MATHEMATIC O OPERATION RESEARC Vo  No  Novembe  Prtnted m U

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SA INTEGE PROGRAMMIN WIT A FIXE NUMBE VARIABLES W LENSTRA JR Universiteit van Amsterdam i show tha th intege linea programmin proble wit a fixe numbe o variable i polynomiall solvable Th proo depend o method fro geometr o numbers Th integer linear pr ID: 28926 Download Pdf

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MATHEMATIC O OPERATION RESEARC Vo No Novembe Prtnted m U

SA INTEGE PROGRAMMIN WIT A FIXE NUMBE VARIABLES W LENSTRA JR Universiteit van Amsterdam i show tha th intege linea programmin proble wit a fixe numbe o variable i polynomiall solvable Th proo depend o method fro geometr o numbers Th integer linear pr

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MATHEMATIC O OPERATION RESEARC Vo No Novembe Prtnted m U




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MATHEMATIC O OPERATION RESEARC Vo 8 No 4 Novembe 198 Prtnted m U.S.A. INTEGE PROGRAMMIN WIT A FIXE NUMBE VARIABLES W LENSTRA JR Universiteit van Amsterdam i show tha th intege linea programmin proble wit a fixe numbe o variable i polynomiall solvable Th proo depend o method fro geometr o numbers Th integer linear programming problem i formulate ä follows Le n an m b positiv integers A a m X «-matri wit integra coefficients an b e T". Th questio i t decid whethe ther exist a vecto e / satisfyin th Syste o m inequalitie Ax < b. N algorith fo th solutio o thi proble i know whic

ha runnin tim tha i bounde b a polynomia functio o th length o th data Thi lengt may fo ou purposes b define t b n · m · log(i + 2) wher a denote th maximu o th absolut value o th coefficient o A an b. Indeed n suc polynomial algorithm i likel t exist sinc th proble i questio i NP-complete [3] [12] thi pape w conside th intege linea programmin proble wit a fixe valu n. I th cas n = l i i trivia t desig a polynomia algorith fo th solutio o th problem Fo n = 2 Hirschber an Won [5 an Kanna [6 hav give polynomia algorithm i specia cases A complet treatmen o th cas n = 2 wa give b Scar [10] I wa

conjecture [5] [10 tha fo an fixe valu o « ther exist polynomia algorith fo th solutio o th intege linea programmin problem I th presen pape w prov thi conjectur b exhibitin suc a algorithm. Th degre th polynomia b whic th runnin tim o ou algorith ca b bounde i a exponentia functio o n. Ou algorith i describe i §1 Usin tool fro geometr o number [1 w sho tha th proble ca b transforme int a equivalen on havin th followin additiona property eithe th existenc o a vecto Eil" satisfyin Ax < b i obvious o i i know tha th las coordinat o an suc belong t a interva whos lengt i bounde b a constan onl

dependin o n. I th latte case th proble i reduce t a bounde numbe o lowe dimensiona problems i th origina proble eac coordinat o i require t b i {0,1} n transformaüo o th proble i neede t achiev th conditio jus stated Thi suggest tha i thi cas ou algorith i equivalen t complet enumeration W remar tha th {0,1 linea programmin proble i TV/'-complete th genera cas w nee tw auxiliar algorithm fo th constructio o th require transformation Th firs o these whic "remodels th conve se (x e R" Ax < b}, i give i §2 L Loväs observe tha m origina algorith fo thi coul mad polynomia eve fo varyin n, b

employin th polynomia solvabilit o th linea programmin proble [8] [4] I a indebte t Loväs fo permissio t describ th improve algorith i §2 *Receive Novembe 13 1981 revise M 2 1982 AMS 1980 subject classification. Primary 68C25 Secondary 90C10 OR/MS Index 1978 subject classification. Primary 62 Programming/integer/algorithms Key words. Intege programming polynomia algorithm geometr o numbers. 53 0364-765X/83/0804/0538S01.2 Copyiigh 1983 Th Institut o Managemen Science
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INTEGC PROGRAMMIN WIT I 1XE NUMBE O VARIABLE 53 Th secon auxihar algonth i a leductio proces fo w-dimensiona

