Job shop scheduling with unit processing times

Transcription

1 Job shop schduling with unit procssing tims Nikhil Bansal Tracy Kimbrl Maxim Sviridnko Abstract W considr randomizd algorithms for th prmptiv job shop problm, or quivalntly, th cas in which all oprations hav unit lngth. W giv an α- approximation for th cas of two machins whr α < 1.45, an improvd approximation ratio of O( log m log log m ) for an arbitrary numbr m of machins, and th first (2+ε)- approximation for a constant numbr of machins. Th first rsult is via an approximation algorithm for a string matching problm which is of indpndnt intrst. 1 Introduction Job shop schduling is a widly studid and difficult combinatorial optimization problm [11]. In this problm, w ar givn a collction of jobs and a st of machins. Each job consists of a squnc of oprations, which must b prformd in ordr. Each opration has a particular siz and must b prformd on a spcific machin. Th goal is to minimiz th makspan, dfind as th compltion tim of th last job to finish. Th problm is dfind formally in Sction 2. W considr th prmptiv cas with th objctiv to minimiz makspan. This problm is strongly NPhard vn for two machins [9]. If th numbr of machins is part of th input, thn th rsult of Williamson t al. [20] implis that thr is no polynomial-tim approximation algorithm with prformanc guarant bttr than 5/4 unlss P = NP. For th gnral nonprmptiv job shop schduling problm, th bst known approximation algorithm has prformanc guarant of O(log 2 mµ/ log 2 log mµ) whr m is th numbr of machins and µ is th maximum numbr of oprations in a job [19, 7]. If for vry job thr is at most on opration on ach machin, this bound can b improvd to O(log 1+ε m) for vry ɛ > 0 [5, 6]. This variant is calld th acyclic job shop schduling problm. In th cas of th acyclic job shop with unit procssing tims for vry opration, th famous paprs of Lighton t al. [12, 13] giv constant factor approximation algorithms. In th cas of th prmptiv job shop an approximation with ratio O(log mµ/ log log mµ) is known [12, IBM T.J. Watson Rsarch Cntr, P.O. Box 218, Yorktown Hights, NY mail:{nikhil,kimbrl,sviri}@us.ibm.com 19, 7]. A polynomial-tim approximation schm (PTAS) is known for th spcial cas of a constant numbr of machins and a constant numbr of oprations pr job [10] both for th prmptiv and nonprmptiv problms. In this papr w giv th following rsults for th prmptiv job shop schduling problm: 1. For m = 2: It is asy to s that any rasonabl schdul has an approximation ratio of 2. Svastianov and Wogingr [15] gav th first non-trivial 1.5-approximation algorithm for m = 2. An algorithm with a tightr approximation guarant in trms of th maximum machin load L and th maximum job lngth l, but still 1.5 in th worst cas, was givn by Andrson t al. [3]. All known approximation rsults for shop schduling (othr than approximation schms) us as a lowr bound th maximum of L and l. Svastianov and Wogingr [15] not than any approximation algorithm with ratio bttr than 1.5 for th two machin cas would rquir a nw, non-trivial lowr bound on th optimal makspan. In this papr, w giv such a rsult basd on th rlationship btwn th prmptiv job shop schduling problm and a string matching problm ovr th binary alphabt. Our algorithm has an approximation ratio of lss than For arbitrary m: W giv an algorithm with an approximation ratio of O(log m/ log log m) for m machins. This liminats th dpndnc on µ in th rsults of [12, 19, 7]. W giv anothr vry simpl algorithm that constructs a schdul of lngth (1 + ɛ)l + O(µ log m)p max in th non-prmptiv cas, and hnc a schdul of lngth (1 + ɛ)l + O(l log m) in th prmptiv cas (s Sction 2 for rlationship btwn prmptiv job shop and nonprmptiv job shop with unit procssing tims), whr p max is th maximum lngth of an opration. In som cass this givs a substantially strongr guarant than Svastianov s guarant [16, 17] of L + O(mµ 3 )p max. In particular, if th instanc is such that µ ɛl/ log m, thn our algorithm givs a (1 + ɛ) approximation. 1

