : W disinguish hr sub-cass, according o h cardinaliy of h inrscion N U () N U ( ), (c-8(i)) N U () N U ( ) = 1; (c-8(ii)) N U () N U ( ) = 2; and (c-8(iii)) N U () N U ( ) = 3. Cas c-8(i). Wihou loss of gnraliy, assum ha N U () N U ( ) = { 2 } (s Figur 12): W branch on h dg. No ha N U ( ) \ {} = { 8, 9 } (a) forc( ) in c-8(i) (b) dl(1) in c-8(i) : unforcd : nwly dld : forcd Figur 12: Illusraion of branching rul c-8(i), whrrx V f8 and N U (; V f5 ). In h branch of forc( ), h dg will b addd o F by h branching opraion, and 2, 3, 4, 5,,, 2, 8 and 9 will b dld from G by h rducion ruls. So h wigh of rx dcrass by w 8, and h wigh of rx dcrass by w 5. Vrx 2 can only b on of yps f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and is wigh dcrass by a las m 8 = min{ 5,3, 5,3, 6,4, 6,4, 7,5, 7,5, 8,6, 8,6}. Each of h rics 3, 4, 5, and can only b on of yps f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 7 = min{ 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Each of hrics 8 and 9 can b any of h possiblrx yps f3, u3, f4, u4, f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 1 = min{w 3, w 3, 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 +w 5 +m 8 +5m 7 +2m 1. 16

17 In h branch of dl( ), h dg will b dld from G by h branching opraion. So h wigh of rx dcrass by 8,7, and h wigh of rx dcrass by 5,4. Thus h oal wigh dcras for his cas in h branch of dl() is a las w 8 w 7 + w 5 w 4. As a rsul, w g h following branching cor: (w 8 + w 5 + m 8 + 5m 7 + 2m 1, w 8 w 7 + w 5 w 4). (36) Cas c-8(ii). Wihou loss of gnraliy, assum ha N U () N U ( ) = { 2, 3 } (s Figur 13): W branch on h dg. No ha N U ( ) \ {} = { 8 } (a) forc( ) in c-8(ii) (b) dl( ) in c-8(ii) : unforcd : nwly dld : forcd Figur 13: Illusraion of branching rul c-8(ii), whrrx V f8 and N U (; V f5 ). In h branch of forc( ), h dg will b addd o F by h branching opraion, and 2, 3, 4, 5,,, 2, 3 and 8 will b dld from G by h rducion ruls. So h wigh of rx dcrass by w 8, and h wigh of rx dcrass by w 5. Each of hrics 2 and 3 can only b on of yps f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 8 = min{ 5,3, 5,3, 6,4, 6,4, 7,5, 7,5, 8,6, 8,6}. Each of hrics 4, 5, and can only b on of yps f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 7 = min{ 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Vrx 8 can b any of h possiblrx yps f3, u3, f4, u4, f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and is wigh dcrass by a las m 1 = min{w 3, w 3, 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 + w 5 + 2m 8 + 4m 7 + m 1. In h branch of dl( ), h dg will b dld from G by h branching opraion. So h wigh of rx dcrass by 8,7, and h wigh of rx dcrass by 5,4. Thus h oal wigh dcras for his cas in h branch of dl() is a las w 8 w 7 + w 5 w 4. As a rsul, w g h following branching cor: (w 8 + w 5 + 2m 8 + 4m 7 + m 1, w 8 w 7 + w 5 w 4). (37) Cas c-8(iii). Wihou loss of gnraliy, assum ha N U () N U ( ) = { 2, 3, 4 } (s Figur 14): W branch on h dg. In h branch of forc( ), h dg will b addd o F by h branching opraion, and 2, 3, 4, 5,,, 2, 3 and 4 will b dld from G by h rducion ruls. So h wigh of rx dcrass by w 8, and h wigh of rx dcrass by w 5. Each of hrics 2, 3 and 4 can only b on of yps f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 8 = min{ 5,3, 5,3, 6,4, 6,4, 7,5, 7,5, 8,6, 8,6}. Each of hrics 5, and can only b on of yps f5, u5, f6, 17

