On Approximation Lower Bounds for TSP with Bounded Metrics

Transcription

1 O Approxmato Lowr Bouds for TSP wth Boudd Mtrcs Mark Karpsk Rchard Schmd Abstract W dvlop a w mthod for provg xplct approxmato lowr bouds for TSP problms wth boudd mtrcs mprovg o th bst up to ow kow bouds. Thy almost match th bst kow bouds for uboudd mtrc TSP problms. I partcular, w prov th bst kow lowr boud for TSP wth boudd mtrcs for th mtrc boud qual to. Itroducto W gv frst th basc dftos ad a ovrvw of th kow rsults. Travlg Salsprso (TSP) Problm W ar gv a mtrc spac (V, d) ad th task cossts of costructg a shortst tour vstg ach vrtx xactly oc. Th TSP problm mtrc spacs s o of th most fudamtal NP-hard optmzato problms. Th dcso vrso of ths problm was show arly to b NP-complt by Karp [K7]. Chrstofds [C76] gav a algorthm approxmatg th TSP problm wth 3/,.., a algorthm that producs a tour wth lgth bg at most a factor 3/ from th optmum. As for lowr bouds, a rducto du to Papadmtrou ad Yaakaks [PY93] ad th PCP Thorm [ALM + 98] togthr mply that thr xsts som costat, ot bttr tha + 0 6, such that t s NP-hard to approxmat th TSP problm wth dstacs thr o or two. For dscusso of boudd mtrcs TSP, s also [T00]. Ths hardss rsult was mprovd by Egbrts [E03], who provd that t s NP-hard to approxmat th TSP problm rstrctd to dstacs o ad two wth a approxmato factor bttr tha 538/5380 (.0008). Böckhaur ad Sbrt [BS00] studd th TSP problm wth dstacs o, two ad thr, ad obtad a approxmato lowr boud of 383/38 (.0006). Th bst Dpt. of Computr Scc ad th Hausdorff Ctr for Mathmatcs, Uvrsty of Bo. Supportd part by DFG grats ad th Hausdorff Ctr grat EXC59-. Emal: mark@cs.u-bo.d Dpt. of Computr Scc, Uvrsty of Bo. Work supportd by Hausdorff Doctoral Fllowshp. Emal: schmd@cs.u-bo.d

2 kow approxmato lowr boud for th gral vrso of ths problm s du to Papadmtrou ad Vmpala [PV06]. Thy provd that th TSP problm s NPhard to approxmat wth a approxmato factor lss tha 0/9 (.0056). Th rstrctd vrso of th TSP problm, whch th dstac fucto taks valus {,..., B}, s rfrrd to as th (, B) TSP problm. Th (, ) TSP problm ca b approxmatd polyomal tm wth a approxmato factor 8/7 du to Brma ad Karpsk [BK06]. O th othr had, Egbrts ad Karpsk [EK06] provd that t s NPhard to approxmat th (, B) TSP problm wth a approxmato factor lss tha 7/70 (.0035) for B = ad 389/388 (.0057) for B = 8. I ths papr, w prov that th (, ) TSP ad th (, ) TSP problm ar NPhard to approxmat wth a approxmato factor lss tha 535/53 ad 337/336, rspctvly. Asymmtrc Travlg Salsprso (ATSP) Problm W ar gv a asymmtrc mtrc spac (V, d),.., d s ot cssarly symmtrc, ad w would lk to costruct a shortst tour vstg vry vrtx xactly oc. Th bst kow algorthm for th ATSP problm approxmats th soluto wth O( log / log log ), whr s th umbr of vrtcs th mtrc spac [AGM + 0]. O th othr had, Papadmtrou ad Vmpala [PV06] provd that th ATSP problm s NP-hard to approxmat wth a approxmato factor lss tha 7/6 (.0086). It s cocvabl that th spcal cass wth boudd mtrc ar asr to approxmat tha th cass wh th dstac btw two pots grows wth th sz of th stac. Clarly, th (, B) ATSP problm, whch th dstac fucto s takg valus th st {,..., B}, ca b approxmatd wth B by ust pckg ay tour as th soluto. Wh w rstrct th problm to dstacs o ad two, t ca b approxmatd wth 5/ du to Bläsr [B0]. Furthrmor, t s NP-hard to approxmat ths problm wth a approxmato factor bttr tha 3/30 [BK06]. For th cas B = 8, Egbrts ad Karpsk [EK06] costructd a rducto yldg th approxmato lowr boud 35/3 for th (, 8) ATSP problm. I ths papr, w prov that t s NP-hard to approxmat th (, ) ATSP ad th (, ) ATSP problm wth a approxmato factor lss tha 07/06 ad /0, rspctvly. Maxmum Asymmtrc Travlg Salsprso (MAX-ATSP) Problm W ar gv a complt drctd graph G ad a wght fucto w assgg ach dg of G a ogatv wght. Th task s to fd a tour of maxmum wght vstg vry vrtx of G xactly oc. Ths problm s wll-kow ad motvatd by svral applcatos (cf. [BGS0]). A good approxmato algorthm for th MAX ATSP problm ylds a good approxmato algorthm for may othr optmzato problms such as

3 th Shortst Suprstrg problm, th Maxmum Comprsso problm ad th (, ) ATSP problm. Th MAX (0, ) ATSP problm s th rstrctd vrso of th MAX-ATSP problm, whch th wght fucto w taks valus th st {0, }. Vshwaatha [V9] costructd a approxmato prsrvg rducto provg that ay (/α) approxmato algorthm for th MAX (0, ) ATSP problm problm trasforms a ( α) approxmato algorthm for th (, ) ATSP problm. Du to ths rducto, all gatv rsults cocrg th approxmato of th (, ) ATSP problm mply hardss rsults for th MAX (0, ) ATSP problm. Sc th (, ) ATSP problm s APX-hard [PY93], thr s lttl hop for polyomal tm approxmato algorthms wth arbtrary good prcso for th MAX (0, ) ATSP problm. Du to th xplct approxmato lowr boud for th (, ) ATSP problm gv [EK06], t s NP-hard to approxmat th MAX (0, ) ATSP problm wth a approxmato factor lss tha 30/39. Th bst kow approxmato algorthm for th rstrctd vrso of ths problm s du to Bläsr [B0] ad achvs a approxmato rato 5/. For th gral problm, Kapla t al. [KLS + 05] dsgd a algorthm for th MAX ATSP problm yldg th bst kow approxmato uppr boud of 3/. O th approxmato hardss sd, Karpsk ad Schmd [KS] costructd a rducto yldg th approxmato lowr boud 07/06 for th MAX ATSP problm. I ths papr, w prov that approxmatg th MAX (0, ) ATSP problm wth a approxmato rato lss tha 06/05 s NP-hard. Ovrvw of Kow Explct Approxmato Lowr Bouds ad Our Rsults (, B) ATSP problm B = B = B = 8 uboudd Prvously kow 3/30 3/30 35/3 7/6 rsults (.003) (.003) (.0076) (.0086) [EK06] [EK06] [EK06] [PV06] Our rsults 07/06 /0 (.0085) (.007) (, B) TSP problm B = B = B = 8 uboudd Prvously kow 7/70 7/70 389/388 0/9 rsults (.0035) (.0035) (.0057) (.0056) [EK06] [EK06] [EK06] [PV06] Our rsults 535/53 337/ /336 (.0087) (.0097) (.0097) Fgur : Kow xplct approxmato lowr bouds ad th w rsults. 3

4 MAX (0, ) ATSP problm MAX ATSP problm Prvously kow 30/39 07/06 rsults (.003) (.0085) [EK06] [KS] Our rsults 06/05 06/05 (.0087) (.0087) Fgur : Kow xplct approxmato lowr bouds ad th w rsults. Prlmars I ths scto, w df th abbrvatos ad otatos usd ths papr. Gv a atural umbr k ad a ft st V, w us th abbrvato [k] for {,..., k} ad ( ) V for th st { S V S = }. Gv a asymmtrc mtrc spac (V, d) wth V = {v,..., v } ad d : V V R 0, a Hamltoa cycl V or a tour V s a cycl vstg ach vrtx v V xactly oc. I th rmadr, w spcfy a tour σ V by σ = {(v, v ),..., (v, v )} V V or xplctly by σ = v v v v. Gv a tour σ (V, d), th lgth of a tour l(σ) s dfd by l(σ) = a σ d. Aalogously, gv a mtrc spac (V, d) wth V = {v,..., v } ad d : ( ) V R >0, w spcfy a tour σ V by σ = { {v, v },..., {v, v } } ( ) V or xplctly by σ = v v v v. I ordr to spcfy, a stac (V, d) of th (, ) ATSP problm, t suffcs to dtfy th arcs a V V wth wght o. Th sam stac s spcfd by a drctd graph D = (V, A), whr a A f ad oly f d =. Aalogously, th (, ) TSP problm, a stac s compltly spcfd by a graph G = (V, E). I th rmadr, w rfr to a arc ad a dg wth wght x N as a x-arc ad x-dg, rspctvly. 3 Rlatd Work 3. Hybrd Problm Brma ad Karpsk [BK99], s also [BK0] ad [BK03] troducd th followg Hybrd problm ad provd that ths problm s NP-hard to approxmat wth som costat. Dfto (Hybrd problm). Gv a systm of lar quatos mod cotag varabls, m quatos wth xactly two varabls, ad m 3 quatos wth xactly