2 3. For constant m: W show how our additiv approximation of (1+ɛ)L+O(l log m) can b usd to giv a polynomial tim (2 + ε)-approximation algorithm for any constant numbr of machins. (Not that w allow numbr of oprations pr job to b part of th input). Prviously, no algorithm with an approximation ratio indpndnt of m was known for th problm. 2 Modl and notation In th job shop schduling problm thr is a st J = {J 1,..., J n } of n jobs that must b procssd on a givn st M = {M 1,..., M m } of m machins. Each job J j consists of a squnc of µ j oprations O 1j,..., O µjj that must b procssd in ordr. Opration O kj must b procssd on machin M πkj, during p kj tim units. A machin can procss at most on opration at a tim, and ach job may b procssd by at most on machin at any tim. For a givn schdul, lt C kj b th compltion tim of opration O kj. Th objctiv is to find a schdul that minimizs th maximum compltion tim, C max = max kj C kj. Th valu of C max is also calld th makspan or th lngth of th schdul. For a givn instanc of th job shop schduling problm, th valu of th optimum makspan will b dnotd Cmax. For ach job j and machin i, l ij is th total amount of work in job j dsignatd for machin i. Lt l j = i M l ij dnot th total lngth of job j, and lt l = max j J l j. Lt L i = j J l ij dnot th load on machin i, and lt L dnot max i M L i. Clarly, max{l, l} Cmax. Lt µ = max j µ j b th maximum numbr of oprations in any job. In this papr w considr th prmptiv variant of th job shop schduling problm, in which vry opration can b prmptd during its xcution and rsumd latr without any pnalty. Of cours, w must still oby prcdnc constraints btwn oprations, and for vry opration O kj, th total tim allocatd on machin π kj must b qual to its procssing tim p kj. It is wll-known that thr xists an optimal schdul for th prmptiv job shop problm whr prmptions occur at intgral tims (s [4] for mor gnral rsults of this sort). W may assum that all opration lngths ar polynomially boundd sinc this assumption can b rmovd with only ε loss for any ε > 0 by standard scaling and rounding tchniqus. Thus, w will considr th prmptiv job shop schduling problm to b quivalnt to th nonprmptiv job shop schduling problm with unit procssing tims; i.., w split vry opration O kj with procssing tim p kj into p kj unit lngth oprations and assum that all inputs p kj, L i, l j ar polynomially boundd. W us ε throughout th papr to dnot a constant that can b mad arbitrarily small. W assum without loss of gnrality that for all i M, L i = L; this is asily achivd as follows. W add a dummy job for ach machin i if ndd, comprising L L i unit lngth oprations on machin i only. It is asy to s that this dos not chang th optimal makspan, sinc any schdul has lngth at last L and at last L L i idl tim stps in which th dummy job can b insrtd. 3 Th two-machin cas In this sction w considr th prmptiv two-machin job shop problm J2 pmtn C max. Th prviously bst known algorithms for this problm hav a worst cas approximation ratio of 1.5 [15, 3]. As mntiond prviously, Svastianov and Wogingr obsrvd that it is impossibl to achiv a bttr ratio if w us th trivial lowr bound of max{l, l} only. To s this, considr th instanc with two idntical jobs J 1 and J 2. Each of thm consists of L/2 oprations that rquir machin 1 followd by L/2 oprations that rquir machin 2. Clarly, th trivial lowr bound on th makspan is L. Howvr, it is asy to s that any fasibl schdul has makspan at last 1.5L. In ordr to gt a ratio strictly bttr than 1.5, w adopt th following approach: W first not that in th cass whn l < 0.88L or whn l > 1.16L th trivial lowr bound is nough to giv a ratio bttr than In th hard cas with a big job B of lngth about L (i.. l L), w attmpt to maximiz th numbr of oprations of othr jobs that ar prformd concurrntly with B. This is most clarly statd in trms of a string matching matching problm which w dscrib in Sctions 3.1 and 3.3 blow. W show how to solv this string matching problm so that w can prform concurrntly with B at last a (1 1/) fraction of th most possibl. W thn show that combining ths diffrnt cass givs a worst cas approximation ratio of about in th gnral cas. 3.1 Maximizing disjoint matchs btwn a st of binary strings and on long on. Considr th following problm. Lt S b a binary string and C = {S 1, S 2,..., S n } b a collction of binary strings. Lt l dnot th lngth of S, and lt l i dnot th lngth of S i. For a string X, lt X(i) dnot th i th charactr of X. W say that S i has a matching E i of valu k in S if thr xist indics a 1 < a 2 <... < a k and b 1 < b 2 <... < b k such that S i (a j ) = S(b j ) for all 1 j k. In othr words, S i has a matching E i of valu k in S if som subsqunc of k charactrs of S i can b matchd with a subsqunc in S. W dnot th valu of E i by E i. W also associat E i with th st 2

