An FPTAS for the iniu total weighted tardiness proble with a fixed nuber of distinct due dates George Karaostas Stavros G. Kolliopoulos Jing Wang February 3, 2009 Abstract Given a sequencing of jobs on a single achine, each one with a weight, processing tie, and a due date, the tardiness of a job is the tie needed for its copletion beyond its due date. We present an FPTAS for the basic scheduling proble of iniizing the total weighted tardiness when the nuber of distinct due dates is fixed. Previously, an FPTAS was nown only for the case where all jobs have a coon due date. 1 Introduction The iniu total weighted tardiness proble for a single achine is defined as follows. We are given n jobs, each with a weight w j > 0, processing tie p j, and due date d j. When these jobs are sequenced on a single achine, each job j will have a copletion tie C j. The tardiness T j of job j is defined as ax{0,c j d j }. If T j = 0, the job is early, otherwise it is tardy. The objective is to iniize the total weighted tardiness, i.e., we loo for a schedule that iniizes j w jt j. The proble is very basic in scheduling (see surveys [1, 10] and the references in [4, 5]) and is nown to be NP-hard [8] even in the case of unit weights [3]. Despite the attention it has received, frustratingly little is nown on it approxiability. The best nown approxiation algorith has a perforance guarantee of n 1 [2]. For the unit weight case, Lawler gave early on a fully polynoialtie approxiation schee (FPTAS) [7], which is a odification of his pseudopolynoial dynaic prograing algorith in [6]. For general weight values, the proble reains NP-hard even when all jobs have a coon due date [11]. Kolliopoulos and Steiner [5] gave a pseudopolynoial dynaic prograing algorith for the case of a fixed nuber of distinct due dates. Using essentially Lawler s rounding schee fro [7], they obtained an FPTAS only for the case of polynoially bounded weights. Kellerer and Strusevich [4] gave an FPTAS for general weights in the case where all jobs have a coon due date. The existence however of an FPTAS for the case of general weights and a fixed nuber of distinct due dates has reained open. We note that for a general nuber of distinct due dates the proble becoes strongly NP-hard [6]. In this wor, we settle the case of a fixed nuber of distinct due dates by giving an FPTAS. We design first a pseudopolynoial algorith and then apply the rounding schee of [4] to obtain the Dept. of Coputing & Software, and School of Coputational Engineering & Science, McMaster University, Hailton, ON, Canada. E-ail: araos@caster.ca. Research supported by an NSERC Discovery grant. Dept. of Inforatics and Telecounications, National and Kapodistrian University of Athens, Athens 157 84, Greece. URL: wwww.di.uoa.gr/ sg School of Coputational Engineering & Science, McMaster University, Hailton, ON, Canada. E-ail: wang257@caster.ca. Research supported by an NSERC Discovery grant. 1
desired approxiation schee. We exploit two crucial properties of the algoriths in [4]. The first is that the optial choice is feasible at every job placeent the FPTAS perfors (cf. Lea 10). This step-by-step iicing of the optial chain of coputation is crucial for bounding the approxiation error. Of course, the schedule we output ay be suboptial due to our approxiate ( rounded ) estiation of tardiness. The second property is that the rounding schee of [4] produces values which correspond to actual schedules; therefore by rounding up the processing tie of tardy jobs with due date d, one rounds down the processing tie of early jobs with the sae due date by the sae aount. Since the total tie needed for these jobs reains the sae, this eans that there is epty space that allows our algorith to push bac the extra tardy processing tie towards the past. This need for preeption, i.e., allowing the processing of a job to be interrupted and later restarted, did not arise in [4] where the extra tardy processing tie past the coon due date D could always be accoodated in the tie interval [D, ). In addition to these basic facts, we need a nuber of other new ideas, soe of which we outline next. Our algorith wors in two stages. First, via dynaic prograing