18 (a) forc( ) in c-8(iii) (b) dl( ) in c-8(iii) : unforcd : nwly dld : forcd Figur 14: Illusraion of branching rul c-8(iii), whrrx V f8 and N U (; V f5 ). u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 7 = min{ 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 + w 5 + 3m 8 + 3m 7. In h branch of dl( ), h dg will b dld from G by h branching opraion. So h wigh of rx dcrass by 8,7, and h wigh of rx dcrass by 5,4. Thus h oal wigh dcras for his cas in h branch of dl() is a las w 8 w 7 + w 5 w 4. As a rsul, w g h following branching cor: (w 8 + w 5 + 3m 8 + 3m 7, w 8 w 7 + w 5 w 4). (38) Cas c-9. Non of h prious cass ar applicabl, and hr xis rics V f8 and N U (; V u5 ) (s Figur 15): W branch on h dg (a) forc( ) in c (b) dl( ) in c-9 : unforcd : nwly dld : forcd Figur 15: Illusraion of branching rul c-9, whrrx V f8 and N U (; V u5 ). In h branch of forc( ), h dg will b addd o F by h branching opraion, and 2, 3, 4, 5, and will b dld from G by h rducion ruls. So h wigh of rx dcrass by w 8, and h wigh of rx dcrass by 5. Each of hrics 2, 3, 4, 5, and can only b on of yps u5, f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 9 = min{ 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 + w 5 w 5 + 6m 9. In h branch of dl( ), h dg will b dld from G by h branching opraion. So h wigh of rx dcrass by 8,7, and h wigh of rx dcrass by 5,4. Thus h oal wigh dcras for his cas in h branch of dl( ) is a las w 8 w 7 + w 5 w 4. 18

19 As a rsul, w g h following branching cor: (w 8 + w 5 w 5 + 6m 9, w 8 w 7 + w 5 w 4 ). (39) Cas c-10. Non of h prious cass ar applicabl, and hr xis rics V f8 and N U (; V f6 ) (s Figur 16): W branch on h dg. No ha N U ( ) \ {} = { 8, 9, 0, 1 } (a) forc( ) in c (b) dl( ) in c-10 : unforcd : nwly dld : forcd Figur 16: Illusraion of branching rul c-10, whrrx V f8 and N U (; V f6 ). In h branch of forc( ), h dg will b addd o F by h branching opraion, and 2, 3, 4, 5,,, 8, 9, 0 and 1 will b dld from G by h rducion ruls. So h wigh of rx dcrass by w 8, and h wigh of rx dcrass by w 6. Each of hrics 2, 3, 4, 5, and can only b on of yps f6, u6, f7, u7, f8 and a u8-rx, and ach of hir wighs dcrass by a las m 10 = min{ 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Each of hrics 8, 9, 0 and 1 can b any of h possibl rx yps f3, u3, f4, u4, f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 1 = min{w 3, w 3, 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7 }. Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 + w 6 + 6m m 1. In h branch of dl( ), h dg will b dld from G by h branching opraion. So h wigh of rx dcrass by 8,7, and h wigh of rx dcrass by 6,5. Thus h oal wigh dcras for his cas in h branch of dl() is a las w 8 w 7 + w 6 w 5. As a rsul, w g h following branching cor: (w 8 + w 6 + 6m m 1, w 8 w 7 + w 6 w 5). (40) Cas c-11. Non of h prious cass ar applicabl, and hr xis rics V f8 and N U (; V f6 ) such ha N U () N U ( ) : W disinguish four sub-cass, according o h cardinaliy of h inrscion N U () N U ( ), (c-11(i)) N U () N U ( ) = 1; (c-11(ii)) N U () N U ( ) = 2; (c-11(iii)) N U () N U ( ) = 3; and (c-11(iv)) N U () N U ( ) = 4. Cas c-11(i). Wihou loss of gnraliy, assum ha N U () N U ( ) = { 2 } (s Figur 17): W branch on h dg. No ha N U ( ) \ {} = { 8, 9, 0 }. In h branch of forc( ), h dg will b addd o F by h branching opraion, and 2, 3, 4, 5,,, 2, 8, 9, 0 will b dld from G by h 19