5 thr varabls, fd a assgmt to th varabls that satsfs as may quatos as possbl. I [BK99], Brma ad Karpsk costructd spcal stacs of th Hybrd problm wth boudd occurrcs of varabls, for whch thy provd th followg hardss rsult. y 9 y 8 y 0 y 7 y 6 y 5 y y 3 y y y y y y3 y y 5 y 6 y 7 y 0 y 9 y 8 x 9 x 8 x 7 x 6 x 5 x x 3 x x x 0 x x x 3 x 9 x x 0 x x 5 x 6 x 7 x 8 hyprdg z 9 z 7 z 8 z 6 z 5 z z 3 z z z 0 z z z z 3 z z 5 z 6 z 7 z 0 z 9 z 8 Fgur 3: A xampl of a Hybrd stac wth crcls C x, C y, C z, ad hyprdg = {z 7, y, x }. Thorm ([BK99]). For ay costat ɛ > 0, thr xsts stacs of th Hybrd problm wth ν varabls, 60ν quatos wth xactly two varabls, ad ν quatos wth xactly thr varabls such that: () Each varabl occurs xactly thr tms. () Ethr thr s a assgmt to th varabls that lavs at most ɛν quatos usatsfd, or ls vry assgmt to th varabls lavs at last ( ɛ)ν quatos usatsfd. () It s NP-hard to dcd whch of th two cass tm () abov holds. Th stacs of th Hybrd problm producd Thorm hav a v mor spcal structur, whch w ar gog to dscrb. Th quatos cotag thr varabls ar of th form x y z = {0, }. Ths quatos stm from th Thorm of Håstad [H0] dalg wth th hardss of approxmatg quatos wth xactly thr varabls. W rfr to t as 5

6 th MAX E3 LIN problm, whch ca b s as a spcal stac of th Hybrd problm. Thorm (Håstad [H0]). For ay costat δ (0, ), thr xsts systms of lar quatos mod wth m quatos ad xactly thr ukows ach quato such that: () Each varabl th stac occurs a costat umbr of tms, half of thm gatd ad half of thm ugatd. Ths costat grows as Ω( /δ ). () Ethr thr s a assgmt satsfyg all but at most δ m quatos, or vry assgmt lavs at last ( δ)m quatos usatsfd. () It s NP-hard to dstgush btw ths two cass. For vry varabl x of th orgal stac E 3 of th MAX-E3-LIN problm, Brma ad Karpsk troducd a corrspodg st of varabls V x. If th varabl x occurs t x tms E 3, th, V x cotas 7t x varabls x,..., x 7tx. Th varabls cotad Co(V x ) = {x {7ν ν [t x ]}} ar calld cotact varabls, whras th varabls C(V x ) = V x \Co(V x ) ar calld chckr varabls. All varabls V x ar coctd by quatos of th form x x + = 0 wth [7t x ] ad x x 7tx = 0. I addto to t, thr xsts quatos of th form x x = 0 wth {, } M x, whrby th st M x ( ) C(V x) dfs a prfct matchg o th st of chckr varabls. I th rmadr, w rfr to ths costructo as th crcl C x cotag th varabls x V x. Lt E 3 b a stac of th MAX E3 LIN problm ad H b ts corrspodg stac of th Hybrd problm. W dot by V (E 3 ) th st of varabls whch occur th stac E 3. Th, H ca b rprstd graphcally by V (E 3 ) crcls C x wth x V (E 3 ) cotag th varabls V (C x ) = {x,..., x tx } as vrtcs. Th dgs ar dtfd by th quatos cludd H. Th quatos wth xactly thr varabls ar rprstd by hyprdgs wth cardalty = 3. Th quatos x x + = 0 duc a cycl cotag th vrtcs {x,..., x tx } ad th matchg quatos x x = 0 wth {, } M x duc a prfct matchg o th st of chckr varabls. A xampl of a stac of th Hybrd problm s dpctd Fgur 3. I summary, w otc that thr ar four typ of quatos th Hybrd problm () th crcl quatos x x + = 0 wth [7t x ], () crcl bordr quatos x x 7tx, () matchg quatos x x = 0 wth {, } M x, ad (v) quatos wth thr varabls of th form x y z = {0, }. I th rmadr, w may assum that quatos wth thr varabls ar of th form x y z = 0 or x y z = 0 du to th trasformato x y z = 0 x y z =. 3. Approxmato Hardss of TSP Problms Rducg th Hybrd Problm to th (, ) (A)TSP Problm Egbrts ad Karpsk [EK06] costructd a approxmato prsrvg rducto from th Hybrd problm to th (, ) ATSP problm to prov xplct ap- 6

7 v 3 c s c v c s c+ v c Graph for quatos wth thr varabls. Varabl graph. Fgur : Gadgts usd [EK06]. proxmato lowr bouds for th lattr problm. Thy troducd graphs (gadgts), whch smulat varabls, quatos wth two varabls ad quatos wth thr varabls. I partcular, th graphs corrspodg to varabls ad to quatos of th form x y z = 0 ar dpctd Fgur ad, rspctvly. For th graph corrspodg to a quato wth thr varabls, thy provd th followg statmt. Proposto ([EK06]). Thr s a Hamltoa path from s c to s c+ Fgur f ad oly f a v umbr of tckd dgs s travrsd. A smlar rducto was costructd ordr to prov xplct approxmato lowr bouds for th (, ) TSP problm. Th corrspodg graphs ar dpctd Fgur 5. Th graph cotad th dashd box Fgur 5 wll play a crucal rol our rducto ad w rfr to t as party graph. I partcular, our varabl gadgt cossts of a party graph. I th rducto of th (, ) TSP problm, th followg statmt was provd for th graph corrspodg to quatos of th form x y z = 0. Proposto ([EK06]). Thr s a smpl path from s c to s c+ Fgur 5 cotag th vrtcs v {v c, v c } f ad oly f a v umbr of party graphs s travrsd. Ths rductos combd wth Thorm yld th followg xplct approxmato lowr bouds. Thorm 3 ([EK06]). It s NP-hard to approxmat th (, ) ATSP ad th (, ) TSP problm wth a approxmato rato lss tha 3/30 (.003) ad 7/70 (.0035), rspctvly. 7

8 v c s c sc+ v c Graph corrspodg to x y z = 0. Varabl graph. Fgur 5: Gadgts usd [EK06] to prov approxmato hardss of th (, ) TSP problm. Explct Approxmato Lowr Bouds for th MAX ATSP problm By rplacg all dgs wth wght two of a stac of th (, ) ATSP problm by dgs of wght zro, w obta a stac of th MAX (0, ) ATSP problm, whch rlats th (, ) ATSP problm to th MAX ATSP problm th followg ss. Thorm ([V9]). A (/α) approxmato algorthm for th MAX (0, ) ATSP problm mpls a ( α) approxmato algorthm for th (, ) ATSP problm. Ths rducto trasforms vry hardss rsult addrssg th (, ) ATSP problm to a hardss rsult for th MAX (0, ) ATSP problm. I partcular, Thorm 3 mpls th bst kow xplct approxmato lowr boud for th MAX (0, ) ATSP problm. Corollary. It s NP-hard to approxmat th MAX (0, ) ATSP problm wth ay bttr tha 30/39 (.003). Our Cotrbuto W ow formulat our ma rsults. Thorm 5. Suppos w ar gv a stac H of th Hybrd problm wth crcls, m quatos wth two varabls ad m 3 quatos wth xactly thr varabls wth th proprts dscrbd Thorm. () Th, t s possbl to costruct polyomal tm a stac D H of th (, ) ATSP problm wth th followg proprts: 8

