O Ht Ttrrzts O Cvx Pyr Frs C 1 Q-Hu D 2 C A W 3 1 Dprtt Cputr S T Uvrsty H K, H K, C. E: @s.u. 2 R & TV Trsss Ctr, Hu, C. E: q@163.t 3 Dprtt Cputr S, Mr Uvrsty Nwu St. J s, Nwu, C A1B 35. E: w@r.s.u. Astrt Lt T p t y ttrrzt vx pyr P t G T t u rp T p su tt G T rrsps t ttrr T p tw s r t y G T y t tw rrsp ttrr sr t T p. T p s Ht ttrrzt G T ts Ht pt (HP). A ww p pr putt try s: vry pytp 3D prtt t ttrr su tt t u rp s HP? I ts t, w s sw tt tr xsts 92-vrtx pyr w t pu t s t y Ht ttrrzt, r t pu t s t spst t t sur r-sz pst s t st y us ttrrzt ts r vx pyr. Furtrr, w strut vx pyr wt vrts su tt t st pt t u rp qust s srt s O(1). Ts t susts tt t y t pss t pprxt HP r vx pyr us t pu t. 1 Itrut A ttrrzt vx pyr P, t y T p, s prtt P y st sps ttrr. Ttrrzt T p s ur trst prprts. Fr xp, t ur ttrr rt ttrrzts pyr wt vrts y vry r θ() t w t O( 2 ) t. T st y us ttrrzt ts r pu s ( t ttr us p-swp s sp s). I t pu t, vrtx v P, px, s t t tr vrts P y s t r ttrrzt P, t s t, p s 206
O Ht Ttrrzts O Cvx Pyr 207 rv r P ttrrz t stp, wr p s sp tw t vx us CH(P) CH(P {v}) r vrtx v P. Nt tt pu surs r ur ttrr w s y rt Θ( ) ttrr t wrst s [3]. Furtrr, ttrrzt P wt t u ur ttrr, pt ttrrzt, s prv t NP-Cpt [2], t st pprxt rt t s 2 1 [4]. Ts ut try strutur s s y ppts. Fr xp, putr rps, t prr rt rr rts s sy rt t t quty ttrrzt, prrssv trsr urs putr twr, t prr s sy rt t t t rt. I prtur, ps tt t u rp t ttrrzt ts HP s tt t t rt t t rts r t t twrs ru. W vry 2D py s trut wt HP, t s t w ts s v r 3D vx pyr. It s tur tt tr wys xsts Ht ttrrzt r y vx pyr [1], t sst pr tr s st t p pr prt [5]. I ts t, w sw tt tr xsts vx pyr Q w pu s t y Ht ttrrzt. T utt sut t ts pr s st usv. W s s sw tt ttrrzt t y pu ts pprxt t HP, trs rt vrts. As r rsut, w prst 94-3-rur 3-t pr rp wtut HP. 2 A vx pyr w pu s t y Ht ttrrzt W rst sr s vx pyr u ts prprts. W t strut vx pyr us ts s sw tt t rsut pyr s t v Ht ttrrzt y pu. Ts u uss t s- Tutts -HC rp pt [6]. 2.1 T 2D s s Lt us sr trut 10 vrts 2D ts u rp. Ts u rp 25 s s pt 3-rur 3-t rp w s t v Ht r [6]. Osrv tt t u rp G T10 s xty tr r-2 s tr s HP r rt pr strt s, prtur, r-2 s (rr t Fur 1). W us ts rp t u rr rp 45 s, w Ht pts xst. T rrsp trut s 25 vrts t p (rr t t t s Fur 2). It w vr L 2 tt y pr rp G t ts 45-s surp wu v t v t st HP ts surp G s y HP (rr t t rt s Fur 2). L 1. Tr s HP G T25.