3 of indics b 1,..., b k. Matchings E i and E j of S i and S j, rspctivly, in S ar said to b disjoint if E i E j =. Problm: Find a collction of matchings E 1,..., E n of S 1,..., S n in S such that th E i ar pairwis disjoint, i.. E i E j = for all 1 i < j n. Th goal is to maximiz th cardinality of n i=1 E i. In othr words, w sk to match as much of S 1,...,S n as possibl, but with th rstriction that th matchings for any two S i and S j ar disjoint. W show th following: Thorm 3.1. Thr is a randomizd approximation algorithm for our string matching problm with xpctd ratio at last (1 1/). To prov Thorm 3.1, w considr an LP rlaxation of a natural intgr program formulation and show that it can b roundd to giv th dsird approximation ratio. Th proof is dfrrd until sction 3.3. W now show how Thorm 3.1 implis a 1.44 approximation for th job shop problm with two machins. 3.2 Solving job shop using th string matching algorithm Thorm 3.2. For vry ε > 0, thr is a randomizd (α + ε)-approximation algorithm for J2 pmtn C max, whr α = , with running tim polynomial in th numbr n of jobs, th maximum numbr µ of oprations in a job, and 1/ε. corrspondnc is invrtd: oprations on th first machin corrspond to 0 s and oprations on th scond machin corrspond to 1 s. W can obtain a schdul for a job shop instanc from a solution to th matching problm such that a match in th string problm corrsponds to an opration of a big job bing prformd in paralll with an opration of anothr job. Thus, th numbr of matchs is qual th numbr of tim units in which two oprations ar xcutd in paralll. Lt V dnot th optimal valu of th string matching instanc. V is th maximum possibl ovrlap btwn job B and th rmaining jobs, i.., th maximum numbr of unit-lngth oprations in B that can b xcutd concurrntly with oprations in othr jobs. Not this is not ncssarily th amount of ovrlap btwn B and th othr jobs in any optimal schdul, but is an uppr bound. This allows us to lowr bound th optimum makspan as follows: Assum that th maximum ovrlap btwn B and th othr jobs is achivd, furthr w allow th maximum ovrlap btwn th rmaining oprations in othr jobs. In particular, th rmaining oprations in othr jobs ar j B l j V, and w assum that optimum can schdul ths in paralll. Thus, th optimal makspan Cmax is at last l ( j B l j V ) = L + l/2 V/2 Proof. Lt 0 δ 1 b a constant to b dtrmind latr. W considr sparatly th following cass: 1. l 2δL. In this cas, w us th algorithm of Andrson t al. [3] which finds a schdul of lngth at most L + l/2, so th trivial lowr bound of L in this cas is nough to giv an approximation ratio of 1 + δ. 2. l 2 1L. Again w us th algorithm of Andrson t al. [3], which givs a schdul of lngth at most L + l/2. In this cas th trivial lowr bound of l implis a ratio of at most L/l + 1/2 / L > l > 2δL. In this cas, w us Thorm 3.1. W rduc J2 pmtn C max to th string matching problm as follows. String S corrsponds to a big job B job of lngth l (An arbitrary choic can b mad if thr ar two or mor such jobs). All othr jobs corrspond to strings in th st C. W crat a binary strings corrsponding to th jobs. For th job B, w crat a string of lngth l, whr oprations of B procssd on th first machin corrspond to 1 s in S and oprations on th scond machin corrspond to 0 s. For th othr jobs, th 3 Th quality follows by our assumption that ach machin has load xactly L and hnc that = 2L l. j B Using Thorm 3.1 w can obtain a schdul that has at last (1 1/)V matchs. In th worst cas, w assum that no othr oprations ar xcutd in paralll. Thus, in th worst cas, th xpctd lngth of th schdul at most 2L (1 1/)V. Thus th approximation ratio is at most f(v ) = 2L (1 1/)V L + l/2 V/2 It can b vrifid that for l 2 1L, f(v ) in monotonically incrasing in V. Thus sinc V l th approximation ratio has maximum valu at most 2 2(1 1/)δ with l = V = 2δL. Choosing δ = , w s that 2 2(1 1/)δ = 1 + δ and hnc in ithr of th thr cass abov, w obtain a schdul of lngth at most min{2 2(1 1/)δ, 1 + δ, /2}Cmax. This implis th dsird rsult.