9 If thr xsts a assgmt φ to th varabls of H whch lavs at most u quatos usatsfd, th, thr xst a tour σ φ D H wth lgth at most l(σ φ ) = 3m + 3m u. From vry tour σ D H wth lgth l(σ) = 3m + 3m u, w ca costruct polyomal tm a assgmt ψ σ to th varabls of H that lavs at most u quatos H usatsfd. () Furthrmor, t s possbl to costruct polyomal tm a stac (V H, d H ) of th (, ) ATSP problm wth th followg proprts: If thr xsts a assgmt φ to th varabls of H whch lavs at most u quatos usatsfd, th, thr xst a tour σ φ (V H, d H ) wth lgth at most l(σ φ ) = m + 0m u +. From vry tour σ (V H, d H ) wth lgth l(σ) = m +0m 3 ++u+, w ca costruct polyomal tm a assgmt ψ σ to th varabls of H that lavs at most u quatos H usatsfd. Th formr thorm ca b usd to drv a xplct approxmato lowr boud for th (, ) ATSP problm by rducg stacs of th Hybrd problm of th form dscrbd Thorm to th (, ) ATSP problm. Corollary. It s NP-hard to approxmat th (, ) ATSP problm wth a approxmato factor lss tha 07/06 (.0085). Proof. Lt E 3 b a stac of th MAX E3 LIN problm. W df k to b th mmum umbr of occurcs of a varabl E 3. Accordg to Thorm, w 07 δ may choos δ > 0 such that 07 ɛ holds. Gv a stac E 06+δ of th k MAX E3 LIN problm wth δ (0, δ), w grat th corrspodg stac H of th Hybrd problm. Th, w costruct th corrspodg stac D H of th (, )-ATSP problm wth th proprts dscrbd Thorm 5. W coclud accordg to Thorm that thr xst a tour D H wth lgth at most 3 60ν + 3 ν + δ ν + + (06 + δ + + )ν (06 + δ ν k )ν or th lgth of a tour D H s boudd from blow by 3 60ν + 3 ν + ( δ )ν + + (06 + ( δ ))ν (07 δ )ν. From Thorm, w kow that th two cass abov ar NP-hard to dstgush. Hc, for vry ɛ > 0, t s NP-hard to fd a soluto to th (, )-ATSP problm wth a approxmato rato 07 ɛ δ 06+δ + 7 k For th symmtrc vrso of th problms, w costruct rductos from th Hybrd problm wth smlar proprts. Thorm 6. Suppos w ar gv a stac H of th Hybrd problm wth crcls, m quatos wth two varabls ad m 3 quatos wth xactly thr varabls wth th proprts dscrbd Thorm. 9

10 () Th, t s possbl to costruct polyomal tm a stac G H of th (, ) TSP problm wth th followg proprts: If thr xsts a assgmt φ to th varabls of H whch lavs at most u quatos usatsfd, th, thr xst a tour σ φ G H wth lgth at most l(σ φ ) = 8m + 7m u. From vry tour σ G H wth lgth l(σ) = 8m + 7m u, w ca costruct polyomal tm a assgmt ψ σ to th varabls of H that lavs at most u quatos H usatsfd. () Furthrmor, t s possbl to costruct polyomal tm a stac (V H, d H ) of th (, ) TSP problm wth th followg proprts: If thr xsts a assgmt φ to th varabls of H whch lavs at most u quatos usatsfd, th, thr xst a tour σ φ (V H, d H ) wth lgth at most l(σ φ ) = 0m + 36m u. From vry tour σ (V H, d H ) wth lgth l(σ) = 0m +36m u, w ca costruct polyomal tm a assgmt ψ σ to th varabls of H that lavs at most u quatos H usatsfd. Aalogously, w comb th formr thorm wth th xplct approxmato lowr boud for th Hybrd problm of th form dscrbd Thorm yldg th followg approxmato hardss rsult. Corollary 3. It s NP-hard to approxmat th (, ) TSP ad th (, ) TSP problm wth a approxmato factor lss tha 535/53 (.0087) ad 337/336 (.0097), rspctvly. From Thorm ad Corollary, w obta th followg xplct approxmato lowr boud. Corollary. It s NP-hard to approxmat th MAX (0, ) ATSP problm wth a approxmato factor lss tha 06/05 (.0087). 5 Approxmato Hardss of th (, ) ATSP problm 5. Ma Idas As mtod abov, w prov our hardss rsults by a rducto from th Hybrd problm. Lt E 3 b a stac of th MAX-E3-LIN problm ad H th corrspodg stac of th Hybrd problm. Evry varabl x l th orgal stac E 3 troducs assocatd crcl C l th stac H as llustratd Fgur 3. Th ma da of our rducto s to mak us of th spcal structur of th crcls H. Evry crcl C l H corrspods to a graph D l th stac D H of th (, )-ATSP problm. Morovr, D l s a subgraph of D H, whch bulds almost a cycl. A assgmt to th varabl x l wll hav a atural trprtato ths 0

11 outr loop r loop D D 3 D 3 D D 6 D 3 D D 5 Fgur 6: A llustrato of th stac D H ad a tour D H. rducto. Th party of x l corrspods to th drcto of movmt D l of th udrlyg tour. Th crcl graphs of D H ar coctd ad buld togthr th r loop of D H. Evry varabl x l a crcl C l posssss a assocatd party graph P l (Fgur 7) D l as a subgraph. Th two atural ways to travrs a party graph wll b calld 0/-travrsals ad corrspod to th party of th varabl x l. Som of th party graphs D l ar also cotad graphs Dc 3 (Fgur ad Fgur 9 for a mor dtald vw) corrspodg to quatos wth thr varabls of th form gc 3 x y z = 0. Ths graphs ar coctd ad buld th outr loop of D H. Th whol costructo s llustratd Fgur 6. Th outr loop of th tour chcks whthr th 0/-travrsals of th party graphs corrspod to a satsfyg assgmt of th quatos wth thr varabls. If a udrlyg quato s ot satsfd by th assgmt dfd va 0/-travrsal of th assocatd party graph, t wll b pushd by usg a costly -arc. 5. Costructg D H from a Hybrd Istac H Gv a stac of th Hybrd problm H, w ar gog to costruct th corrspodg stac D H = (V (D H ), A(D H )) of th (, )-ATSP problm. For vry typ of quato H, w wll troduc a spcfc graph or a spcfc way to coct th so far costructd subgraphs. I partcular, w wll dstgush btw graphs corrspodg to crcl quatos, matchg quatos, crcl bordr quatos ad quatos wth thr varabls. Frst of all, w troduc graphs corrspodg to th varabls H. Varabl Graphs Lt H b a stac of th hybrd problm ad C l a crcl H. For vry varabl x l th crcl C, w troduc th party graph P l cosstg of th vrtcs v l, v l ad v l0 dpctd Fgur 7.

12 Fgur 7: Party graph P l corrspodg to th varabl x l crcl C l. Matchg ad Crcl Equatos Lt H b a stac of th hybrd problm, C l a crcl H ad M l th assocatd prfct matchg. Furthrmor, lt x l x l = 0 wth = {, } M l ad < b a matchg quato. Du to th costructo of H, th crcl quatos x l x l + = 0 ad x l x l + = 0 ar both cotad C l. Th, w troduc th assocatd party v+ l0 v+ l v+ l + v l(+) + + Fgur 8: Coctg th party graph P l graph P l cosstg of th vrtcs, v l th party graphs P l, P l ad v l(+) +, P, l P+ l ad P l as dpctd Fgur 8. Graphs Corrspodg to Equatos wth Thr Varabls. I addto to t, w coct Lt gc 3 x l x s x k t = 0 b a quato wth thr varabls H. Th, w troduc th graph Dc 3 (Fgur ) corrspodg to th quato gc 3. Th graph Dc 3 cluds th vrtcs s c, vc, vc, vc 3 ad s c+. Furthrmor, t cotas th party graphs P, l Pb s ad Pa k as subgraphs, whrby w usd th abbrvatos = {, + }, b = {, + } ad a = {t, t + }. Exmplary, w dsplay Dc 3 wth ts coctos to th graph corrspodg to th crcl quato x l x l + = 0 Fgur 9. I cas of g 3 c x l x s x u k = 0, w coct th party graphs wth arcs (, ( +, ) ad (, +). Graphs Corrspodg to Crcl Bordr Equatos Lt C l ad C l+ b crcls H. I addto, lt x l x l = 0 b th crcl bordr quato of C l. Th, w troduc th vrtx b l ad coct t to v l0 ad v l. Lt ),

13 v 3 c v k a v k a v k0 a v c s c s c v c v s0 b v s b v s b Fgur 9: Th graph D 3 c corrspodg to g 3 c x l x s x u k = 0 coctd to graphs corrspodg to x l x l + = 0. b l+ v c {,} {,} v c b v l a {,} b l Fgur 0: Th graph corrspodg to x l x l = 0 b l+ b th vrtx corrspodg to th crcl C l+. W draw a arc from v l0 to b l+. Fally, w coct th vrtx v{,} l0 to b l+. Rcall that x also occurs th quato gc 3, whch s a quato wth thr varabls H. Ths costructo s llustratd Fgur 0, whr oly a part of th corrspodg graph Dc 3 s dpctd. Lt C b th last crcl H. Th, w st b + = s as s s th startg vrtx of th graph D 3 corrspodg to th quato g. 3 Ths s th whol dscrpto of th graph D H. Nxt, w ar gog to dscrb th assocatd tour σ φ D H gv a assgmt to th varabls H. 5.3 Costructg th Tour σ φ from a Assgmt φ Lt H b a stac of th Hybrd problm ad D H = (V H, A H ) th corrspodg stac of th (, )-ATSP problm as dfd Scto 5.. Gv a assgmt 3