208 Itrt Sypsu OR Its Appts 2005 x z A trut 10 pts, T. 10 y T 10 T u rp t trut G. T sy r t u rp. Fur 1: A u rp wtut Ht pt x- tw s. 2 5 6 7 24 3 34 8 43 25 26 4 33 44 29 23 2827 30 31 32 35 36 9 22 u 10 21 20 37 11 19 18 17 38 42 39 12 41 16 40 13 14 15 45 A trut 25 pts, T, 25 w s tr T 10 s rs 45 r ts. tr T T sy r G 25. T T u rp T, G 25, w 25 HP xsts r y pr 45 s. Fur 2: A u rp wtut Ht pt tw y pr ts 45 s. Pr. Nt tt y pt w s trvrs t rst G T10 tr t s G T10 ust strt t (rsptvy, ) t s G T10. H, t pt t trvrs t s G T10 wt (rsptvy, ) s t st y ur prvus srvt. Csquty, t st pt L G T25 vr t st tw G T10 s, L s t HP G T25. L 2. Ay rp G t G T25 s surp t G G T25 t G T25 tru tr s r 2 ( ) s t v t st HP G T25 G s su HP. Pr. Lt L st pt G. I L strts s G T25, t s. Otrws, y t v t rstrt tw G G T25 G T25 y L 1, t st st L s G T25 y vr tw t G T10 G T25. T, t rst L ust rtur t G G T25 r-trs G T25 tru t tr t tr G T10. T L ust trt s G T25 s tw tr
O Ht Ttrrzts O Cvx Pyr 209 trs r ry us. Nw w tr s t r w trut T 70 (rr t Fur 3). W s sw tt ts u rp G T70 s t v HP y rp G t G T70 wu t v HP tr. T 70 W T 70 G Fur 3: Du rp G T70 s t v Ht pt. L 3. Tr s HP G T70. Mrvr, rp G t G T70 s surp t G G T25 G T25 tru tr r-2 s G T70 s HP. Pr. Lt L st pt G. By L 2, t st st L vr t st tw G T10 s G T25. Pt L ust tr v G T70 r tr t tw r G T25 s tru tr. Tus, L s t v G T70, t st st L vr t st sv G T10 s v L s G T70 s tt G T70 s s tw uvr st G T10 s. Nt tt w L vs G T70 rtrs G T70, t y vr ts tw st G T10 s. Tus, vtu L vrs G G T70, t s st t HP G. 2.2 Cstrut t pyr I ts st, w sr w t strut vx pyr Q wtut Ht ttrrzt rtv t t pu t us t 2D uts srv St 2.1. Lt,,, ur r trs t sur vx pyr. T u ttrr sst wt tr, sy, s w t r tw qus t r tw t p xt, sr quty s r t rs rt t r t rs rt t, rsptvy. (Rr t prt () Fur 4.)
210 Itrt Sypsu OR Its Appts 2005 α u ttrr α t r s v p t w u q q p r t w u v q () () () Fur 4: T sur rp t strut vx u 10 pts s rp t T 10. T vrts t tw rps v t w rrsp: ( 3)( 2) ( 1)(v 4) (q 5)(u 6) (w 7)(p 8) (r 9)(t 10). Ruy sp, w rst u rur ttrr. Fr t ur trur s t ttrr, w pt s u s ts rrsp u ttrr, wr t s 3D vx u 25 vrts wt trur (rr t prt () Fur 5). E s u ts tr sr s prts (sr s urs), s prt ts vx u 10 vrts s ts rrsp u ttrr. T rp ts vx u s rp t tt t sr trut (rr t prt () Fur 5). v q r x V p u w A trut 10 pts, T 10, t vx sur. () U A trut 25 pts, T, 25 t vx sur, w s t t 3 T s. 10 () W A pyr Q wt 92 vrts. () Fur 5: T sur rp t vx u 10 pts s rp t tt T 10 ; Tr T 10 surs wt s vrpp ury s vrts r sur s u. Fur su s pt rur ttrr r Q. I r t struts 10-vrtx s prt, y rr t F-
O Ht Ttrrzts O Cvx Pyr 211 ur 4. Lt u ttrr. Lt t pt t tr r--strs tr. Lt t t pt t st w s vrt t, s s, s t. Lt r C, tr t t pt st t, vrt t t tu t ury ttrr t. Lt p, q, r t rsptv tt pts C r t p t, t p t, t p t. Lt u v t trst pts tw t r pq p t uv tw t r qr ts p, rsptvy. Nt tt u v r vry s t q. Lt t pt t st t t t tr C. Lt q pt s t uv. T pts t q s s p tt st t q s rt w st uv. W w p vrts pts,,, p, q, r, u, v, w t s sr v. It vr tt t vx u ts t vrts, xu tr, rs sur rp srp t T 10. Lt CH t ts vx u (rr t prt () Fur 4). T u s u, r trur t rur ttrr, sy, w p vrtx U t pt t st (strut sry t x t s prt) s t u ttrr. T, r trr s U, U, U, w tt s prt s t rrsp u ttrr. T vx u t 25-vrtx u rs sur rp rp t t trut s sw t t s Fur 2. Pyr Q s r y tt t ur trur s rur ttrr wt s u s t rrsp u ttrr. T v strut s tw prprts: Prprty 1: Lt CH s prt pt trur vx pyr Q. T, t w pyr Q = Q CH s vx. Ts s us t s prt s p ts rrsp u ttrr wt rspt t Q. T rstrt u ttrr surs tt t vxty t r ps t sur Q s t vt. Lt Q r vx pyr,.., t s Q r pr trs. Prprty 2: T w pyr Q = Q CH s s r vx pyr. Ts s us t sp tw t u ttrr CH s wys -pty u t ur strut. H, t suqut s r trs tr rrsp u ttrr r -pty t. Prprty 2 ps tt t v vx u pt prss rursvy xut, t ur vrts Q rs t s r s ssry. 2.3 T trut surs vw r rt pu ps I ts st, w s sr t tr t-rp su vx pyr Q s vw r pu px. T rrsp u rp s s prst.