lattices Suc a algonthm als du t Loväsz appeare m [9 §1] an a bne sketc i give § o th presen paper Thi algonth i polynomia eve fo varym n. I supersede th muc inferio algonth tha wa describe m a earhe Versio o thi paper § w prove, followm a Suggestio o P va Emd Boas tha th intege linea programmm proble wit a fixe valu o m is als polynomiall solvabl Thi i a immediat consequenc o ou ma result i devote t th mixed integer linear programmmg problem. Combinin ou method wit Khachiyan' result [8] [4 w sho tha thi proble i polynomiall solvabl fo an fixe valu o th numbe o intege variable Thi generahze

bot ou ma lesul an Khachiyan' theore Th algonthm presente i thi pape wer designe fo theoretica purpose only an ther ar severa modification tha migh improv thei practica performance I t b expecte tha th practica valu o ou algonthm i restncte t smal value n. i a pleasur t acknowledg m mdebtednes t P va Emd Boas no onl fo permissio t mclud §4 bu als fo suggestm th proble solve m thi pape an fo severa inspmn an stimulatin discussions Descriptio o th algorithm Le K denot th close conve se = [xfER" · Ax < b] Th questio t b decide i whethe K Z = 0 I th descriptio o th algorith tha follows w mak th

followm tw simphfym assumption abou Ä' (1 K i bounded; (2 K ha positive volume. Th firs assumpüo i justifie b th followm result whic i obtame b combmm a theore o Vo zu Gathe an Sievekm [12 wit Hadamard' determman mequaht (cf (6 below) th se K T" i nonempt i an onl i n 2 contam a vecto whos coefficient ar bounde b (n + \)n" /2 a" m absolut value wher a is ä i th mtroducüon Addin thes mequahtie t th Syste make bounded Fo th justificatio o conditio (2 w refe t §2 Unde th assumption (1 an (2) § describe ho t construc a nonsmgula endomorphis o th vecto spac R" suc tha ha a "spherical appearance Mor

precisely le | denot th Euchdea lengt m R", an pu B(p,z)= (x<=W .\x-p\ < z] fo p e R" zGR >0 th close bal wit cente p an radiu z Wit thi notation th constructe wil satisf B(p, )CrKCB(p,R) (3 fo som p e r K, wit r an R satisfym 7< ( wher c i a constan onl dependm o n. Le suc a b fixed an pu L = τϊ". Thi i a lattice i R" i e , ther exist a basi bj,b . , b o R suc tha (5
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54 H W LENSTR J ca take fo example b = r(e wit e, denotm th U Standar basi vecto o R" W cal b ,b . . , b a basis fo L i (5 holds I b\,b' . . , b' i anothe basi fo L, the ^, = 2"=i ;/ f° som « X «-matri

M = (m !/ 1<(>/< wit integra coefficient an det(Af)=±l I follow tha th positiv rea numbe |det(£, . , b„)\ (th ft, bein wntte ä colum vectors onl depend o L, an no o th choic o th basis i i calle th determmant o L, notation d(L). W ca Interpre d(L) ä th volum o th parallelepipe Σ"=ι[° 0 ' *, wher [0 1 = {zElR-0 Thi Interpretatio lead t th mequahty of Hadamard d(L) < \b\. (6 = l Th equaht sig hold i an onl i th basi b\,b . . . , b i orthogonal I i a classica theore tha L ha a basi b ,b . . . , b tha i nearl orthogona m th sens tha th followm mequaht holds fl\b,\ -d(L) (7 /= wher c i a