4 3.3 Proof of Thorm 3.1W first considr a linar programming formulation for a rlaxation of th problm. W viw th rlaxd problm as follows: Considr a charactr S i (j) and imagin travrsing th string S on charactr at a tim starting from S(1) and going all th way to S(l). At stp k w match a fraction x i,j,k of S i (j) to S(k), and thn w discard (i.., lav unmatchd) a fraction y i,j,k of S i (j) bfor moving on to stag k + 1. W also assum that for ach i, j, thr is a stp 0, whr w can discard a fraction y i,j,0 of S i (j). Considr th following linar program: (3.1) k 1 (3.2) (3.3) (3.4) (3.5) (3.6) Maximiz k 1 (x i,j 1,t + y i,j 1,t ) t=0 n l i l i=1 j=1 k=1 x i,j,k (x i,j,t + y i,j,t ) x i,j,k 0 {i, j, k} t=0 l= S x i,j,k 1 k i,j x i,j,k = 0, S i (j) S(k) y i,j,k 0 x i,j,k 0 (x i,j,t + y i,j,t ) = 1 {i, j} t=0 Th first st of constraints modls th prcdnc constraints implid by th ordring of th charactrs in th strings. Thy say that th amount of S i (j) matchd up to S(k) and discardd up to S(k 1) is no mor than total amount of S i (j 1) matchd and discardd up to S(k 1). Th scond st of constraints nsurs that vry charactr is S is matchd to a total amount of at most 1. Th third st of constraint nsurs that 0s ar not matchd to 1s and vic vrsa. Th fourth and fifth sts of constraints nsur that x i,j,k and y i,j,k ar non-ngativ. Th final st of constraints say that ach charactr S i (j) is ithr matchd or discardd. W now show that th LP optimum is an uppr bound on th intgral optimum solution. Considr any intgral solution (I), and suppos that S i (j a ) is matchd to S(k a ), for a = 1,..., b. Thn for a = 1,..., b w st x i,ja,k a = 1 and x i,ja,t = 0 for all t k a. W also st y i,ja,t = 0 for all t. For j such that j a < j < j a+1, w st y i,j,ka = 1 and x i,j,t = 0 for all t. If j < j 1, w st y i,j,0 = 1. Clarly th valu of th objctiv function for this assignmnt of variabls is xactly th numbr of matchs in I. W nd only to show that this assignmnt satisfis all th constraints. Sinc x i,j,k and y i,j,k ar {0, 1} variabls in this assignmnt and vry S i (j) is ithr matchd or discardd, th last thr sts of constraints ar trivially satisfid. Th third st of constraints is satisfid sinc I dos not match a 0 to a 1 or vic vrsa. Th scond st of constraints holds bcaus ach S(k) is matchd to at most on charactr among th S i. For th first st of constraints, sinc x i,j,k and y i,j,k ar {0, 1}, such a constraint is violatd only if S i (j 1) has not bn matchd or discardd by stp k 1 and ithr 4 1. S i (j) is matchd at stp k. 2. S i (j) is matchd or discardd by stp k 1 or soonr. Howvr, it is asy to s that if S i (j) is matchd at to S(k), thn ithr j 1 was matchd to S(h) for h < k or discardd at stp g, whr g is th last stp bfor k whr som charactr from S i was matchd. If S i (j) is matchd or discardd by stp k 1 or soonr, it is trivial to s by construction that S i (j 1) is matchd or discardd bfor stp k 1 as wll. W now giv a randomizd procdur to round this LP solution. W will show that it producs a fasibl intgral solution which has a numbr of matchs qual to at last 1 1/ tims th optimum LP valu in xpctation. It is usful to viw th LP solution in th following quivalnt way. Lt 0 s i,j,k 1 dnot th xtnt to which S i (j) has bn matchd to S(1),..., S(k 1) or discardd during th first k 1 stps, i.. s i,j,k = k 1 t=1 (x i,j,t + y i,j,t ). Lt v i,j,k = s i,j,k + x i,j,k ; thus v i,j,k s i,j,k is xactly th xtnt to which S i (j) is matchd to S(k). Also not that y i,j,k = s i,j,k+1 v i,j,k. Not that th first st of constraints implis that s i,j 1,k v i,j,k. Rounding Procdur: 1. For ach string S i choos u i [0, 1] uniformly at random. 2. For ach i, j assign S i (j) to S(k) if and only if s i,j,k u i < v i,j,k. 3. Lt N(k) dnot th numbr of charactrs assignd to S(k) at th nd of th prvious stp. N(k) is a random variabl. If N(k) = 0, S(k) is not matchd to any charactr. If N(k) = 1, w match S(k) to th uniqu S i (j) assignd to it. If N(k) 2, w arbitrarily match S(k) to on of th charactrs, and discard th rmaining N(k) 1 charactrs (nvr to b matchd again).