20 (a) forc( ) in c-11(i) (b) dl(1) in c-11(i) : unforcd : nwly dld : forcd Figur 17: Illusraion of branching rul c-11(i), whrrx V f8 and N U (; V f6 ). rducion ruls. So h wigh of rx dcrass by w 8, and h wigh of rx dcrass by w 6. Vrx 2 can only b on of yps f6, u6, f7, u7, f8, and a u8-rx, and is wigh dcrass by a las m 11 = min{ 6,4, 6,4, 7,5, 7,5, 8,6, 8,6}. Each of h rics 3, 4, 5, and can only b on of yps f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 10 = min{ 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Each of hrics 8, 9 and 0 can b any of h possiblrx yps f3, u3, f4, u4, f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 1 = min{w 3, w 3, 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 + w 6 + m m m 1. In h branch of dl( ), h dg will b dld from G by h branching opraion. So h wigh of rx dcrass by 8,7, and h wigh of rx dcrass by 6,5. Thus h oal wigh dcras for his cas in h branch of dl() is a las w 8 w 7 + w 6 w 5. As a rsul, w g h following branching cor: (w 8 + w 6 + m m m 1, w 8 w 7 + w 6 w 5). (41) Cas c-11(ii). Wihou loss of gnraliy, assum ha N U () N U ( ) = { 2, 3 } (s Figur 18): W branch on h dg. No ha N U ( ) \ {} = { 8, 9 } (a) forc( ) in c-11(ii) (b) dl( ) in c-11(ii) : unforcd : nwly dld : forcd Figur 18: Illusraion of branching rul c-11(ii), whrrx V f8 and N U (; V f6 ). In h branch of forc( ), h dg will b addd o F by h branching opraion, and 2, 3, 4, 5,,, 2, 3, 8 and 9 will b dld from G by h rducion ruls. So h wigh of rx dcrass by w 8, and h wigh of rx dcrass by w 6. Each of hrics 2 and 3 can only b on of yps f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 11 = min{ 6,4, 6,4, 7,5, 7,5, 8,6, 8,6}. Each of hrics 4, 5, and can only b on of yps f6, u6, f7, 20

21 u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 10 = min{ 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Each of hrics 8 and 9 can b any of h possiblrx yps f3, u3, f4, u4, f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 1 = min{w 3, w 3, 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7 }. Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 + w 6 + 2m m m 1. In h branch of dl( ), h dg will b dld from G by h branching opraion. So h wigh of rx dcrass by 8,7, and h wigh of rx dcrass by 6,5. Thus h oal wigh dcras for his cas in h branch of dl() is a las w 8 w 7 + w 6 w 5. As a rsul, w g h following branching cor: (w 8 + w 6 + 2m m m 1, w 8 w 7 + w 6 w 5). (42) Cas c-11(iii). Wihou loss of gnraliy, assum ha N U () N U ( ) = { 2, 3, 4 } (s Figur 19): W branch on h dg. No ha N U ( ) \ {} = { 8 } (a) forc( ) in c-11(iii) (b) dl( ) in c-11(iii) : unforcd : nwly dld : forcd Figur 19: Illusraion of branching rul c-11(iii), whrrx V f8 and N U (; V f6 ). In h branch of forc( ), h dg will b addd o F by h branching opraion, and 2, 3, 4, 5,,, 2, 3, 4 and 8 will b dld from G by h rducion ruls. So h wigh of rx dcrass by w 8, and h wigh of rx dcrass by w 6. Each of hrics 2, 3 and 4 can only b on of yps f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 11 = min{ 6,4, 6,4, 7,5, 7,5, 8,6, 8,6}. Each of hrics 5, and can only b on of yps f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 10 = min{ 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Vrx 8 can b any of h possiblrx yps f3, u3, f4, u4, f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and is wigh dcrass by a las m 1 = min{w 3, w 3, 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 + w 6 + 3m m 10 + m 1. In h branch of dl( ), h dg will b dld from G by h branching opraion. So h wigh of rx dcrass by 8,7, and h wigh of rx dcrass by 6,5. Thus h oal wigh dcras for his cas in h branch of dl() is a las w 8 w 7 + w 6 w 5. As a rsul, w g h following branching cor: (w 8 + w 6 + 3m m 10 + m 1, w 8 w 7 + w 6 w 5). (43) Cas c-11(iv). Wihou loss of gnraliy, assum ha N U () N U ( ) = { 2, 3, 4, 5 } (s Figur 20): W branch on h dg. 21