constan onl dependm o n, cf [l Chapte VIII] [11] I § w shal mdicat a reduction process, i.e. a algonth tha change a give basi fo L mt on satisfym (7) LFMMA Let b\,b . . . ,b be any basis for L. Then VxeR :3 eL - y\ <\(\b,\ + ··· + \bf). (8 PROO W us mductio o n, th cas n=\ (o n = 0 bein obvious Le = Σ7~ Zfe, thi i a lattic m th (n - l)-dimensiona hyperplan H = ^"=1 R*, Denot b A th distanc o b t // Clearl w hav h<\b (9 No t prov (8) le G R" W ca fin m G Z suc tha th distanc o - mfc t //i < i/z Wnt - mb = x + x wit x G / an x perpendicula t H. The \x < < \b . B th mductio hypothesi ther

exists/ L suc tha -_y,| 2 + · · · + |6„„_,| Sinc x i orthogona to/ th elemen y =y + mb o L nowsaüsfies|x-j| x -J,| + |x + ·· +|*„-i| ^„| Thi prove th lemma Notic tha th proo give a effectiv constructio o th elemen y E. L tha i asserte t exist w numbe th b, suc tha \b„\ = max{|^ :\ < < n}, the (8 imphe VxeW:3yeL:\x-y\<±Tln\b„\. (10 No assum tha b ,b . . . , b i a reduced basi fo L m th sens tha (7 holds an le L' an A hav th sam meamn ä m th proo o th lemma I i easil see tha d(L) = h · d(L'). (11 Fro (7) ( 1 an (6) apphe t L' w ge fl \b,\ < c · d(L} = c · h · d(L') -h- \b,\
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INTEGE PROGRAMMIN WIT FIXE NUMBE VARIABLE ^4 an therefore wit (9) \b„\ (12 Afte thes preparation w descnb th procedur b whic w decid whethe n Z 0 01 equivalently rK n L = 0 W assum tha b ,b . . · , b i a basi fo fo whic (7 holds numbere suc tha \b max{|Z> . l Applym (10 wit = p we fin a vecto j L wit p - y\ <,\^n\b \. I 7 the TA n L i= 0, an w ar done Suppos therefor tha y & τΚ. Then & B(p,r), (3) s \p - y\> r, an thi implie tha r <±Jn\b \. Le no H, L', h hav th sam meanm ä m th proo o th lemma W hav = L' + U C H + U = U (/ + fci ). Henc L i contame m th unio o countabl man paralle

hyperplanes whic hav successiv distance h fro eac other W ar onl mtereste m thos hyper plane tha hav a nonempt mtersectio wit τΚ, thes have b (3) als a nonempt mtersecüo wit B (p, R). Suppos tha precisel t o th hyperplane H + kb mter sec B(p,R). The w hav cleail / - l < IR/h. B (4 an (12 w hav t - l < c\c {n . Henc th numbe o value fo k tha hav t b considere i bounde b a constan onl dependm o n Whic value o k nee b considere ca easil b deduced fro a representaüo o p ä a linea combmatio o Z?, Z> ...,& w fi th valu o k the w restnc attentio t thos = 2" y b, fo whic k; an thi lead t a

intege programmm pioble wit n l variable y\->y-i-> · · · >y -i- I 1 stiaightforwar t sho tha th lengt o th dat o thi ne proble i bounde b a polynomia functio o th lengt o th origina data i th direction o § hav bee followe fo th constructio o r. Eac o th lowe dimensiona problem i treate recursively Th cas o dimen sio n l (o eve n = 0 ma serv ä a basi fo th recursion Thi fmishe ou descnptio o th algonthm observ tha m th cas tha K Z i nonempty ou algonth actuall produce a elemen K Z" 2. Th conve se K. Le K = (x e R : Ax < b], an assum tha K i bounded thi sectio w descnb a algonth tha ca b use t