14 φ : V (H) {, 0} to th varabls of H, w ar gog to costruct th assocatd Hamltoa tour σ φ D H. I addto to t, w aalyz th rlato btw th lgth of th tour σ φ ad th umbr of satsfd quatos by φ. Lt H b a stac of th Hybrd problm cosstg of crcls C, C,..., C m ad quatos wth thr varabls g 3 wth [m 3 ]. Th assocatd Hamltoa tour σ φ D H starts at th vrtx b. From a hgh lvl vw, σ φ travrss all graphs corrspodg to th quatos assocatd wth th crcl C dg wth th vrtx b. Succssvly, t passs all graphs for ach crcl H utl t rachs th vrtx b m = s as s s th startg vrtx of th graph D. 3 At ths pot, th tour bgs to travrs th rmag graphs Dc 3 whch ar smulatg th quatos wth thr varabls H. By ow, som of th party graphs apparg graphs Dc 3 alrady hav b travrsd th r loop of σ φ. Th outr loop chcks whthr for ach graph G 3 c, a v umbr of party graphs has b travrsd th r loop. I vry stuato, whch φ dos ot satsfy th udrlyg quato, th tour ds to us a -arc. Ths paths, whch ar coctd va -arcs, wll b algd by mas of -arcs ordr to buld a Hamltoa tour D H. For ach crcl C l, w us -arcs to obta a Hamltoa path from b l to b l+ travrsg all graphs assocatd wth C l som ordr xcpt th cas wh all varabls th crcl hav th sam party. Sc w spcfy oly a part of th tour σ φ w rathr rfr to a rprstatv tour from a st of tours havg th sam lgth ad th sam spcfcato. I ordr to aalyz th lgth of th tour rlato to th umbr of satsfd quato, w ar gog to xam th part of σ φ passg th graphs corrspodg to th udrlyg quato ad accout th local lgth to th aalyzd parts of th tour. Lt us bg to dscrb σ φ passg through party graphs assocatd to varabls H. Travrsg Party Graphs Lt x l b a varabl H. Th, th tour σ φ travrss th party graph P l usg th path v l[ φ(xl )] v l v lφ(xl ). I th rmadr, w call ths part of th tour a φ(x l )-travrsal of th party graph. I Fgur, w dpctd th corrspodg travrsals of th graph P l gv th assgmt φ(x l ), whrby th travrsd arcs ar llustratd by thck arrows. I both cass, w assocat th local lgth wth ths part of th tour. -travrsal of P l gv φ(x l ) =. 0-travrsal of P l gv φ(x l ) = 0. Fgur : Travrsal of th graph P l gv th assgmt φ.

15 Travrsg Graphs Corrspodg to Matchg Equatos Lt C l b a crcl H ad x l x l = 0 wth = {, } M l a matchg quato. Gv x x + = 0, x x = 0, x x + = 0 ad th assgmt φ, w ar gog to costruct a tour through th corrspodg party graphs dpdc of φ. W bg wth th cas φ(x ) φ(x + ) = 0, φ(x ) φ(x ) = 0 ad φ(x ) φ(x + ) = 0.. Cas φ(x ) φ(x + ) = 0, φ(x ) φ(x ) = 0 ad φ(x ) φ(x + ) = 0: I ths cas, w travrs th corrspodg party graphs as dpctd Fgur v l(+) + v l(+) Fgur :. Cas φ(x ) φ(x + ) = 0, φ(x ) φ(x ) = 0 ad φ(x ) φ(x + ) = v l(+) + v l(+) Fgur 3:. Cas φ(x ) φ(x + ) = 0, φ(x ) φ(x ) = ad φ(x ) φ(x + ) = 0. 5

16 I Fgur, w hav φ(x ) = φ(x + ) = φ(x ) = φ(x + ) =, whras Fgur, w hav φ(x ) = φ(x + ) = φ(x ) = φ(x + ) = 0. I both cass, ths part of th tour has local lgth 5.. Cas φ(x ) φ(x + ) = 0, φ(x ) φ(x ) = ad φ(x ) φ(x + ) = 0: Th tour σ φ s pcturd Fgur 3 ad. I th cas φ(x ) = φ(x + ) = 0 ad φ(x ) = φ(x + ) = dpctd Fgur 3, w ar forcd to tr ad lav th party graph P l va -arcs. So far, w assocat th local lgth 6 wth ths part of th tour. I Fgur 3, w hav φ(x ) = φ(x + ) = ad φ(x ) = φ(x + ) = 0. Ths part of th tour σ φ cotas o -arc yldg th local lgth Cas φ(x ) φ(x + ) = 0, φ(x ) φ(x ) = 0 ad φ(x ) φ(x + ) = : I dpdc of φ, w travrs th corrspodg party graphs th way as dpctd Fgur v l(+) + v l(+) Fgur : 3. Cas φ(x ) φ(x + ) = 0, φ(x ) φ(x ) = 0 ad φ(x ) φ(x + ) =. Th stuato, whch φ(x ) = φ(x + ) = 0 ad φ(x ) = 0 φ(x + ) holds, s dpctd Fgur. O th othr had, f w hav φ(x ) = φ(x + ) = ad φ(x ) = φ(x + ), th tour s pcturd Fgur. I both cass, w assocat th local lgth 6.. Cas φ(x ) φ(x + ) = 0, φ(x ) φ(x ) = ad φ(x ) φ(x + ) = : Th tour σ φ s dsplayd Fgur 5. I Fgur 5, w ar gv φ(x ) = φ(x + ) = 0 ad φ(x ) = φ(x + ) =, whras Fgur 5, w hav φ(x ) = φ(x + ) = ad φ(x ) = φ(x + ) = 0. I both cass, w assocat th local lgth 6 wth ths part of th tour. 6

17 + + + v l(+) + v l(+) Fgur 5:. Cas φ(x ) φ(x + ) = 0, φ(x ) φ(x ) = ad φ(x ) φ(x + ) =. 5. Cas φ(x ) φ(x + ) =, φ(x ) φ(x ) = ad φ(x ) φ(x + ) = : I ths cas, w travrs th corrspodg party graphs as dpctd Fgur v l(+) + v l(+) Fgur 6: 5. Cas φ(x ) φ(x + ) =, φ(x ) φ(x ) =, φ(x ) φ(x + ) =. I th lft had sd of Fgur 6, w st φ(x ) = φ(x + ) = 0 φ(x ) = φ(x + ) =, whras, w hav φ(x ) = φ(x + ) = ad φ(x ) = φ(x + ) = 0. Ths part of th tour has local lgth 7. 7

18 6. Cas φ(x ) φ(x + ) =, φ(x ) φ(x ) = 0 ad φ(x ) φ(x + ) = : I ths cas, w travrs th corrspodg party graphs as dpctd Fgur 7. I th lft had sd of Fgur 7, w hav φ(x ) = φ(x + ) = 0 ad v l(+) v l v+ l0 v l(+) Fgur 7: 6. Cas φ(x ) φ(x + ) =, φ(x ) φ(x ) = 0, ad φ(x ) φ(x + ) =. φ(x ) = φ(x + ) =, whras, w hav φ(x ) = φ(x + ) = ad φ(x ) = φ(x + ) = 0. Ths part of th tour has local lgth Cas φ(x ) φ(x + ) =, φ(x ) φ(x ) = 0 ad φ(x ) φ(x + ) = 0: + v+ l0 + v l(+) + v l(+) Fgur 8: 7. Cas φ(x ) φ(x + ) =, φ(x ) φ(x ) = 0 ad φ(x ) φ(x + ) = 0. I ths cas, w travrs th corrspodg party graphs as dpctd Fgur 8. 8