22 (a) forc( ) in c-11(iv) (b) dl( ) in c-11(iv) : unforcd : nwly dld : forcd Figur 20: Illusraion of branching rul c-11(iv), whrrx V f8 and N U (; V f6 ). In h branch of forc( ), h dg will b addd o F by h branching opraion, and 2, 3, 4, 5,,, 2, 3, 4 and 5 will b dld from G by h rducion ruls. So h wigh of rx dcrass by w 8, and h wigh of rx dcrass by w 6. Each of hrics 2, 3, 4 and 5 can only b on of yps f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 11 = min{ 6,4, 6,4, 7,5, 7,5, 8,6, 8,6}. Each of hrics and can only b on of yps f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 10 = min{ 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 + w 6 + 4m m 10. In h branch of dl( ), h dg will b dld from G by h branching opraion. So h wigh of rx dcrass by 8,7, and h wigh of rx dcrass by 6,5. Thus h oal wigh dcras for his cas in h branch of dl() is a las w 8 w 7 + w 6 w 5. As a rsul, w g h following branching cor: (w 8 + w 6 + 4m m 10, w 8 w 7 + w 6 w 5). (44) Cas c-12. Non of h prious cass ar applicabl, and hr xis rics V f8 and N U (; V u6 ) (s Figur 21): W branch on h dg (a) forc( ) in c (b) dl( ) in c-12 : unforcd : nwly dld : forcd Figur 21: Illusraion of branching rul c-12, whrrx V f8 and N U (; V u6 ). In h branch of forc( ), h dg will b addd o F by h branching opraion, and 2, 3, 4, 5, and will b dld from G by h rducion ruls. So h wigh of rx dcrass by w 8, and h wigh of rx dcrass by 6. Each of rics 2, 3, 4, 5, and can only b on of yps u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 12 = min{ 6,5, 7,6, 7,6, 8,7, 8,7}. Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 +w 6 w 6 +6m

23 In h branch of dl( ), h dg will b dld from G by h branching opraion. So h wigh of rx dcrass by 8,7, and h wigh of rx dcrass by 6,5. Thus h oal wigh dcras for his cas in h branch of dl( ) is a las w 8 w 7 + w 6 w 5. As a rsul, w g h following branching cor: (w 8 + w 6 w 6 + 6m 12, w 8 w 7 + w 6 w 5 ). (45) Cas c-13. Non of h prious cass ar applicabl, and hr xis rics V f8 and N U (; V f7 ) (s Figur 22): W branch on h dg. No ha N U ( ) \ {} = { 8, 9, 0, 1, 2 } (a) forc( ) in c (b) dl( ) in c-13 : unforcd : nwly dld : forcd Figur 22: Illusraion of branching rul c-13, whrrx V f8 and N U (; V f7 ). In h branch of forc( ), h dg will b addd o F by h branching opraion, and 2, 3, 4, 5,,, 8, 9, 0, 1 and 2 will b dld from G by h rducion ruls. So h wigh of rx dcrass by w 8, and h wigh of rx dcrass by w 7. Each of hrics 2, 3, 4, 5, and can only b on of yps f7, u7, f8 and a u8-rx, and ach of hir wighs dcrass by a las m 13 = min{ 7,6, 7,6, 8,7, 8,7 }. Each of hrics 8, 9, 0, 1 and 2 can b any of h possiblrx yps f3, u3, f4, u4, f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 1 = min{w 3, w 3, 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 +w 7 +6m 13+5m 1. In h branch of dl( ), h dg will b dld from G by h branching opraion. So h wigh of rx dcrass by 8,7, and h wigh of rx dcrass by 7,6. Thus h oal wigh dcras for his cas in h branch of dl() is a las w 8 w 6. As a rsul, w g h following branching cor: (w 8 + w 7 + 6m m 1, w 8 w 6). (46) Cas c-14. Non of h prious cass ar applicabl, and hr xis rics V f8 and N U (; V f7 ) such ha N U () N U ( ) : W disinguish fi sub-cass, according o h cardinaliy of h inrscion N U () N U ( ), (c-14(i)) N U () N U ( ) = 1; (c-14(ii)) N U () N U ( ) = 2; (c-14(iii)) N U () N U ( ) = 3; (c-14(iv)) N U () N U ( ) = 4; and (c-14(v)) N U () N U ( ) = 5. 23