5 W now study th proprtis of th obtaind solution. Givn two opn intrvals of numbrs I 1 = (l 1, u 1 ) and I 2 = (l 2, u 2 ), w say that I 1 > I 2 if l 1 u 2, i.. ach lmnt in I 1 is gratr than vry lmnt than I 2. Fix i and j. Sinc x i,j,k 0 and y i,j,k 0 it trivially follows that v i,j,k s i,j,k+1 and hnc th intrvals I k = (s i,j,k, v i,j,k ) ar pairwis disjoint. Thus, no S i (j) can b assignd to two or mor charactrs in S. At th nd of th third rounding stp, no S(k) is matchd to two or mor charactrs. Thus, to show th validity of th solution producd aftr rounding, w nd only to show that no prcdnc constraints ar violatd for any S i. Lmma 3.1. For a fixd i and k, lt I j dnot th (possibly mpty) intrval (s i,j,k, v i,j,k ). Thn, I j1 > I j2 for all j 1 < j 2 or quivalntly, v i,j2,k s i,j1,k. In particular this implis that I j1 I j2 = for all 1 j 1 < j 2 l i Proof. It suffics to show that I j 1 > I j for all j. By th first st of constraints w know that v i,j,k s i,j 1,k for all i, j, k. This implis th lmma. W now show that no prcdnc constraints will b violatd for any S i. Suppos S i (j 1 ) is assignd to S(k 1 ) and S i (j 2 ) to S(k 2 ) for j 1 < j 2 and k 1 k 2. Sinc w us th sam random u i for all charactrs in th string S i, it must b th cas that u i (s i,j1,k 1, v i,j1,k 1 ) and u i (s i,j2,k 2, v i,j2,k 2 ), which implis that v i,j2,k 2 > s i,j1,k 1. Sinc v i,j,k is monotonically incrasing in k, and k 2 k 1, this implis that v i,j2,k 1 > s i,j1,k 1. But this violats lmma 3.1. Thus w hav shown that th rounding producs a fasibl solution. W now analyz th quality of th solution. Lmma 3.2. Lt V dnot th optimum LP valu. Th rounding procdur matchs at last V (1 1/) charactrs in S in xpctation. Proof. Lt us considr th solution obtaind aftr th first two stps of rounding. Lt N(i, k) b a random variabl that dnots th numbr of charactrs from S i that ar assignd to th charactr S(k). Lt p i,k dnot j x i,j,k = j (v i,j,k s i,j,k ). By Lmma 3.1, w know that th intrvals I j = (s i,j,k, v i,j,k ) ar disjoint. Hnc th probability that som charactr of S i is matchd to S(k) is xactly qual to th probability that u i j I j, which is xactly p i,k. Thus N(i, k) is a Brnoulli random variabl with paramtr p i,k, that is, it is 1 with probability p i,k and is 0 othrwis. Thus, N(k) is th sum of m Brnoulli random variabls with paramtrs p 1,k,..., p n,k. Lt I 0 dnot th 5 random variabl that dnot th numbr of charactrs in S that hav zro matchs. Thus, I 0 = {k : N(k) = 0}. Clarly, th numbr of matchs in th LP is i,j,k x i,j,k = i,k p i,k. Th numbr of matchs in th intgral solution is th lngth of S minus th numbr of non-matchs. Thus, th numbr of matchs is l I 0. Now P r [N(k) = 0] = Π n i=1(1 p i,k ) n i=1 p i,k. Thus, E[I 0 ] l numbr of matchs is k=1 n l E[I 0 ] l (1 n k=1 (1 1/) i=1 p i,k Thus th xpctd l k=1 i=1 i=1 p i,k ) n p i,k Th last stp holds sinc 1 x x(1 1/) for 0 x 1. Not that n i=1 p i,k 1 follows from constraint st 2. This complts th proof of thorm An improvd bound for an arbitrary numbr of machins 4.1 An O(log m/ log log m)-approximation algorithm. In this sction w considr a simpl randomizd algorithm prviously considrd [12, 19, 7]. Namly, th algorithm chooss an intgr numbr t j btwn 0 and L 1 indpndntly at random for ach job J j J. It thn constructs an infasibl schdul procssing ach job J j in no-wait fashion in th tim intrval [t j, l j + t j 1]. Th problm with this schdul is that in som tim stps, som machins must procss mor than on job ach. Lt τ it b th numbr of oprations assignd to a tim stp t on machin M i by th abov randomizd procdur. To obtain a fasibl schdul from th infasibl on constructd by th randomizd shifting procdur, w xpand vry tim stp t into max i=1,...,m τ it stps. Sinc all oprations assignd to th sam stp blong to diffrnt jobs, w can fasibly schdul all such oprations using this intrval of lngth max i=1,...,m τ it. Finally, w concatnat th schduls so obtaind for all tim stps. Th total lngth of th final fasibl schdul is at most L+l t=1 max i=1,...,m τ it (whr th uppr limit of summation corrsponds to th uppr bound of L + l on th lngth of th infasibl schdul). By Chrnoff bounds it can b shown that with high probability max i,t τ it = O(log ml/ log log ml) [12,