19 I Fgur 8, w hav φ(x ) = φ(x + ) = 0 ad φ(x ) = φ(x + ) =, whras, w hav φ(x ) = φ(x + ) = ad φ(x ) = φ(x + ) = 0. Ths part of th tour has local lgth Cas φ(x ) φ(x + ) =, φ(x ) φ(x ) = ad φ(x ) φ(x + ) = 0: I th fal cas, w travrs th corrspodg party graphs as dpctd Fgur v l(+) + v l(+) Fgur 9: 8. Cas φ(x ) φ(x + ) =, φ(x ) φ(x ) = ad φ(x ) φ(x + ) = 0. I Fgur 9, w st φ(x ) = φ(x + ) = 0 ad φ(x ) = φ(x + ) =, whras, w hav φ(x ) = φ(x + ) = ad φ(x ) = φ(x + ) = 0. Ths part of th tour has local lgth 6. Our aalyss ylds th followg proposto. Proposto 3. Lt x l x l + = 0, x l x l = 0 x l x l + = 0 b quatos H. Gv a assgmt φ to th varabls H, th assocatd tour σ φ has local lgth at most 5 + u, whr u dots th umbr of usatsfd quatos by φ. Travrsg Graphs Corrspodg to Equatos wth Thr Varabls Lt gc 3 x l x s x k t = 0 b a quato wth thr varabls H. Furthrmor, lt x l x l + = 0, x s x s + = 0 ad x k t x k t+ = 0 b crcl quatos H. For otatoal smplcty, w troduc = {, +}, a = {t, t+} ad b = {, +}. I Fgur 0, w dsplay th costructo volvg th graphs Dc, 3 Pa k, Pb s, P, l P l ad P+. l Exmplary, w dpctd th coctos of th graphs Dc, 3 P, l P l ad P+ l ths fgur. W ar gog to costruct th tour σ φ travrsg th corrspodg graphs ad aalyz th dpdcy of th local lgth of σ φ ad th umbr of satsfd quatos, amly x l x s x k t = 0, x l x l + = 0, x s x s + = 0 ad x k t x k t+ = 0. 9

20 v 3 c v k a v k a v k0 a v c s c s c v c v s0 b v s b v s b Fgur 0: Th graph D 3 c wth ts coctos to P l ad P l +. Rcall from Proposto that thr s a Hamltoa path from s c to s c+ cotag th vrtcs v c, v c ad v 3 c G 3 c f ad oly f a v umbr of party graphs P {P k a, P s b, P l } s travrsd. Th outr loop travrss th graph G 3 c startg at s c ad dg at s c+. Furthrmor, t cotas th vrtcs v c, v c ad v 3 c som ordr. If σ φ travrss a v umbr of party graphs P {P k a, P s b, P l } th r loop, t s possbl to costruct a Hamltoa local lgth wth ths part. I th othr cas, w hav to us a -arc yldg th local lgth. v 3 c v c Fgur : A part of th graphs corrspodg to x l x l + = 0 ad x l x s x k t = 0. Lt us aalyz th part of σ φ travrsg graphs corrspodg to x l x l + = 0. For ths raso, w wll xam th stuato dpctd Fgur. Lt us bg wth th cas φ(x l ) φ(x l +) = 0.. Cas φ(x l ) φ(x l +) = 0 : If φ(x l ) = φ(x l +) = holds, th tour σ φ uss th arc (v l, v+). l0 Aftrwards, th party graph P l wll b travrsd wh th tour lads through th graph G 3 c. Mor prcsly, t wll us th path vc 3 v l0 v l v l vc. I Fgur, w llustratd ths part of th tour. 0

21 v 3 c v c Fgur : Cas φ(x l ) = φ(x l +) =. I th othr cas φ(x l ) = φ(x l +) = 0, w us th path v+ l0 v l v l. Aftrwards, th tour σ φ cotas th arc (vc, vc 3 ). I both cass, w assocat th local lgth wth ths part of th tour.. Cas φ(x l +) φ(x l +) = : Assumg φ(x l ) φ(x l +) =, th tour σ φ uss a -arc trg v l ad th path v l. Furthrmor, w d aothr -arc ordr to tr v+. l0 v l v l v l0 Th stuato s dpctd Fgur 3. v 3 c v c Fgur 3: Cas φ(x l ) φ(x l +) =. I th othr cas, amly φ(x l ) φ(x l +) = 0, w us -arcs lavg v+ l0 ad v l rspctvly. Aftrwards, th tour uss th path vc 3 v l0 v l v l vc whl travrsg th graph G 3 c. I both cass, w assocat th local lgth wth ths part of th tour. W obta th followg proposto. Proposto. Lt x l x s x k t = 0 b a quato wth thr varabls H. Furthrmor, lt x l x l + = 0, x s x s + = 0 ad x k t x k t+ = 0 b crcl quatos H. Gv a assgmt φ to th varabls H, th assocatd tour σ φ has local lgth at most u, whr u dots th umbr of usatsfd quatos by φ.

22 Travrsg Graphs Corrspodg to Crcl Bordr Equatos Lt C l b a crcl H ad x l x l = 0 ts crcl bordr quato. Rcall that th varabl x l s also cludd a quato wth thr varabls. W ar gog to dscrb th part of th tour passg through th graphs dpctd Fgur dpdc of th assgd valus to th varabls x l ad x l. Lt us start wth th cas φ(x l ) φ(x l ) = 0. b l+ v 3 c {,} {,} v c {,} b v l a b l Fgur : Travrsg Graphs Corrspodg To Crcl Bordr Equatos.. Cas φ(x l ) φ(x l ) = 0 Th startg pot of th tour σ φ passg through th graph corrspodg to x l x l = 0 s th vrtx b l. Gv th valus φ(x l ) = φ(x l ), w us ach cas th φ(x l )-travrsal of th party graphs P l ad P l dg b l l+. Not that th cas φ(x l ) = φ(x l ) = 0, w us th -travrsal of th party graph P l {,}. Exmplary, w dsplay th stuato φ(x l ) = φ(x l ) = Fgur 5. b l+ v 3 c {,} {,} v c {,} b v l a b l Fgur 5: Cas φ(x l ) = φ(x l ) =. I both cass, w assocatd th local lgth wth ths part of th tour.

23 . Cas φ(x l ) φ(x l ) = Gv th assgmt φ(x l ) φ(x l ) = 0, w travrs th arc (b l, v l0 ). Du to th costructo, w hav to us -arcs to tr b l+ ad v l as dpctd Fgur 6. b l+ v 3 c {,} {,} v c b v l a {,} b l Fgur 6: Cas φ(x l ) φ(x l ) = 0. I th othr cas, w hav to us a -arc ordr to lav th vrtx b l. I addto, th tour cotas th path v l v{,} l0 vl {,} vl {,} b l+. Hc, both cass, w assocat th local lgth 3 wth ths part of σ φ. W obta th followg proposto. Proposto 5. Lt x l x l = 0 b a crcl bordr quatos H. Gv a assgmt φ to th varabls H, th assocatd tour σ φ has local lgth f th quato x l x l = 0 s satsfd by φ ad othrws, th local lgth Costructg th Assgmt ψ σ from a Tour σ Lt H b a stac of th Hybrd problm, D H = (V H, A H ) th assocatd stac of th (, )-ATSP problm ad σ a tour D H. W ar gog to df th corrspodg assgmt ψ σ : V (H) {0, } to th varabls H. I addto to t, w stablsh a cocto btw th lgth of σ ad th umbr of satsfd quatos by ψ σ. Lt us df th corrspodg assgmt ψ σ gv a tour σ D H. Dfto ( Assgmt ψ σ ). Lt H b a stac of th Hybrd problm, D H = (V H, A H ) th assocatd stac of th (, )-ATSP problm. Gv a tour σ D H, whch all party graphs ar cosstt wth rspct to σ, th corrspodg assgmt ψ σ : V (H) {0, } s dfd as follows. ψ σ (x l ) = f σ uss a -travrsal of P l = 0 othrws I ordr to obta a wll-dfd assgmt, w hav to trasform th udrlyg tour σ. 3

24 Cosstcy of Party Graphs Frst of all, w troduc th oto of a cosstt tour. Dfto 3 (Cosstt Tour). Lt H b a stac of th Hybrd problm ad D H th assocatd stac of th (, ) ATSP problm. A tour D H s calld cosstt f th tour uss oly 0/-travrsals of all D H cotad party graphs. Du to th followg proposto, w may assum that th udrlyg tour s cosstt. Proposto 6. Lt H b a stac of th Hybrd problm ad D H th assocatd stac of th (, ) ATSP problm. Ay tour σ D H ca b trasformd polyomal tm to a cosstt tour σ wth l(σ ) l(σ ). Proof. For vry party graph cotad D H, t ca b s by cosdrg all possblts xhaustvly that ay tour D H that s ot usg th corrspodg 0/-travrsals ca b modfd to a tour wth at most th sam umbr of -arcs. Th lss obvous cass ar show Fgur 5 (s 0. Fgur Appdx). Lt us start wth th aalyss. I th rmadr, w assum that th udrlyg (, )-tour σ s cosstt wth all party graphs D H. Trasformg σ Graphs Corrspodg to Matchg Equatos Gv th quatos x x + = 0, x x = 0, x x + = 0 ad a tour σ, w ar gog to costruct a assgmt dpdc of σ. I partcular, w aalyz th rlato btw th lgth of th tour ad th umbr of satsfd quatos by ψ σ.