it coputes an assignent of the job copletion ties to the tie horizon, where only a subset of the jobs is explicitly paced and the rest are left floating fro their copletion tie bacwards. This is what we call an abstract schedule. In the second stage, a greedy procedure allocates the actual job lengths, possibly also with preeption. As in previous algoriths, the jobs that straddle a due date in a schedule, the so-called straddlers, play an iportant role. We observe that only the placeent of the tardy straddlers is critical. The tie intervals, called superintervals, between consecutive tardy straddlers, for the basic tie unit on our tie horizon. The scheduling of a job j as early can then be localized within only one of these superintervals, depending on the actual d j value (cf. The Braceting Lea 3). This helps to shrin the state space of the dynaic progra. It is well-nown that the preeptive and non-preeptive optia coincide when iniizing tardiness on a single achine [9]. This powerful fact has found only liited use in approxiation algoriths so far, for exaple through the preeptive scheduling of early jobs in [5]. We tae the opposite view fro [5] and insist on the non-preeptive scheduling of early jobs. Moreover, all early jobs are paced explicitly in the abstract schedule. This is necessary since early jobs are particularly difficult to handle: enuerating their total length is prohibitive coputationally and distorting their placeent even by a tiny aount ight result in a severely suboptial schedule. We allow instead preeptive scheduling of the tardy jobs. As explained above, preeption will allow us to flexibly push bac the extra tardy processing tie, introduced by the rounding, towards the past. Following this idea to its natural conclusion, we allow even straddlers to be preepted. In the final schedule, it could be that only the copletion tie of a tardy job happens in the interval in which it was originally assigned by the dynaic progra, while all the processing happens earlier. The algebraic device we introduce that allows the abstract schedule to eep soe of the jobs floating, without pinning down anything but their copletion tie, is the potential epty space within a prefix of a schedule (cf. Eq. (3) below). To ensure that preeptions can be ipleented into actual epty space is perhaps the largest technical difficulty in our proof. The approxiability of total weighted tardiness proble with an arbitrary nuber of distinct due dates reains as the ain open proble. 2 Structural properties of an optial schedule We are given n jobs j = 1,...,n, each with its own processing tie p j and weight w j and a due date fro a set of K possible distinct due dates {d 1,d 2,...,d K }, where K will be assued to be a constant for the rest of this paper. For convenience, we are also going to define the artificial 2
due date d 0 = 0. The due dates partition the tie horizon into K + 1 intervals I l = [d l 1,d l ) for l = 1,...,K, and I K+1 = [d K, ). We partition the jobs into K classes C 1,C 2,...,C K according to their due dates. A crucial concept for the algoriths we describe is the grouping of intervals I l in the following anner: for any i u,i u+1, intervals I iu+1,i iu+2,...,i iu+1 are grouped into a superinterval G iui u+1 = I iu+1 I iu+2... I iu+1 = [d iu,d iu+1 ), if straddlers S iu and S iu+1 are consecutive tardy straddlers, i.e., there is no other tardy straddler in between due dates d iu,d iu+1. Note that it ay be the case that i u+1 = i u + 1, i.e., G iui u+1 I iu+1 if both S iu,s iu+1 are tardy. Also, since straddler S K is tardy, the last superinterval is G K,K+1 = I K+1. In any schedule of the n jobs, a job that finishes before or on its due date will be an early job, otherwise it will be tardy. We also call any job that starts before or on a due date but finishes after it a straddler. It is well-nown [9] that the optial values of the preeptive and the non-preeptive version of the proble are the sae. Therefore we can assue that the optial schedule is a nonpreeptive one. In it the straddlers will appear as contiguous blocs, crossing one or ore due dates. For easiness of exposition, we will assue that there is an optial schedule with