6 19, 7]. Thrfor, with high probability th lngth of th final schdul is O(log ml/ log log ml) max{l, l}. Instad of using this high probability rsult, w will try to stimat th xpctd schdul lngth which is E( L+l t=1 max i=1,...,m τ it ) = L+l t=1 E(max i=1,...,m τ it ). W will do this using th following lmma. Th tchniqus ar standard and similar rsults can b found in [8, 14]. W dfr th proof of this lmma to Sction 6. Lmma 4.1. (Non-uniform Balls and Bins) Suppos w hav m bins and n balls. Evry ball j chooss a bin i at random with probability λ ij, i.. m i=1 λ ij 1. Th xpctd numbr of balls in vry n bin is at most on, i.. j=1 λ ij 1. Thn th xpctd maximum numbr of balls in any of th m bins is O(log m/ log log m). Lmma 4.1 immdiatly implis an uppr bound for L+l t=1 E(max i=1,...,m τ it ). In vry tim intrval of unit lngth w hav an instanc of th balls and bins problm with m bins and n balls. Ball j landing in bin i corrsponds to an opration of job J j bing schduld on machin M i at tim stp t by th randomizd shifting procdur. Lt p ijt b th probability of that vnt. Sinc our algorithm procsss at most on opration of vry job in any tim unit w hav that m i=1 p ijt 1. On th othr sid p ijt l ij /L and thrfor n j=1 p ijt 1. Thrfor, E(max i=1,...,m τ it ) = O(log m/ log log m) and w obtain th following. Thorm 4.1. Th xpctd makspan of th fasibl schdul obtaind by th abov randomizd algorithm is O(log m/ log log m) max{l, l}. 4.2 An approximation algorithm with an additiv prformanc guarant.our scond randomizd algorithm also chooss random intgr shifts t j in th intrval [0, L 1] for vry job J j. Th diffrnc is that instad of procssing vry job in th tim intrval [t j, t j + l j ], w procss job J j in th tim intrval [t j, t j + l j K ε log m 1] with qual dlays of K ε log m btwn conscutiv oprations of th sam job whr K ε = 4/ε 2 and ε > 0 is an arbitrary prcision paramtr. This schdul has lngth at most L + (K ε log m)l. This schdul also may b infasibl, but unlik th nowait schdul from th prvious sction w will show it can b transformd into a fasibl schdul of lngth (1 + ε)l + (1 + ε)(k ε log m)l. W now show how to transform an infasibl schdul into a fasibl on with (1 + ε) factor incras in th xpctd lngth of th schdul. Considr an infasibl schdul obtaind by th modifid randomizd shifting procdur. This schdul has lngth at most L + (K ε log m)l. W split th tim intrval 6 [0, L + (K ε log m)l] into conscutiv intrvals of lngth K ε log m; th last intrval may hav smallr lngth. Th main ida is that sinc ach intrval can contain at most on opration from ach job, th ordr of oprations within an intrval dos not mattr. In particular, any opration within an intrval can b schduld in any ordr in that intrval. Thus, it suffics to show that in any intrval, th xpctd numbr of oprations on th machin with maximum oprations is (1+ɛ)K ε log m. This will imply that all th oprations in an intrval can b fasibly schduld in xpctd in O(1 + ɛ)k ε log m tim stps. To bound th xpctd numbr of oprations in an intrval on th maximally loadd machin, w us th following variant of th balls and bins lmma, th proof of which can b found in Sction 6. Lmma 4.2. Lt 0 < ε < 1 and K ε = 4/ε 2. Suppos w hav m bins and n balls. Evry ball j chooss a bin m i at random with probability λ ij, i.. i=1 λ ij 1. Th xpctd numbr of balls in vry bin is at most n j=1 λ ij K ε log m. Thn th xpctd maximum numbr of balls in any of th m bins is (1+2ε)K ε log m. Thus w obtain th following. Thorm 4.2. Th xpctd makspan of th fasibl schdul obtaind by th abov randomizd algorithm is (1 + ε)l + O(log m)l. Rmark. Applying th sam algorithm for th nonprmptiv job shop schduling problm with gnral