25 x z y + x z y + + v l(+) v + v l(+) v Fgur 7:.Cas ψ σ (x ) ψ σ (x + ) = 0, ψ σ (x ) ψ σ (x ) = 0 & ψ σ (x ) ψ σ (x + ) = 0.. Cas ψ σ (x ) ψ σ (x + ) = 0, ψ σ (x ) ψ σ (x ) = 0 ad ψ σ (x ) ψ σ (x + ) = 0: Gv ψ σ (x ) = ψ σ (x ) = ψ σ (x ) = ψ σ (x + ) =, t s possbl to trasform th udrlyg tour such that o -arcs tr or lav th vrtcs v l, v+, l0 v+, l0, ad v l. Exmplary, w dsplay Fgur 7 such a trasformato, whrby v l(+) Fgur 7 ad Fgur 7 llustrat th udrlyg tour σ ad th trasformd tour σ, rspctvly. Th cas ψ σ (x ) = ψ σ (x + ) = ψ σ (x ) = ψ σ (x + ) = 0 ca b dscussd aalogously. I both cass, w obta th local lgth 5 for ths part of σ whl ψ σ satsfs all 3 quatos.. Cas ψ σ (x ) ψ σ (x + ) = 0, ψ σ (x ) ψ σ (x ) = ad ψ σ (x ) ψ σ (x + ) = 0: Assumg ψ σ (x ) = ψ σ (x + ) = ad ψ σ (x ) = ψ σ (x + ) = 0, w ar abl to trasform th tour such that t uss th arcs (v l, v+) l0 ad (v+, l0 v l ). Du to th costructo ad our assumpto, th tour caot travrs th arcs (v l, ), (, v l ), (v l(+), v+) l0 ad (v+, l0 v l(+) ). Cosqutly, w hav to us -arcs trg ad lavg th party graph P. l Th stuato s dsplayd Fgur 8. W assocat oly th cost of o -arc yldg th local lgth 6, whch corrspods to th fact that ψ σ lavs th quato x x = 0 usatsfd. Not that a smlar stuato holds cas of ψ σ (x ) = ψ σ (x + ) = 0 ad ψ σ (x ) = ψ σ (x + ) = (cf. Fgur 8 ). 3. Cas ψ σ (x ) ψ σ (x + ) = 0, ψ σ (x ) ψ σ (x ) = 0 ad ψ σ (x ) ψ σ (x + ) = : Lt us start wth th cas ψ σ (x ) = ψ σ (x + ) = ad ψ σ (x ) ψ σ (x + ) = 0. Th stuato s dsplayd Fgur 9. 5

26 + + + v l(+) + v l(+) Fgur 8:.Cas ψ σ (x ) ψ σ (x + ) = 0, ψ σ (x ) ψ σ (x ) = & ψ σ (x ) ψ σ (x + ) = 0. Du to th costructo, w ar abl to trasform σ such that t uss th arc (v l, v+). l0 Not that th tour caot travrs th arcs (v l(+), v+) l0 ad (v+, l0 v l ) v l(+) + v l(+) Fgur 9: 3.Cas ψ σ (x ) ψ σ (x + ) = 0, ψ σ (x ) ψ σ (x ) = 0 & ψ σ (x ) ψ σ (x + ) =. Hc, w ar forcd to us two -arcs crasg th cost by. All all, w obta th local lgth 6. 6

27 + + + v l(+) v l v+ l0 v l(+) Fgur 30:. Cas ψ σ (x ) ψ σ (x + ) =, ψ σ (x ) ψ σ (x ) = 0 & ψ σ (x ) ψ σ (x + ) =. Th cas ψ σ (x ) = ψ σ (x + ) = 0 ad ψ σ (x ) ψ σ (x + ) = ca b aalyzd aalogously (cf. Fgur 9 ). A smlar argumtato holds for ψ σ (x ) ψ σ (x + ) =, ψ σ (x ) ψ σ (x ) = 0 ad ψ σ (x ) ψ σ (x + ) = 0.. Cas ψ σ (x ) ψ σ (x + ) =, ψ σ (x ) ψ σ (x ) = 0 ad ψ σ (x ) ψ σ (x + ) = : Gv ψ σ (x ) ψ σ (x + ) = 0 ad ψ σ (x ) ψ σ (x + ) = 0, w ar abl to trasform th tour such that t uss th arc (v+, l0 v l(+) ). Ths stuato s dpctd Fgur 30. Notc that w ar forcd to us four -arcs ordr to coct all vrtcs. Cosqutly, t ylds th local lgth 7. Th cas, whch ψ σ (x ) ψ σ (x + ) = 0 ad ψ σ (x ) ψ σ (x + ) = 0 holds, s dsplayd Fgur 30 ad ca b dscussd aalogously. 7

28 + z v l(+) + x + z v l(+) + x Fgur 3: 5.Cas wth ψ σ (x ) = ψ σ (x + ) = ad ψ σ (x ) ψ σ (x + ) =. 5. Cas ψ σ (x ) ψ σ (x + ) = 0, ψ σ (x ) ψ σ (x ) = ad ψ σ (x ) ψ σ (x + ) = : Lt th tour σ b charactrzd by ψ σ (x ) = ψ σ (x + ) = ad ψ σ (x ) ψ σ (x + ) =. Th, w trasform σ such a way that w ar abl to us th arc (v l, v+). l0 Th corrspodg stuato s llustratd Fgur 3. I addto to t, w travrs th party graph P l th othr drcto ad obta ψ σ (x ) =. Ths trasformato ducs a tour wth at most th sam cost. O th othr had, th corrspodg assgmt ψ σ satsfs at last mor quatos sc x l x l = 0 mght gt usatsfd. I ths cas, w assocat th local costs of 6 wth σ. I th othr cas, whch ψ σ (x ) = ψ σ (x + ) = 0 ad ψ σ (x ) ψ σ (x + ) = 0 holds, w argu smlarly. Th trasformato s dpctd Fgur 3. 8

29 + + z v l(+) x v lk + v l(+) v lk + z x Fgur 3: 5.Cas wth ψ σ (x ) = ψ σ (x + ) = 0 ad ψ σ (x ) ψ σ (x + ) = Cas ψ σ (x ) ψ σ (x + ) =, ψ σ (x ) ψ σ (x ) = ad ψ σ (x ) ψ σ (x + ) = : Gv a tour σ wth ψ σ (x ) ψ σ (x + ) = ad ψ σ (x ) ψ σ (x + ) = 0, w trasform σ such that t travrss th party graph P l th oppost drcto mag ψ σ (x ) = 0 (cf. Fgur 33). Ths trasformato abls us to us th arc (v+, l0 v l ). Furthrmor, t ylds at last o mor satsfd quato H. I ordr to coct th rmag vrtcs, w ar forcd to us at last two -arcs. I summary, w assocat th local lgth 7 wth ths stuato coformty wth th at most usatsfd quatos by ψ σ. If w ar gv a tour σ wth ψ σ (x ) ψ σ (x + ) = 0 ad ψ σ (x ) ψ σ (x + ) =, w obta th stuato dsplayd Fgur 3. By applyg local trasformatos wthout crasg th lgth of th udrlyg tour, w achv th scaro dpctd Fgur 3. W argu that th assocatd local lgth of th tour s 7. 9

30 + + v l(+) + + v l(+) Fgur 33: 6.Cas wth ψ σ (x ) ψ σ (x + ) = ad ψ σ (x ) ψ σ (x + ) = 0. Th cas, whch ψ σ (x ) ψ σ (x + ) =, ψ σ (x ) ψ σ (x ) = ad ψ σ (x ) ψ σ (x + ) = 0 holds, ca b dscussd aalogously. 30

31 + v v l(+) + l0 v l + v l(+) + Fgur 3: 6.Cas wth ψ σ (x ) ψ σ (x + ) = 0 ad ψ σ (x ) ψ σ (x + ) =. W obta th followg proposto. Proposto 7. Lt x l x l + = 0, x l x l = 0 ad x l x l + = 0 b quatos H. Th, t s possbl to trasform polyomal tm th gv tour σ passg through th graph corrspodg to x l x l + = 0, x l x l = 0 x l x l + = 0 such that t has local lgth 5 + u, whr u dots th umbr of usatsfd quato by ψ σ. Trasformg σ Graphs Corrspodg to Equatos Wth Thr Varabls Lt gc 3 x l x s x r k = 0 b a quato wth thr varabls H. Furthrmor, lt C l b a crcl H ad x l x l + = 0 a crcl quato. For otatoal smplcty, w st = {, + }. W ar gog to aalyz th th umbr of satsfd quatos by ψ σ dpdc to th local lgth of σ th graphs P l, P l Frst, w trasform th tour travrsg th graphs P l, P l th ψ σ (x l )-travrsal of P. l +, P l ad Dc. 3 + ad P l such that t uss 3