verif tha K satisfie conditio (2 o §1 t leduc th numbe o variable i tha conditio i foun no t satisfied an t fin th ma use i §1 Th algonth i bette tha wha i strictl neede m §1 i th sens tha i i polynomia eve fo varym n. l a indebte t L Loväs fo pointm out t ni ho thi ca b achieved th firs stag o th algonth on attempt t construc vertice Ü ,ü . . . , v o whos conve hül i a «-Simple o positive volume B maximizm a arbitrar hnea functio o K, employin Khachiyan' algonth [8] [4] on fmd a verte u o K, unles K is empty Suppose mductively tha vertice u u , . . . , v o K hav bee foun fo whic o u . . . , v

o ar hnearl independent wit d < n. The w ca construc n d lineari independen linea functions/, . . ,/„_ o R suc tha th of-dimensiona subspac V = 2 = i R( ~ % 1 give b K={ R ./,(*) -· =/„_,(* = 0}
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54 H W LENSTRA J Agai employin Khachiyan' algorithm w maximiz eac o th linea function /, /|,/ —/2 · · · ,f -d> ~J„-d o K, unti vertc ü d+ o K is foun fo whic f](v d+ \) ^ fj(v fo some e {1,2 ...,« i/} I thi occurs the - u . . . , v v d+ ü ar linearl independent an th inductiv ste o th constructio i complcted If o th othe band n suc v d+ i foun afte eac o th 2(n d) function /, —/, . . .

,/„_ —/„_< ha bee maximized the w mus hav ^(x) ^(VQ) fo al G K an al j = l 2 . . . , n d, an therefor K C u + F I thi cas w reduc th proble t a intege programmin proble wit onl d variables follows Choose for = l 2 . . . , d, a nonzer scala multipl Wj o v - v suc tha w e Z" an denot b W th ( X ä?)-matri whos column ar th w . Notic tha W ha ran d. Employin th Hermite normal form algorith o Kanna an Bacher [7 w ca find i polynomia time a integra n X «-matri U wit det(f/ = ± l suc tha UW = (%) 1<1 i wit = i i>j, (\3) k ^ 0 fo l { ' Denot b M,, . . . , u th column o th integra matri U~ Thes for a

basi R", an als o th lattic Z" Z = Σ"= 2 M, Th subspac F o R i generate b th column o W' = U~ · (k,j), s (13 implie tha κ= (14 = i Defin r, r . . . , r e R b = Σ" \^> s (Γ$ , = [/« No suppos tha j» G Ä Z" The = 2" i J,^ w it G Z an e K implie tha ü G F B (14 thi mean tha j = r fo d < n. S i a leas on o O+ > · · · ' r IB no a integer the K n Z = 0 Suppose therefore tha r ,,...,/· ar al integral Substitutin = Σ^= Τ/" + Σ"=^+ O m ou or ig ma Syste Ax < b the se tha th proble i equivalen t a intege programmin proble wit d variable y ,, y . . . , y ä required Th

vertice v v ,,..., v o K giv ris t d + l vertice v' ,v\, . . . ,v' of th conve se i R belongin t th ne problem an O' ,V'I, . · . , v' spa a i/-dimensiona simple o positiv volume Thi mean tha fo th new d-dimensiona proble th firs stag o th algorith tha w ar describin ca b bypassed conclud th firs stag o th algorithm w ma no suppos tha fo eac = 0 l,...,« l th constructio o v d+ i successful The afte n Step w hav + l vertice u ,«,,... v o K fo whic u - u . . . , u - i> ar linearl indepen dent Th «-simple spanne b v ,v . . . , v i containe i K, an it volum equal |detM|/n wher M i th matri wit

colum vector v - v . . . , v - v Thi i positive s conditio (2 o § l i satisfied J th secon stag o th algorith w construc th coordinat transformatio neede i §1 T thi en w firs tr t fin a simple o "large volum i K. Thi i , ] don b a iterativ applicatio o th followin procedure startin fro th simple spanne b D ,Ü, . . . , v„. Th volum o tha simple i denote b vol(« «, . . ,,ü„) Construc n + l linea function g g ,,..., g : R." -> suc tha g, i constan on [v :0 < j < n, i =£ i], (15 »* fo Q<
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INTEGE PROGRAMMIN WIT FIXE NUMBE O VARIABLE 54 fo / = 0 l . . . , n Maximizin th function g -g