procssing tims and using dlays of (K ε log m)p max btwn conscutiv oprations of th sam job, whr p max is maximum procssing tim in th instanc, w can gt a fasibl schdul of lngth at most (1+2ε)L+ O(µ log m)p max, whr µ is maximum numbr of oprations pr job and a hiddn constant dpnds on ε. 5 A (2 + ε)-approximation for any constant numbr of machins W split th st of jobs into two sts: L = {J j l j εl/(k ε log m) is th st of big jobs and S = J \ L is th st of small jobs. Obsrv that sinc th numbr of machins is constant th numbr of big jobs is O(log m) which is a constant. Th st of small jobs is schduld by using th randomizd algorithm from th prvious sction. Thorm 4.2 guarants that th schdul lngth is at most (1 + O(ε)) max{l, l}. On th othr hand, th st of big jobs can b schduld optimally using a straightforward dynamic program, that for ach tim stp and for ach job stors how many oprations hav bn schduld thus far in th partial schdul [1]. In gnral, this givs a psudopolynomial tim algorithm as th procssing tims could

7 b xponntially larg. Howvr, for th prmptiv job shop problm, th lngth of th job is qual to th numbr of oprations and hnc polynomial in th siz of th input. Evn if w assum a tightr ncoding whr th numbr of conscutiv unit lngth of oprations on a machin is ncodd in binary, th job lngth can b mad polynomial using standard rounding tchniqus. Thrfor, w can schdul th big jobs with makspan at most Cmax (or (1 + ε)cmax if w includ th factor w los whn w apply rounding and scaling to dcras numbr of unit lngth oprations). Concatnating th two schduls, w obtain a schdul of lngth at most (2 + ε)cmax. 6 Proofs of Lmmas 4.1 and 4.2 W us th following vrsion of Chrnoff bounds as givn on pag 267, Corollary A.1.10, [2]. Lmma 6.1. Suppos X 1,..., X n, ar 0-1 random variabls, such that P r[x i = 1] = p i. Lt X = n i=1 p i and X = n i=1 X i. Thn P r[x X a] a (a+ X) ln(1+a/ X) W will also nd th following corollary. Lmma 6.2. P r[x X a min(1/5,a/4 X) a] Proof. Lt x = X/a. Thn th right-hand sid of th inquality in Lmma 6.1 can b writtn as a((x+1) ln(1+1/x) 1). Now, (x + 1) ln(1 + 1/x) 1 is dcrasing in x. At x = 2, its valu is 3 ln 5/3 1 1/5. For x > 2, it is at last (x + 1)(1/x 1/2x 2 ) 1 = 1/2x 1/2x 2 1/4x. Hnc, by th union bound, a ln ln m/2 P r[b > 1 + a] mp r[b i 1 + a] = m Now w hav E[B] = P r[b x] x=1 3 ln m ln ln m + 3 ln m ln ln m + m a 3 ln m ln ln m a 3 ln m ln ln m 3 ln m ln ln m + 2 m 1/2 ln ln m = O Proof. (of Lmma 4.2) By Lmma 6.2 P r[b 1 + a] a ln ln m/2 ( ) ln m ln ln m P r[b i K ε ln m + δ] P r[b i E[B i ] + δ] δ min(1/5,δ/4 X) W will only b intrstd in δ εk ε ln m and small ε > 0. Thus, P r[b i K ε ln m+δ] δ ε/4. and hnc by th union bound P r[b K ε ln m + δ] m δ ε/4. Thus, w hav that E[B] (1 + ε)k ε ln m + m δ ε/4 δ>εk ε ln m (1 + ε)k ε ln m + m 4 ε ε2 K ε ln m/4 Rcalling that K ε = 4/ε 2, w hav that E[B] (1 + ε)k ε ln m + 4/ɛ (1 + ɛ)k ε ln m + ɛk ε (1 + 2ε)K ε ln m Proof. (of Lmma 4.1) Lt B ij b a 0-1 random variabl that is 1 iff ball j gos to bin i. Lt B i dnot th numbr of balls in bin i. Finally, lt B = max i B i. As E[B i ] 1, by Lmma 6.1 w hav that P r[b i 1 + a] P r[b i E[B i ] + a] For a 3 ln m ln ln m, w hav that a (a+e[bi]) ln(1+a/e[bi]) a a ln(1+a) ln(1 + a) 1 ln(ln m/ ln ln m) (ln ln m)/2 which implis that, a ln ln m/2 P r[b i 1 + a] 7 7 Opn Problms Two outstanding opn qustions that rmain for th gnral prmptiv job shop problm ar: 1. Is thr an O(1)-approximation for th gnral prmptiv job shop problm with an arbitrary numbr of machins? As all lowr bounds on th makspan ar ssntially O(max{L, l}), a vry intrsting rsult would b to show an instanc such that th optimal makspan is not within a constant factor of max{l, l}. It is known that for nonprmptiv job shop schduling thr is no O(1)-approximation algorithm with rspct to max{l, l} vn for acyclic instancs [6]. W bliv that rsolving this qustion for prmptiv schduls would rquir significant nw insights into thir structur.