distinct straddlers for every due date, i.e., there are K distinct straddlers S 1,...,S K corresponding to due dates d 1,...,d K. After the description of the algoriths, it should be clear how to odify the in order to deal with the special case of soe straddlers crossing ore than one due dates. For convenience, let also S 0 be an artificial tardy straddler for d 0 with w S0 = p S0 = 0. In any optial schedule, the achine has clearly no idle tie. Hence, wlog, due dates that are greater than j p j, can be set to. Accordingly, we can assue that there is a straddler for every due date. Tardy straddlers are going to be of particular interest to what our algoriths do. We will assue that we have guessed the nuber M K of tardy straddlers and these tardy straddlers S i1,...,s im of the optial schedule (also S i0 = S 0 ). By guessing, we ean the exhaustive enueration of all cobinations of jobs with due dates (with repetition in the general case where a job can be straddler of ore than one due dates), which produces a polynoial nuber of possibilities, since K is constant. Let = n M be the nuber of the reaining jobs, which are ordered according to p 1 w 1 p 2 their weighted shortest processing ties (WSPT), i.e., w 2... p w. With soe abuse of terinology, we will call these jobs non-straddling, although soe of the are the early straddlers. We will also assue that we have guessed a bound Z ub such that for the optial value OPT we have Z ub /2 OPT Z ub. 1 It should be obvious that, in any interval I l, the tardy jobs in that interval are processed before the early ones. It is also well-nown (e.g., see Lea 2.1 in [5]) that the tardy jobs ust be processed in WSPT order. With respect to a given partial schedule we define the following quantities, which are going to be iportant throughout this wor: y (i 1)t, 1 t < i K + 1, 1 : the total processing tie of those (tardy) jobs aong the first (in WSPT order) jobs, that belong to class C t and are in I i. Also define y 0t = 0 for all t. W (i 1)t, 1 t < i K + 1, 1 : the total weight of the jobs in the previous ite. A t, 1 t K, 1 : the total processing tie of the class C t jobs aong the first jobs. Notice that these quantities can be calculated in advance. e it, 1 i t K, 1 : the total processing tie of those (early) jobs aong the first (in WSPT order) jobs, that belong to class C t and are in I i. 1 This can be done by running the algorith with Z ub = 2 x, for all x = 0, 1,..., U, with U being a trivial upper bound of OPT, e.g. U = log(n 2 w axp ax) = O(log n + log w ax + log p ax). 3
The following leas are iportant properties of an optial schedule: Lea 1 In the optial schedule and for any 1 i K, if S i is tardy, then for any 1 l i and any i + 1 u K, we have e lu = 0. Lea 2 In the optial schedule and for any 2 i K, if S i 1 is early, then y (i 1)u any 1 u i 1, i.e., there are no tardy jobs in I i. = 0 for Lea 2 iplies that the only non-zero y s are the ones that correspond to the first interval of each superinterval. Therefore, fro now on we will use only the values y iut, 1 u M, 1 t i u, 1. Leas 1 and 2 iply that for every 1 and for every 1 t K s.t. i s 1 < t i s for soe 1 s M we have A t = M u=s y iut + t q=i s 1 +1 e qt (1) A direct consequence of Lea 1 and the definition of a superinterval is the following. Lea 3 (Braceting Lea for early jobs) Let u M. In an optial schedule only jobs fro classes C t, with i u 1 < t i u can be assigned as early in the superinterval G iu 1 i u. 3 A dynaic prograing algorith to find an abstract schedule An abstract schedule is an assignent of the the non-straddling jobs to superintervals so that (i) early jobs are feasibly and non-preeptively paced within their assigned superinterval (ii) there is enough epty space so that tardy jobs that coplete in their assigned superinterval can be preeptively paced and (iii) there is enough epty space so that the M tardy straddlers can be preeptively paced. An abstract -schedule,, is an abstract schedule for the first nonstraddling jobs. In this section we describe a pseudopolynoial dynaic prograing algorith (DP) that coputes a suitable abstract schedule. In the next section we show how to pin down the actual processing of the tardy