g -g, . . . , g„, -g o K b Khachiyan' algorith w ca decid whethe ther exis / G {0 l ...,« an a verte suc tha &(*-°,)l>! &(»,-»,) j = i (th choic o j i immaterial b (15)) Suppos tha suc a pai i, i found The w replac u b x. Thi replacemen enlarge vol(ü ,ü, . . . , v„) b a facto \g,(x - ü,)|/|g,(u - u (for ^ /) whic i mor tha 3/2 W no retur t th beginnin o th procedur ("Construc n + l linea function . . . ") ever Iteratio ste vol(t> ,«,,... u increase b a facto > 3/2 O th othe hand thi volum i bounde b th volum o K. Henc afte a polynomiall bounde nurabe o iteration w reac a Situatio i whic th

abov procedur discover tha l&(*-°,)l,) (16 fo al G K an al i,j G {0 l ...,« wit i^=j. I tha cas w le b a nonsingula endomorphis o R wit th propert tha T(U τ(υ,) . . . , T(Ü spa a regulär n-simplex Wit p = (n + 1)~'Σ/= w no clai tha B(p,r)CrK CB(p,R) fo certai positiv rea number r,R satisfyin R/r<2n 3/2 i.e. tha condition (3 an (4 o § ar satisfied wit c = 2« 3/2 Thi finishe th descriptio ou algorithm prov ou claim w writ z = r(Vj), fo 0 < j < « w writ S fo th regulä «-simple spanne b z z,,... z an w define ior c > 1 (xeR" :vol(z . . .,*,_,,*,£ + , ...,z C-VO\(Z . . . , z„)forall

G (0,1 ...,«}} Conditio (16 (fo al e K an al / =£j) mean precisel tha C. T Further i clea tha S C τΚ. Ou clai no follow fro th followin lemma LEMMA Let c > 1 With the above noiation we have B(p,r) C S C T C B(p,R)for two positive real numbers r, R satisfying \ \cV + (c + l)n if n is even, >' * \cV + (2c - 2 + l)n + (c - 2c) / n i orfd PROOF Usin a similarit transformatio w ca identif W wit th hyperplan [(>})" G R" +1 Σ"= = 1 i K" suc tha z„,z,,... z i th Standar basi o +l .Thenwehav l l n+\ f^ an " } = (r)" G M" : |r, < c fo 0 < 7 < n, an r = l . y= J a straightforwar analysi on

prove tha T i th conve hül o th se o point
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54 H W LENSTRA JR obtaine b permutin th coordinate o th poin n ~ c z + c z if n = 2m, j=\ j = m+ l n - c)z - c 2 Zj + c 2 Zj i n = 2m+l. j= l }-m+ I follow tha T C B(p,R), wher R i th distanc o p t th abov point n i even «isodd + l Further B(p,r) C S, wher i th distanc o p t (0 l/n, l/n, . . . , l/n): ~ n(n+ 1 ' Thi prove th lemma REMARKS (a T th constructio o i th abov algorith on migh rais th objectio tha nee no b give b a matri wit rational coefficients Indeed fo = 2,4,5,6,10,.. ther exist n regulä n-simple al o whos vertice hav

rationa coordinates Thi objectio ca b answere i severa ways On migh replac th regulä simple b a rationa approximatio o it o indee b an fixe «-simple wit rationa vertice an positiv volume a th cos o gettin a large valu fo c, Alternatively on migh etnbe R i M" +1 ä wa don i th proo o th lemma Finally i ca b argue tha i i no necessar tha th matri M definin b rational bu onl th Symmetri matri M definin th quadrati for (rx, rx); an thi ca easil b achieve i th abov constructio o Ί. (b Th proo tha th algorith describe i thi sectio i polynomial eve fo varyin n, i entirel straightforward W indicat