24 Cas c-14(i). Wihou loss of gnraliy, assum ha N U () N U ( ) = { 2 } (s Figur 23): W branch on h dg. No ha N U ( ) \ {} = { 8, 9, 0, 1 } (a) forc( ) in c-14(i) (b) dl( ) in c-14(i) : unforcd : nwly dld : forcd Figur 23: Illusraion of branching rul c-14(i), whrrx V f8 and N U (; V f7 ). In h branch of forc( ), h dg will b addd o F by h branching opraion, and 2, 3, 4, 5,,, 2, 8, 9, 0 and 1 will b dld from G by h rducion ruls. So h wigh of rx dcrass by w 8, and h wigh of rx dcrass by w 7. Vrx 2 can only b on of yps f7, u7, f8 and a u8-rx, and is wigh dcrass by a las m 14 = min{ 7,5, 7,5, 8,6, 8,6}. Each of hrics 3, 4, 5, and can only b on of yps f7, u7, f8 and a u8-rx, and ach of hir wighs dcrass by a las m 13 = min{ 7,6, 7,6, 8,7, 8,7}. Each of hrics 8, 9, 0 and 1 can b any of h possiblrx yps f3, u3, f4, u4, f5, u5, f6, u6, f7, u7, f8, and a u8-rx, and ach of hir wighs dcrass by a las m 1 = min{w 3, w 3, 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 + w 7 + m m m 1. In h branch of dl( ), h dg will b dld from G by h branching opraion. So h wigh of rx dcrass by 8,7, and h wigh of rx dcrass by 7,6. Thus h oal wigh dcras for his cas in h branch of dl() is a las w 8 w 6. As a rsul, w g h following branching cor: (w 8 + w 7 + m m m 1, w 8 w 6). (47) Cas c-14(ii). Wihou loss of gnraliy, assum ha N U () N U ( ) = { 2, 3 } (s Figur 24): W branch on h dg. No ha N U ( ) \ {} = { 8, 9, 0 } (a) forc( ) in c-14(ii) (b) dl(1) in c-14(ii) : unforcd : nwly dld : forcd Figur 24: Illusraion of branching rul c-14(ii), whrrx V f8 and N U (; V f7 ). In h branch of forc( ), h dg will b addd o F by h branching opraion, and 2, 3, 4, 5,,, 2, 3, 8, 9 and 0 will b dld from G by 24

25 h rducion ruls. So h wigh of rx dcrass by w 8, and h wigh of rx dcrass by w 7. Each of hrics 2 and 3 can only b on of yps f7, u7, f8 and a u8-rx, and ach of hir wighs dcrass by a las m 14 = min{ 7,5, 7,5, 8,6, 8,6}. Each of hrics 4, 5, and can only b on of yps f7, u7, f8 and a u8-rx, and ach of hir wighs dcrass by a las m 13 = min{ 7,6, 7,6, 8,7, 8,7}. Each of h rics 8, 9 and 0 can b any of h possiblrx yps f3, u3, f4, u4, f5, u5, f6, u6, f7, u7, f8 and a u8-rx, and ach of hir wighs dcrass by a las m 1 = min{w 3, w 3, 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 + w 7 + 2m m m 1. In h branch of dl( ), h dg will b dld from G by h branching opraion. So h wigh of rx dcrass by 8,7, and h wigh of rx dcrass by 7,6. Thus h oal wigh dcras for his cas in h branch of dl() is a las w 8 w 6. As a rsul, w g h following branching cor: (w 8 + w 7 + 2m m m 1, w 8 w 6). (48) Cas c-14(iii). Wihou loss of gnraliy, assum ha N U () N U ( ) = { 2, 3, 4 } (s Figur 25): W branch on h dg. No ha N U ( ) \ {} = { 8, 9 } (a) forc( ) in c-14(iii) (b) dl( ) in c-14(iii) : unforcd : nwly dld : forcd Figur 25: Illusraion of branching rul c-14(iii), whrrx V f8 and N U (; V f7 ). In h branch of forc( ), h dg will b addd o F by h branching opraion, and 2, 3, 4, 5,,, 2, 3, 4, 8 and 9 will b dld from G by h rducion ruls. So h wigh of rx dcrass by w 8, and h wigh of rx dcrass by w 7. Each of hrics 2, 3 and 4 can only b on of yps f7, u7, f8 and a u8-rx, and ach of hir wighs dcrass by a las m 14 = min{ 7,5, 7,5, 8,6, 8,6}. Each of hrics 5, and can only b on of yps f7, u7, f8 and a u8-rx, and ach of hir wighs dcrass by a las m 13 = min{ 7,6, 7,6, 8,7, 8,7}. Each of h rics 8 and 9 can b any of h possiblrx yps f3, u3, f4, u4, f5, u5, f6, u6, f7, u7, f8 and a u8-rx, and ach of hir wighs dcrass by a las m 1 = min{w 3, w 3, 4,3, 4,3, 5,4, 5,4, 6,5, 6,5, 7,6, 7,6, 8,7, 8,7}. Thus h oal wigh dcras for his cas in h branch of forc( ) is a las w 8 + w 7 + 3m m m 1. In h branch of dl( ), h dg will b dld from G by h branching opraion. So h wigh of rx dcrass by 8,7, and h wigh of rx dcrass by 7,6. Thus h oal wigh dcras for his cas in h branch of dl() is a las w 8 w 6. As a rsul, w g h following branching cor: (w 8 + w 7 + 3m m m 1, w 8 w 6). (49) 25