32 Aftrwards, du to th costructo of D 3 c ad Proposto, th tour ca b trasformd such that t has local lgth of f t passs a v umbr of party graphs P {P l, P k a, P s b } whl usg a smpl path through D 3 c. Othrws, t ylds a local lgth of 3+. Lt us start wth th cas ψ σ (x l ) = ad ψ σ (x l +) =. v c v c v c v c Fgur 35:. Cas ψ σ (x l ) = ad ψ σ (x l +) = 3

33 v c v c v c v c Fgur 36:. Cas ψ σ (x l ) = 0 ad ψ σ (x l +) = 0. Cas ψ σ (x l ) = ad ψ σ (x l +) = : I Fgur 35 ad, w dsplay th tour passg through P l, P+ l ad P l wth ψ σ (x l ) = ad ψ σ (x l +) = bfor ad aftr th trasformato, rspctvly. It s possbl to trasform th tour σ wthout crasg th lgth such that t travrss th arc (v l, v+). l0 I th outr loop, th tour wll us at last o of th arcs (vc 3, v l0 ) ad (v l, vc 3 ) dpdg o th party chck of Dc. 3 W assocat th local lgth wth ths part of th tour.. Cas ψ σ (x l ) = 0 ad ψ σ (x l +) = 0 : I Fgur 36, w dsplay th udrlyg scaro wth ψ σ (x l ) = 0 ad ψ σ (x l +) = 0. Th trasformd tour uss th 0-travrsal of th party graph P. l Th vrtcs vc 3 ad vc ar coctd va a -arc. W assg th local lgth to ths part of th tour. 3. Cas ψ σ (x l ) = ad ψ σ (x l +) = 0 : Lt us assum that ψ σ (x l ) ψ σ (x s ) ψ σ (x r k) = 0 holds. Hc, t s possbl to trasform th tour such that t uss th path vc 3 v l0 v l v l vc ad thus, th 0-travrsal of th party graph P l as dsplayd Fgur 37. I th othr cas, amly ψ σ (x l ) ψ σ (x s ) ψ σ (x r k) =, w wll chag th valu of ψ σ (x l ) achvg ths way at last mor satsfd quato. Lt us xam th scaro Fgur 38. Accordgly, th tour uss th 0-travrsal of th party graph P, l whch abls σ to pass th party chck Dc. 3 I both cass, w obta th local lgth coformty wth th at most o usatsfd quato by ψ σ. 33

34 v c v c x z v c v c z x Fgur 37: 3. Cas ψ σ (x l ) =, ψ σ (x l +) = 0 ad ψ σ (x l ) ψ σ (x s ) ψ σ (x r k) = 0 v c v c y x v c v c y x Fgur 38: 3. Cas ψ σ (x l ) =, ψ σ (x l +) = 0 ad ψ σ (x l ) ψ σ (x s ) ψ σ (x r k) =. 3

35 v c v c x + y + + v c v c x + y + + Fgur 39:. Cas ψ σ (x l ) = 0 ad ψ σ (x l +) =. Cas ψ σ (x l ) = 0 ad ψ σ (x l +) = : Assumg ψ σ (x l ) ψ σ (x l ) ψ σ (x l ) = 0 ad th scaro dpctd Fgur 39, th tour wll b modfd such that th party graphs P l ad P l ar travrsd th sam drcto. Sc w hav ψ σ (x l ) ψ σ (x l ) ψ σ (x l ) = 0 w ar abl to ucoupl th party graph P l from th tour through G 3 c wthout crasg ts lgth. W dsplay th trasformd tour Fgur 39. Assumg ψ σ (x l ) ψ σ (x l ) ψ σ (x l ) = ad th scaro dpctd Fgur 0, w trasform σ such that th party graph P l s travrsd wh σ s passg through Dc 3 mag vc 3 v l0 v l v l vc s a part of th tour. I addto, w chag th valu of ψ σ (x l ) yldg at last mor satsfd quatos. Th trasformd tour s dsplayd Fgur 0. I both cass, w assocat th local lgth wth σ. O th othr had, ψ σ lavs at most o quato usatsfd. W obta th followg proposto. Proposto 8. Lt g 3 c x l x s x r k = 0 b a quato wth thr varabls H. Furthrmor, lt x l x l + = 0 x s x s + = 0 ad x r k x r k+ = 0 b crcl quatos H. Th, t s possbl to trasform polyomal tm th gv tour σ passg through th graph corrspodg to x l x l + = 0 x s x s + = 0, x r k x r k+ = 0 ad g 3 c such that t has local lgth u, whr u s th umbr of usatsfd quato by ψ σ. 35

36 v c v c x + y + + v c v c + x + y + Fgur 0: Cas ψ σ (x l ) = 0, ψ σ (x l +) = ad ψ σ (x l ) ψ σ (x s ) ψ σ (x r k) = Trasformg σ Graphs Corrspodg to Crcl Bordr Equatos Lt C l b a crcl H ad x l x l = 0 ts crcl bordr quato. Furthrmor, lt gc 3 x l x s x r k = 0 b a quato wth thr varabls cotad H. W ar gog to trasform a gv tour σ passg through th graph corrspodg to x l x l = 0 such that t wll hav th local lgth + u, whr u s th umbr of usatsfd quato. Lt us bg wth th aalyss startg wth th cas ψ σ (x ) = 0 ad ψ σ (x ) = 0. I ach cas, w modfy σ such that σ uss a ψ(x l )-travrsal of P{,} l. Aftrwards, σ wll b chckd D3 c whthr t passs th party tst.. Cas ψ σ (x ) = 0 ad ψ σ (x ) = 0: Lt us assum that ψ σ lavs gc 3 usatsfd mag ψ σ (x l ) ψ σ (x s ) ψ σ (x r k) =. I addto to t, w assum that th path vc 3 v{,} l vl {,} vl0 {,} v c s a part of σ. Ths mas that σ fals th party chck Dc 3 f vc 3 v{,} l vl {,} vl0 {,} v c s ot usd by σ. Frst, w modfy th tour such that t cluds th arc (b l, v l0 ). For th sam raso, w may assum that v l ad b l+ s coctd va a -arc. W obta th scaro dpctd Fgur. 36

37 b l+ v 3 c {,} {,} v c b {,} v l a b l b l+ v 3 c {,} {,} v c {,} b v l a b l Fgur : Cas ψ σ (x ) = 0 ad ψ σ (x ) = 0 As for th xt stp, w trasform σ such that t cotas th arcs (b l, v l ) ad (v l0, b l+ ). Cosqutly, w us th -travrsal of th party graph P{,} l ad coct vc 3 ad vc va a -arc. Th modfd tour s dpctd Fgur. If ψ σ satsfs gc 3 ad σ cotas th path vc 3 v{,} l vl {,} vl0 {,} v c, w modfy σ Dc 3 such that t passs th party tst Dc 3 ad cotas th arc (vc, vc 3 ). I both cass, w assocat th local lgth wth ths part of σ.. Cas ψ σ (x ) = ad ψ σ (x ) = : Lt us assum that ψ σ (x l ) ψ σ (x s ) ψ σ (x r k) = holds ad σ cotas th arc (vc, vc 3 ). Gv ths scaro, w may assum that (b l, v l ) s cotad σ du to a smpl modfcato. Th, w ar gog to aalyz th stuato dpctd Fgur. W trasform σ th way dscrbd Fgur. Aftrwards, σ wll b modfd Dc 3 such that t uss a smpl path Dc 3 falg th party chck. Th cas, whch ψ σ (x l ) ψ σ (x l ) ψ σ (x l ) = 0 holds ad σ cotas th arc (vc, vc 3 ), ca b dscussd smlarly sc σ passs th party chck by cludg th path vc 3 v{,} l vl {,} vl0 {,} v c. 37

38 y b l+ v 3 c v{,} l {,} vl {,} v c x b v l a b l y b l+ v 3 c v{,} l {,} vl {,} v c x b v l a b l Fgur :. Cas ψ σ (x ) = ad ψ σ (x ) = I both cass, w assocat th lgth wth ths part of σ. 38

39 b l+ v 3 c {,} v c b b l v l a {,} y {,} x b l+ v 3 c {,} v c b b l v l a {,} y {,} x b l+ v 3 c {,} {,} v c (c) b b l v l a y {,} x Fgur 3: 3. Cas ψ σ (x ) = 0 ad ψ σ (x ) = 3. Cas ψ σ (x ) = 0 ad ψ σ (x ) = : Lt us assum that ψ σ (x l ) ψ σ (x s ) ψ σ (x r k) = holds ad σ travrss th path vc 3 v{,} l vl {,} vl0 {,} v c. Th, w trasform th tour σ such that t cotas th arc (v l0, b l+ ). Not that thr (b l, v l0 ) or (b l, v l ) s cludd th tour. Hc, σ cotas a -arc to coct b l. Th sam holds for th vrtx v l. Ths stuato s dsplayd Fgur 3. W ar gog to vrt th valu of ψ σ (x l ) such that ψ σ satsfs gc 3 ad x l x l = 0. I ths way, w ga at last mor satsfd quatos. Th corrspodg trasformato s pcturd Fgur 3. 39