th mai points Th constructio o · · · 'fn-d m tn fr rst Sta an o g gi, . . . , g i th secon stage ca b don Gaussia elimination whic i wel know t b a polynomia algorithm cf [2 §7] follow tha Khachiyan' algorith i onl applie t problem whos length ar bounde b a polynomia functio o th lengt o th origina data Th sam applie th intege programmin proble constructe i th firs stage Furthe detail ar lef t th reader (c W discus t whic exten th valu 2« fo c i (4 i bes possible Replacin th coefficien 3/ i (16 b othe constant c > l w find usin th lemma tha fo an fixe E > 0 w ca tak + e)(« + 2« 1/ i «iseven Cl

1/ [( + e)(« n n) i n i odd on i satisfie wit a algorith tha i onl polynomia fo fixe n on ca als tak e = 0 i thi formula T achiev this on use a lis o al vertice o K t fin th simple o maxima volum insid K, an transform thi simple int a regulä one Th followin resul show tha ther i stil roo fo improvement i K C R i an close conve se satisfyin (1 an (2 the ther exist a nonsingula endomorphis r R suc tha (3 an (4 hol wit c = n. T prov this on choose a ellipsoid E insid K wit maxima volume an on choose suc tha E i a sphere. Th cas tha K i a simple show tha th valu c = n i bes possible Fo fixe n an e

> 0 ther i a polynomia algorith tha achieve c = ( + ε)« I d no kno ho wel th bes possibl valu c\ n ca b approximate b a algorith tha i polynomia fo varyin n.
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INTEGE PROGRAMMIN WIT F1XE NUMBE O VARIABLE 54 (d Th algorith describe i thi sectio applie equall wel t an clas W o compac conve bodie i R" fo whic ther exist a polynomia algorith tha maximize linea function o member K o Cff . Thi remar wil pla a importan rol i §5 I particular w ca tak fo Ctf a "solvable clas o conve bodies i th terminolog o [4 §§ an 3] Th sam remar ca b mad fo th algorith presente i §1 Th reductio

process Le n b a positiv integer an le b ,b . . . , b G R" n linearl independen vectors Pu L = 2" 1 U, ; thi i a lattic i R" I thi sectio w indicat a algorith tha transform th basi b ,b . . . , b fo L int on satisfyin (7 wit c = 2" "~ 1)/ Th algorith i take fro [9 §1] t whic w refe fo a mor detaile description recal th Gram-Schmid orthogonalizatio process Th vector b* (\ < i < n) an th rea number µ, ( < j < i < n) ar inductivel define b b* = b, - µ,/ , µ = (b, , bf}/(b* , b*), = ' wher ( , ) denote th ordinar inne produc o R" Notic tha b* i th projectio b, o th orthogona complemen o ^~ R£> an

tha 2 = Rf = Σ, R*/ fo < i < n. It follow tha &*,&* - . . , b* i a orthogona basi o R". Th followin resul i take fro [9] PROPOSITION Suppose that k,l< 07 for l < j < i < n, and for ! Then i.e., (7 holds with c = 2" (w -"/ PROOF Se [9 Propositio 1.6] explai conditio (18 w remar tha th vector b* + µ, _,£>*_ an b*_ ar th projection o b, an Z>,_ o th orthogona complemen o 2;=^fy Henc i (18 doe not hol fo som / the i doe hol fo th basi obtaine fro b lt 2i . . . , b interchangin b , an b, . chang a give basi b ,b . . · , b fo L int on satisfyin (7 w ma no iterativel appl th followin

transformations First transformation: selec / ! suc tha (18 doe no hold an inter chang ft,_, an b ; Second transformation: selec / y l < j < i < « suc tha (17 doe no hold an replac b, b b, rb , wher r i th intege nearest t µ . ca b show that independentl o th orde i whic thes transformation ar applie an independentl o th choice o / an o / andy tha ar made thi lead afte a finit numbe o Step t a basi b\,b . . . ,b satisfyin (17 an (18) The (7 satisfie ä well b th proposition Thi finishe ou sketc o th algorithm particularl efficien strateg fo choosin whic transformatio t apply an fo whic / o i