8 2. Is thr a polynomial tim approximation schm for th cas of a constant numbr of machins? Our algorithm in this papr givs a PTAS if l ɛl/ log m. On th othr hand, if w rstrict th problm to instancs with only a fw long jobs that compris all but an ε fraction of th load, thn w can gt a PTAS via dynamic programming. Making progrss on th gnral cas, in which both long jobs and short jobs may mak up significant portions of th load, would rquir an undrstanding of how small jobs and long jobs ar mixd togthr in an optimal schdul. A first stp in this dirction would b to undrstand th approximability of th string matching problm from Sction 3. Rfrncs [1] S. B. Akrs, A graphical approach to production schduling problms, Oprations Rsarch 4 (1956), [2] N. Alon, and J. Spncr, Th Probabilistic Mthod. John Wily & Sons, [3] E.J. Andrson, T.S. Jayram, and T. Kimbrl. Tightr Bounds on Prmptiv Job Shop Schduling with Two Machins. Computing 67 (2001), pp [4] P. Baptist, J. Carlir, A. Kononov, M. Quyrann, S. Svastianov and M. Sviridnko, Structural Proprtis of Prmptiv Schduls, submittd for publication. [5] A. Czumaj and C. Schidlr, A Nw Algorithmic Approach to th Gnral Lovasz Local Lmma with Applications to Schduling and Satisfiability Problms, Proc. 32 ACM Symposium on Thory of Computing (STOC), [6] U. Fig and C. Schidlr, Improvd bounds for acyclic job shop schduling. Combinatorica 22 (2002), no. 3, [7] L.A. Goldbrg, M. Patrson, A. Srinivasan, and E. Swdyk. Bttr approximation guarants for jobshop schduling. SIAM J. Discrt Math. 14 (2001), [8] G. Gonnt, Expctd lngth of th longst prob squnc in hash cod sarching. J. Assoc. Comput. Mach. 28 (1981), no. 2, [9] T. Gonzalz and S. Sahni. Flowshop and jobshop schduls: complxity and approximation. Oprations Rsarch 26, pp , [10] K. Jansn, R. Solis-Oba, and M. Sviridnko. Makspan minimization in job shops: a linar tim approximation schm. SIAM J. Discrt Math. 16 (2003), [11] E.L. Lawlr, J.K. Lnstra, A.H.G. Rinnooy Kan, and D.B. Shmoys. Squncing and Schduling: Algorithms and Complxity. In S.C. Gravs, A.H.G. Rinnooy Kan, and P.H. Zipkin (ds.), Logistics of Production and Invntory, Handbooks in Oprations Rsarch and Managmnt Scinc 4, North-Holland, Amstrdam, 1993, [12] F.T. Lighton, B. Maggs, and S. Rao. Packt routing and jobshop schduling in O(congstion + dilation) stps. Combinatorica 14, pp , [13] F.T. Lighton, B.M. Maggs, and A.W. Richa, Fast algorithms for finding O(congstion + dilation) packt routing schduls. Combinatorica 19 (1999) [14] M. Raab and A. Stgr, Balls into bins a simpl and tight analysis. Randomization and approximation tchniqus in computr scinc (Barclona, 1998), , Lctur Nots in Comput. Sci., 1518, Springr, Brlin, [15] S.V. Svastianov and G.J. Wogingr. Makspan minimization in prmptiv two machin job shops. Computing 60 (1998), [16] S. Svast janov, On som gomtric mthods in schduling thory: a survy. Discrt Appl. Math. 55 (1994), no. 1, [17] S. Svastianov, Bounding algorithm for th routing problm with arbitrary paths and altrnativ srvrs, Cybrntics 22 (1986), pp [18] Y. Sotskov and N. Shakhlvich, NP-hardnss of shopschduling problms with thr jobs. Discrt Appl. Math. 59 (1995), no. 3, [19] D. Shmoys, C. Stin, and J. Win. Improvd Approximation Algorithms for Shop Schduling Problms. SIAM Journal on Computing 23:3, , [20] D. P. Williamson, L. A. Hall, J. A. Hoogvn, C. A. J. Hurkns, J. K. Lnstra, S. V. Svast janov and D. B. Shmoys, Short shop schduls, Opration Rsarch 45 (1997), pp