40 O th othr had, f w assum that ψ σ (x l ) ψ σ (x s ) ψ σ (x r k) = 0 holds ad σ travrss th path vc 3 v{,} l vl {,} vl0 {,} v c, w modfy th tour as dpctd Fgur 3 (c). Not that σ passs th party chck Dc 3 ad thrfor, th tour may us a smpl path Dc. 3 W assocat th local lgth 3 wth σ ths cas. v 3 c {,} v{,} l v{,} l0 v c b l+ x b v l a y b l v 3 c {,} v{,} l v{,} l0 v c b l+ x b v l a y b l v{,} l v{,} l v{,} l0 b l+ v 3 c v c x b v l a (c) y b l Fgur :. Cas ψ σ (x ) = ad ψ σ (x ) = 0. Cas ψ σ (x ) = ad ψ σ (x ) = 0: Lt us assum that ψ σ (x l ) ψ σ (x l ) ψ σ (x l ) = holds ad σ uss th arc (vc, vc 3 ). Th, w trasform th tour σ such that t cotas th arc (b l, v l ). W ot that thr (v l0, b l+ ) or (v l, v{,} l0 ) s cludd th tour. For ths raso, σ must us a -arc to coct v{,} l0. Th sam holds for th vrtx vl0. Th corrspodg stuato s dsplayd Fgur. 0

41 W modfy th tour as dsplayd Fgur ad obta at last mor satsfd quatos. O th othr had, f w assum that ψ σ (x l ) ψ σ (x l ) ψ σ (x l ) = 0 holds, w d to clud th path vc 3 v{,} l vl {,} vl0 {,} v c as dpctd Fgur (c). I both cass, w assocat th local lgth 3 wth ths part of th tour. Proposto 9. Lt C l b a crcl H ad x l x l = 0 ts crcl bordr quato. Th, t s possbl to trasform polyomal tm th gv tour σ passg through th graph corrspodg to x l x l = 0 such that t has local lgth, f x l x l = 0 s satsfd by ψ σ ad othrws, local lgth Proof of Thorm 5 () Lt H b a stac of th Hybrd problm cosstg of crcls C,..., C, m quatos wth two varabls ad m 3 quatos wth thr varabls gc 3 wth c [m 3 ]. Th, w costruct polyomal tm th corrspodg stac D H = (V (D H ), A(D H )) of th (, )-ATSP problm as dscrbd Scto 5.. () Lt φ b a assgmt to th varabls H lavg u quatos H usatsfd. Accordg to Proposto 3 5, w t s possbl to costruct polyomal tm th tour σ φ wth lgth l(σ φ ) 3 m + ( + 3 3) m u. () Lt σ b a tour D H wth lgth l(σ) = 3 m + 3 m u. Du to Proposto 6 w may assum that σ uss oly 0/-travrsals of vry party graph cludd D H. Accordg to Dfto, w assocat th corrspodg assgmt ψ σ wth th udrlyg tour σ. Rcall from Proposto 7 9 that t s possbl to covrt σ polyomal tm to a tour σ wthout crasg th lgth such that ψ σ lavs at most u quatos H usatsfd. 6 Approxmato Hardss of th (, )-ATSP Problm I ordr to prov th clamd hardss rsults for th (, )-ATSP problm, w us th sam costructo dscrbd Scto 5. wth th dffrc that all arcs party graphs hav wght, whras all othr arcs cotad th drctd graph D H obta th wght. Th ducd asymmtrc mtrc spac (V H, d H ) s gv by V H = V (D H ) ad dstac fucto dfd by th shortst path mtrc D H boudd by th valu.

42 I othr words, gv x, y V H, th dstac btw x ad y V H s d H (x, y) = m{lgth of a shortst path from x to y D H, }. Th oly dffculty that rmas s to prov that tours rma cosstt. Thus, w hav to prov that gv a tour σ V H, w ar abl to trasform σ polyomal tm to a tour σ, whch uss oly 0/-travrsals party graphs cotad D H, wthout crasg l(σ). Ths statmt ca b provd by cosdrg all possblts xhaustvly. Som cass ar dsplayd Fgur 5 Fgur 5. W ar rady to gv th proof of Thorm 5 (). 6. Proof of Thorm 5 () Gv H a stac of th Hybrd problm cosstg of crcls C,..., C, m quatos wth two varabls ad m 3 quatos wth thr varabls gc 3 wth c [m 3 ], w costruct polyomal tm th assocatd stac (V H, d) of th (, )- ATSP problm. Gv a assgmt φ to th varabls of H lavg u quatos usatsfd H, th thr xsts a tour wth lgth at most m (+)+m 3 (3 + )+ u+(+). O th othr had, f w ar gv a tour σ V H wth lgth m + 0m u, t s possbl to trasform σ polyomal tm to a tour σ wthout crasg th lgth such that th assocatd assgmt ψ σ lavs at most u quatos H usatsfd. 7 Approxmato Hardss of th (, )-TSP Problm I ordr to prov Thorm 6 (), w apply th rducto mthod usd Scto 5 to th (, )-TSP problm. As for th party gadgt, w us th graph dpctd Fgur 5 wth ts corrspodg travrsals. Th travrsd dgs ar pcturd by thck ls. Th party graph P l -travrsal of P l. 0-travrsal of P l. Fgur 5: Travrsal of th graph P l gv th assgmt φ. Lt H b a stac of th hybrd problm. Gv a matchg quato x l x l = 0 H ad th corrspodg crcl quatos x l x l + = 0 ad x l x l + = 0, w coct th assocatd party graphs P l, P+, l P{,} l, P l ad P+ l as dsplayd Fgur 6.

43 P l P l + P l {,} P l + P l Fgur 6: Graphs corrspodg to quatos x l x l = 0, x l x l + = 0 & x l x l + = 0. For quatos wth thr varabls g 3 c x y z = 0 H, w us th graph G 3 c dpctd Fgur 7. Rcall from Proposto that thr s a smpl path from s c to s c+ Fgur 7 cotag th vrtcs v {v c, v c } f ad oly f a v umbr of party graphs s travrsd. v c s c sc+ v c Fgur 7: Th graph G 3 c corrspodg to x y z = 0. Lt C l b a crcl H wth varabls {x l,..., x l }. For th crcl bordr quato of C l, w troduc th path p l = b l b l b 3 l. I addto, w coct b 3 l ad b l+ to th party graphs P l ad P l a smlar way as th rducto from th Hybrd problm to th (, ) ATSP problm. Ths s th whol dscrpto of th corrspodg graph G H = (V (G H ), E(G H )). W ar rady to gv th proof of Thorm 6 (). 3

44 7. Proof of Thorm 6 () Gv H a stac of th Hybrd problm cosstg of crcls C,..., C, m quatos wth two varabls ad m 3 quatos wth thr varabls g 3 c wth c [m 3 ], w costruct polyomal tm th assocatd stac G H of th (, ) TSP problm. Gv a assgmt φ to th varabls of H lavg u quatos usatsfd H, th, thr s a tour wth lgth at most 8 m + ( ) m O th othr had, f w ar gv a tour σ G H wth lgth 8 m +(3 8+3) m 3 +3+, t s possbl to trasform σ polyomal tm to a tour σ such that t uss 0/-travrsals of all cotad party graphs G H wthout crasg th lgth. Som cass ar dsplayd Fgur 50. Morovr, w ar abl to costruct polyomal tm a assgmt to th varabls of H, whch lavs at most u quatos H usatsfd. 8 Approxmato Hardss of th (, )-TSP Problm I ordr to prov th clamd approxmato hardss for th (, )-TSP problm, w caot us th sam party graphs as th costructo th prvous scto sc tours ar ot cssarly cosstt ths mtrc. For ths raso, w troduc th party graph dpctd Fgur 8 wth th corrspodg travrsals. Party graph P l -Travrsal of P l. 0-Travrsal of P l. Fgur 8: 0/-Travrsals of th graph P l. Gv a matchg quato x l x l = 0 H ad th crcl quatos x l x l + = 0 ad x l x l + = 0, w coct th corrspodg graphs as dsplayd Fgur 9. I ordr to df th w stac of th (, ) TSP problm, w rplac all party graphs G H by graphs dsplayd Fgur 8. I th rmadr, w rfr to ths graph as H H. All dgs cotad a party graph hav wght, whras all othr dgs hav wght. Th rmag dstacs th assocatd mtrc spac V H ar ducd by th graphcal mtrc H H boudd by th valu mag d H ({x, y}) = m{lgth of a shortst path from x to y H H, }. Ths s th whol dscrpto of th assocatd stac (V H, d H ) of th (, ) TSP problm. W ar rady to gv th proof of Thorm 6 ().