andy i describe i [9 (1.15)] I w assum th b, t hav integer coordinate the th resultin algorith i polynomial eve fo varyin n, b [9 Propositio 1.26] I follow tha th sam resul i tru i w allo th coordinate o th b, t b rational.
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54 W LENSTRA J REMARKS (a Th algorith sketche abov ca b use t fin th shortes nonzer vecto i L, i th followin way Suppos tha b ,b . . . , b i a basi fo L satisfyin (7) an le G L. The w ca writ = 2? i m,b, wit m, an fro Cramer' rul i i eas t deriv tha \m,\ < c · \x\/\b,\, fo l < i < n. I i th shortes nonzer vecto i L the |jc < \b\ fo al / s \m\ < c S b

searchin th se (27=i : fo ! w ca fin th shortes nonzer vecto i L i polynomia time fo fixe n. Fo variabl n thi proble i likel t b NP-ha.rd. (b W discus t whic exten ou valu fo c i bes possible Th abov algorith yield c = 2" "~ l)//4 W indicat a algorith tha lead t a muc bette valu fo bu th algorith i onl polynomia fo fixe n. (a w showe ho t fin th shortes nonzer vecto i L b a searc procedure B a analogou bu somewha mor complicate searc procedur w ca determin th successive minima \b\\,\b'^, . . . ,\b' o L (se [l Chapte VIII fo th definition) Her b\,b' . . . , b' L ar linearl independent an b [l

Chapte VIII Theore I p 20 an Chapte IV Theore VII p 120 the satisf ι*; < · d(L) 1=\ wher y denote Hermite' constan [l §IX.7 p 247] fo whic i i know tha y„/n fo n->oo Usin a sligh improvemen o [l Chapte V Lemm 8 p 135 w ca chang b\,b' . . . , b' int a basis b" ,b' . . . , b% fo L satisfyin \b','\ ( ',\ (K i < n) d(L) (fo n > 3) conclud that fo fixe n, th basi b ,b . . . ,b produce b th algorith indicate i thi sectio ca b use t find i polynomia time a ne basi satisfyin (7) bu no wit c = (c · n)". Her c denote som absolut positiv constant th othe hand th definitio o implie tha ther exist a

n-dimensiona lattic L suc tha \x\ > y 1/ · d(L)^ /n fo al L, = 0 cf [l Chapte I Lemm 4 21] An basi b\,b · · · , b fo suc a lattic clearl satisfie Therefor th bes possibl valu fo c satisfie c > (c' · n)"/ fo som absolut positiv constan c'. 4. A fixe numbe o constraints I thi sectio w sho tha th intege linea programmin proble wit a fixe valu o m i polynomiall solvable I wa note P va Emd Boa tha thi i a immediat consequenc o ou mai result Le n,m,A,b b ä i th introduction W hav t decid whethe ther exist e Z fo whic Ax < b. Applyin th algorithm o Kanna an Bacher [7 w ca fin a (n X «)-matri U wit

integra coefficient an determinan ± l suc tha th matri ^i = (a;) 1<1 K/< satisfie = 0 fo
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INTEGE PROGRAMMIN WIT FIXE NUMBE O VARIABLE 54 Putting = U~ w se tha th existenc o Z wit Ax < b i equivalen t th existenc o y G Z wit (AU)y < b.If n > m, the th coordinate J OT+ . . . ,y o / not occu i thes inequalities sinc (19 implie tha ^ = 0 fo 7 > m. W conclud tha th origina proble ca b reduce t a proble wit onl min{«,m variables Th latte proble is fo fixe m, polynomiall solvable b th mai resul thi paper 5. Mixe intege linea programming Th mixed integer linear programming prob- lem i

