On Multi-Dimensional Packing Problems

We study the approximability of multi-dimensional generalizations of three classical packing problems: multiprocessor scheduling, bin packing, and the knapsack problem. Specifically, we study the vector scheduling problem, its dual problem, namely, the vector bin packing problem, and a class of packing integer programs. The vector scheduling problem is to schedule n d-dimensional tasks on m machines such that the maximum load over all dimensions and all machines is minimized. The vector bin packing problem, on the other hand, seeks to minimize the number of bins needed to schedule all n tasks such that the maximum load on any dimension accross all bins is bounded by a fixed quantity, say 1. Such problems naturally arise when scheduling tasks that have multiple resource requirements. Finally, packing integer programs capture a core problem that directly relates to both vector scheduling and vector bin packing, namely, the problem of packing a miximum number of vectors in a single bin of unit height. We obtain a variety of new algorithmic as well as inapproximability results for these three problems.  Multi-dimensional packing, vector scheduling, vector bin packing, packing integer programs, multiprocessor scheduling, bin packing, knapsack, approximation algorithms, hardness of approximation, combinatorial optimization Disciplines Computer Engineering Comments Postprint version.  2004, pages 837-851. Publisher URL: http://dx.doi.org/10.1137/S0097539799356265  ! #"$ % &' )(* ,+ . 0/ 1320465#798:4;1329<>=*?98A@CB D#465FEA<G<GH IJ20465#504LK MON#PAQSRUT VWQ XZY []\ ^`_bac\ d Y eWf f g h:ibjlkmeonFjlpqjr\sathou*kt^ pr\ jrvw_bjqk$YSxF[ jlhGxFeop`yWYAx YSgzeopqjl{|eU\ jlhGxF[9hou}\ d g YAY~ peW[ [ jq~Aeop fFeG~zjlx y fFg hGn plYAkm[Skt^ pl\ jqf g hb~SYA[ []hGg [ ~zd YA_b^Fpljqx yFLn jqx f`eW~zjlxFyFLeoxF_\ d Y3xFeofF[ eW~z f g hWn pqYSk,f6YA~Sjr`~AeopqplaGL Y []\ ^`_ba\ d Y$:zSCo! z zob>!fFg hGn plYAkbjr\z[_b^FeWp}f g hGn pqYSkbxFeok$YAplaG\ d Y$U SwUcSmFWz U!f g hWn pqYSk eWxF_¡eZ~SpqeG[ [3hWuFWz U¢FwCGSt6  W U£t%¥¤¦dFY §GYA~ \ hWg$[ ~zd YA_b^Fpljqx y ̈f g hWn pqYSk©j[3\ h¢[ ~zd YA_b^FplYa¬«Wv _ jlk$YSx`[]jqhWxFeWp\zeW[ [hWx ­ kmeW~zdFjlx Y|[O[ ^F~zd¡\ dFeo\O\ d YkmeUibjqkc^ k®plh>eW_¡hU§WYAgeopqp _bjqk$YSxF[ jqhWxF[eWxF_ eopqp kmeG~zd jlxFYA[ ̄j[ ̄k$jlxFjlk$jq{SYA_*-¤¦d Y §GYA~ \ hWg ̄nFjlxJfFeG~zjlx y$f g hWnFplYAk FhWx \ d Y hW\ d YAgdFeWxF_}`[ YSYAb[-\ hmk$jlxFjlk$jq{SY \ d Y ̄x^ kcn6YSg9hWu`n jqxF[9x YAYA_bY|_,\ h![ ~zd Y|_b^ pqY¦eWplpaO\ eW[ b[9[]^F~zdt\ d`eU\9\ d Y¦kmeUibjqkc^ k°plh>eW_,hGx3eWxa,_bjlk$YAxF[]jqhWx eG~ g hG[ [eopqp±n jlx`[Oj[n6hW^ x`_bYA_ na e ̈ ibY|_ 2>^Feox>\ jl\saW [ e:a¬3W ́^`~zd f g hWn pqYSkm[x`eU\ ^Fg eWplpqa¡eWg j[ Y ̄d YSx [ ~zd Y|_b^ pqjlx y\zeW[ b[±\ dFeo\cdFe:§WY$kt^ pr\ jlfFplYmg YA[  jqg YAk$YSx>\ [Aμ#jlxFeWplpqaW#fFeG~z>jqx yjqxG\ YSyGYSg f g hWyGg eWk$[ ~Aeofb\ ^ g Yce~ hGg Ycf g hGn pqYSk¶\ dFeo\±_bjqg Y|~%\ pla g YApqeo\ Y|[ ̄\ hOn`hW\ d§GYA~%\ hWg[ ~zd Y|_b^ pqjlx y eoxF_J§GYA~ \ hWgn jlx fFeW~zjqx yF x`eok$YSpqaWW\ d YfFg hGn plYAk°hWu6fFeG~zjlx y e!kmeUibjqkc^ k·x^ kcn6YSg ̧houL§WYA~ \ hGg [9jqxme []jqx yWpqY ̄n jqxmhouL^ x jl\ ̧d YAjlyGdG\| XZY hGnb\ eWjlxme±§Ueog jlYS\sa,hou6x YS¥eopqyWhGg jl\ dFk3j~-eW[0-YSpqp eW[9jqxFeofFf g h:ibjqkmeon jqpljl\sa g YA[ ^ pr\z[#u1hWg9\ d YA[ Y¦\ d g YSY¦f g hWnFplYAk$[A oJ»>1⁄4}1⁄2c3⁄4 RU¿ P ZÀ^ pr\ jrvw_bjqk$YSxF[ jlhGxFeop fFeW~zjqx yFkt^ pl\ jqf g hb~SYA[ []hGg,[ ~zd Y|_b^ pqjlxFyF#n jlx¢f`eW~zjlxFyF#>x`eofF[ eW~zLÁeWf f g h:ijqkmeU\ jlhGxZeWplyGhWg jr\ d km[S d`eogz_bx YA[ [¦hou9eof fFg h:ibjqk$eo\ jqhWx*b~ShWkcnFjlxFeo\ hGg jeopLhWf \ jqk3jq{Aeo\ jqhWx* M Â ́Ã ÃÁÄÁN}Å » VWQ ÆtÇÈT P|PAÉÈÊ#VGTbQAÉ 3⁄4`Ë¡ÍÌGÎ>Ï!ÐWÑ  Ò%Ó ́ÔGÕsÖz×ØÙ ØÚÛSÕwÜ Ý:Ö ÕsÞwØrßAÚ ̈ßAà ̄ásâGØÞcÔ>ÛAÔbÖ ÕtÛSÔ>ÔbÖ%Û ÕsÖ%ãZØÚZásâ>ÖOä9åwæ%ç]èsèwéAêëUìAíOæCî$ïqð>è ñ|òSïqðôó ̧ëGë>õGö|÷0óø6ùOúÈûoüwó ùýûGþAÿ ¬æSë êqí ç åwè ï è ̄ó ÷ ì:æAå]êïqðoÿ í >Ô>Ô Öz×× 6Û >Þ !#" %$3ß &>ÚUá]ÛSØrÚÓ Ý:Ö '$(&GÕwÕ]ÛzÜ*) Ø×r×+ ',.-/ " " 10 !ÈÙ,ÛAØ×+2436587 9 :<; = > ; 78? 7<@ ; AG3 H IJ K$3ß|ÞCá ßAà*ásâGØrÞML9ßSÕGN(L0ÛSÞ¦ãGßAÚ>Öã &WÕsØÚPOcÛ!ÞQ&>Ù Ù Ö Õ ̧ØÚUásÖ ÕsÚ>ÞwâGØÔÛSá Öz××' 6Û >Þ LÍâGØ×rÖásâ>ÖÛ &Wásâ>ßSÕ L9ÛAÞ ̄Û áSRUá]ÛAÚGàqßAÕ]ãUT-ÚGØÝ:Ö ÕsÞwØ á Ü< V ßAÕGN ÛSá.RUá]ÛAÚGàqßAÕ]ãWL0ÛAÞ ̧ÞQ&>ÔGÔbßAÕwásÖ%ãX UÜcÛAÚ(Y $[Z#ß !Èß|ÔbÖ Õ]ÛSásØÝ:Ö ̄àÖz××ß LÍÞwâ>ØÔ\ ÛAÚmÓM]M^_$(T.]MY ` Õ]ÛSÚUába-Ó Ó )c !d 6!G e!d "8 ÛAÚ>ãf,SR g ÓML9ÛSÕ]ãUZ Zh]b!d i " 8 j a-ÖzÔGá GßAàkZhYlRJ T-Ú>ØÝ:Ö ÕsÞwØá ÜcßAàBmLÖzÚGÚ>ÞCÜW×Ý|ÛSÚ>ØrÛ mÁâGØr×rÛAãGÖz×Ô>âGØrÛ nm Óo 0 !ÈÙ,ÛAØ×+2p? @ q r 7 7 s<>83 = ? AD: t<7 q qBA#7 u : .v9â>ØÞML#ßAÕGN L0ÛSÞ-ãWß|ÚGÖ.LÍâ>ÖzÚtásâ>ØÞÍÛ &Wásâ>ßAÕbL9ÛAÞ ÛSá Öz×× 6Û GÞ n &PwzÖzÚUáMv6Öew âGÚ>ß|×ß O|ØÖzÞ