Lt W sr vrtx tr s Q, t U sr vrtx t tr T 10 s, t V t r vrts T 10 (rr t t tt prt Fur 5). W w sr tr ss suss r t ts tr typs ps. 1. Cs 1 sws t t rp Q vw r px W. T u rp G ts surp G T25 (rr t Fur 6). 2. Cs 2() sws t t rp Q vw r px U. T u rp G ts surp G T70 G G T70 ts tr st pts (rr t Fur 7). 3. Cs 2() sws t t rp Q vw r px V. T u rp G ts surp G T70 G G T70 s t pt (rr t Fur 8). Tr 4. Tr xsts vx pyr Q wt 92 vrts w pu s t y Ht ttrrzt. Pr. T vrts Q ss t tr rups p wt typ surps Q s u rp G TQ ts. I Cs 1, G TQ ts G T25 s w s r s. E pt G TQ G T25 w t t G T25 ust t pt s ty r st. Tr r tr su pts. Trr, t st pt t HP G TQ. I Cs 2 () (), G TQ ts G T70 s surp, y L 3, tr s HP G TQ. I y s, G TQ s t v HP. G(T25) W W W Fur 6: Cs 1: A r sur vw r px w, t u rp t ttrrzt wt pu px w. 212 Itrt Sypsu OR Its Appts 2005
O Ht Ttrrzts O Cvx Pyr 213 u u G(T70) u Fur 7: Cs 2(): A r sur vw r px u, t u rp t ttrrzt wt pu px u. 3 Cstrut vx -vrtx pyr P wt stt-t st pt G T P I ts st, w s strut vx -vrtx pyr P su tt t st pt t u rp T p us pu s stt t trs ur vrts. W wys pt s prt r s u t trur vx pyr Q t r w vx pyr wt r vrts y Prprts 1 2. T rsut pyr P tr su pt prss s t Θ() vrts. W s sw tt t st pt G TP s srt s O(1) t xu ur stt pts s r s Θ(). Hr, t tr stt s tw pts t sr. T s, w s rst sr 3-rur 3-t pr rp, pt s. W t us ts pt s t u pt w s t v y HP ts s t pts y rp tt ts ts pt s surp. It s sy t tt y pt tr t pt s ust v tr (rr t prt () Fur 9). T rp prt () Fur 9 sws tt pt sp t G T109 pt tr t t tr G T109 r uts tru s. Ts s s pt t sr u rp. T strut t sr vx pyr P, w rp t trr s:,,, T 10 Q y s prts. T ur s prts r s p tt tr u rp r pt s. T w trut w ts 109 vrts ts u rp G T109 s s pt. T v pt prss tu s y s ts u t Prprty 2 t rsut pyr P s vx y Prprty 1. Nw P s t st (109 25) (= Θ()) vrts. Nt tt pt G T109 rqurs tw pts t vr. I t pt
214 Itrt Sypsu OR Its Appts 2005 V G(T70) T 10 T 10 U V Fur 8: Cs 2(): r sur vw r px v, t u rp t ttrrzt wt pu px v. strts t s G T109, t t st pt vr t st tw pt ss tw G T10 s, w s 142 s. I t pt strts t uts G T109, t t st pt vr t st tw pt ss, w s 120 s. Tus, t rsut P y v Θ() stt st pts pt s stt t. Tr 5. Tr xsts vx pyr P wt vrts w pu s t y Ht ttrrzt. T st pt G TP vrs 142 s t xu ur stt st pts s Θ(). Crry 6. Tr xsts 3-rur 3-t pr rp 94 s w s t v HP. 4 us I ts t, w strut vx pyr su tt y ttrrzt ts pyr pru y t pu t ts HP ts u rp. W urtr sw tt t st pt su u rp wt Θ() s y s srt s O(1). W s sr 3-rur 3-t pr rp 94 s w s t v HP. T vus p pr s wtr r t tr s HP r y ttrrzt rtrry vx pyr y s. Awt: Ts wr s supprt y NSERC rt OPG0041629.
O Ht Ttrrzts O Cvx Pyr 215 () A pt s wt 60 s A pt s A pt s tr A pt s A pt s () T ur s prts pt t s r t r pt s () A pt r st pts Fur 9: T pt s rp ts 60 s. T rp G T109 ssts G T25 wt tr pt ss. Rrs [1] Ar E., H M., Mt J., S S., (1996), Ht trut r st rr, T Vsu Cputr 12, pp. 429-444. [2] Bw A., D Lr J., Rr-Grt J.,(2000), F truts vx 3-pytps r HP-r, T prs t tt u ACM-SIAM sypsu srt rts, pp.65-66. [3] M. Br, (1993), Cpt Ttrrzts, T prs 9t Au ACM Sypsu Cputt Gtry, 281-288. [4] F. C, S. F, C.A.W, (2001), Apprxt r Mu Truts Sp Cvx 3-pytps, Dsrt & Cputt Gtry. V.26, N.4, pp.499-511. [5] E. D, J. Mt, J. O Rur, Op pr Prt. ttp://s.st.u/ rur/topp/ [6] W. Tutt, (1946), O Ht ruts, Jur t L Mtt Sty, V. 21, pp. 98-101.
216 Itrt Sypsu OR Its Appts 2005 Fur 10: T 3-rur 3-t rp G 94 s ts HP.