formulate ä follows Le k an m b positiv integers an n a intege satisfyin 0 < n < k. Le furthe A b a m X Ä>matri wit integra coefficients an e T". Th questio i t decid whethe ther exist a vecto = ( , . . . , x wit x, e Z fo l < / < n, x, e K fo n + l < / < k satisfyin th syste o m inequalitie Ax < b. thi sectio w indicat a algorith fo th solutio o thi proble tha i polynomia fo an fixe valu o « th numbe o intege variables Thi generalize bot th resul o § l (n = k) an th resul o Khachiya [8] [4 (n = 0) Le K' = (xGR :Ax < b}, = { (x,, . . . , x„) R : ther exis x n+ ,,..., x e M suc tha (* ,x . . .

, x ~J e K'}. Th questio i whethe K Z = 0 Makin us o th argument o Vo zu Gathe an Sievekin [12 w ma agai assum tha K', an henc K, i bounded Nex w appl th algorith o § t th compac conve se K c W. T se tha thi ca b don i suffice t sho tha w ca maximiz linea function o K, se §2 Remar (d) Bu maximizin linea function o K i equivalen l maximizing o K', linea function tha depen onl th firs n coordinate x\,x . . . , x„; an thi ca b don b Khachiyan' algorithm Th res o th algorith proceed ä before A a certai poin i th algorith w hav t decid whethe a give vectc y G R belong t τΚ. Thi ca b don b

solvin a linea programmin proble wit k n variables Thi finishe th descrip tio o th algorithm i § i ca b prove tha th mixe intege linea programmin proble i als polynomiall solvabl i th numbe o inequalitie tha involv on o mor intege variable i fixed or mor generally i th ran o th matri forme b th firs n column o A i bounded Reference [1 Cassels i. W S (1959) An Introduction to the Geometry of Numbers. Springer Berlin Secon printmg 1971 [2 Edmonds 3. (1967) System o Distinc Representative an Lmea Algebra J. Res. Nat. Bur. Standards Sect. B 71 241-245 [3 Garey M R an Johnson D S (1979) Computers

and Intractabihty, A Guide to the Theory of NP-Completeness. Freema & Co. Sa Francisco [4 Grotschel M. Loväsz L an Schnjver A (1981) Th Elhpsoi Metho an It Consequence i Combmatona Optimizatio Combmatonca l 169-197 [5 Hirschberg D S an Wong C K (1976) A PolynomiaJ-tim Algorith fo th Knapsac Proble wit Tw Variables / Assoc. Comput. Mach. 2 147-15
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54 H W LENSTRA J [6 Kannan R (1980) A Polynomia Algonth fo th Two-vanabl Intege Programmin Proble J. Assoc. Comput Mach 27 118-12 [7 _ an Bachern A (1979 Polynomia Algonthm fo Computin th Smit an Heimit Norma Form o a Intege Matrix

SIAM J. Comput. 8 499-50 [8 Khachryan L G (1979) A Polynomia Algonth m Linea Programmin Dokl. Akad Nauk SSSR 24 1093-109 (Englis translation Soviel Math Dokl. 2 (1979) 191-194 [9 Lenstra A K. Lenstra H W. Jr an Loväsz L (1982 Factorin Polynomial wit Rationa Coefficients Math. Ann. 26 515-534 [10 Scarf H E (1981) Producüo Set wit Indivisibihües—Par I Generahtie Econometnca 4 1-3 Par I Th Gas o Tw Activities ibi , 395-423 [11 Va de Waerden B L (1956 Di Reduktionstheori vo positive quadratische Forme Acta Math 9 265-309 [12 Vo zu Gathen J an Sievekmg M (1978) A Boun o Solution o Linea Intege

Equahtie an Inequahtie Proc Amer. Math. Soc. 7 155-15 MATHEMATISC INSTITUUT UNIVERSITEI VA AMSTERDAM ROETERSSTRAA 15 101 W AMSTERDAM TH NETHERLAND