jobs and the straddlers, so that the abstract schedule is converted to an actual schedule of the n jobs with iniu total tardiness. The DP algorith guesses the M tardy straddlers. Extending the dynaic prograing of [4], the states of DP store the following values for a (partial) schedule of the first (in WSPT order) of the non-straddling jobs 2 : (,Z,y i 11,W i 11,y i 21,W i 21,,y i M1,W i M1,y i 12,W i 12,,y i MK ),W i MK, (2) where Z is the total weighted tardiness of the scheduled jobs. Note that soe of the y iuj iuj,w in (2) ay not exist, if i u < j. As in [4], the weight values W iuj will be needed when the tardy straddlers will be re-inserted at the end. The initial state will be (0,0,...,0). A state-to-state transition fro state (2) corresponds to the insertion of the (+1)-th job in a super-interval of the (partial) abstract schedule of the previous jobs. Such a transition corresponds to the choice of inserting this job in a superinterval, and ust 2 Recall that we are looing for schedules that do not include the tardy straddlers, yet they have enough epty space to accoodate the re-insertion of these straddlers in their correct position. Moreover, in every interval I l, the tardy jobs of that interval (if they exist) appear as a bloc starting at d l 1, followed iediately by the bloc of early jobs in this interval. 4
be feasible. The feasibility conditions, described in detail below, require that there is enough epty space to insert the new job in the selected superinterval, and there is still enough epty space for the re-insertion of the straddlers. Note that the cobination of the class C t of the inserted job and the superinterval G iu 1 i u chosen for it by the transition deterines whether this job is early or tardy: if 1 t i u 1 then the job is tardy, otherwise it is early. In order to be able to chec the feasibility of the transitions, we would lie to be able to calculate the epty space in every superinterval fro the inforation stored in states (2). Unfortunately, this is not possible, because essentially there are any possibilities for the placeent of early jobs that yield the sae state and eeping trac of all these possibilities would blow up the state space. As a result of this liited inforation, soe of the space that loos epty will be actually needed to accoodate preepted parts of tardy jobs fro later superintervals. Nevertheless, we can calculate the potential epty space for prefixes of the schedule that start fro tie t = 0. The processing tie for a tardy job is just slated for the prefix that ends at its assigned copletion tie by the first (dynaic prograing) stage of the algorith, without pinning down its exact placeent. This placeent is fixed only during the second stage of the algorith. We introduce the following set of prefix values, which can be calculated given a state (2): L 0l, 1 l K, 1 : the total space fro d 0 to d l inus the space taen by the jobs whose class indices are less than or equal to l. Given 1 l K, let s be such that i s 1 < l i s. Then L 0l can be coputed fro the inforation at hand as follows: = d s 1 l ( L 0l = d l ( i j i j j=1 q=i j 1 +1 h=q i=1 s 1 A i i j j=1 h=1 e qh + y i jh q=i s 1 +1 h=q j=s h=1 s 1 e qh ) ( s 1 y ijh ) ( i j j=1 h=1 i j j=1 h=1 y i jh ) y i jh ) = d l M A i + i=1 j=s h=1 y i jh (3) Recall that there are M tardy straddlers {S iu } M overall. We assue that the (+1)-th job J +1 belongs to class C t, and that we want to schedule it in superinterval G iu 1 i u. Note that Lea 3 iplies that, to even consider such a placeent, t i u ust hold. The three feasibility conditions that ust be satisfied by a DP transition fro state (2) follow. Fro equation (3), given the state inforation, all three can be effectively checed. Condition (1): t i u 1, i.e., J +1 is tardy. 1a. Chec whether L 0l L0i u 1 p +1 holds l s.t. i u 1 l i u. 1b. If 1a doesn t hold, chec whether L 0l p +1 holds l s.t. i u 1 < l i u. 1c. Chec whether L 0i j p +1 holds j s.t. u < j M. Condition (2): i u 1 < t i u., i.e., J +1 is early. 2a. Chec whether L 0l L0i u 1 p +1 holds l s.t. t l i u. 2b. If 2a doesn t hold, chec the following according to which case applies: 2b.1. i u 1 L 0i u 1 : Chec whether d l d iu 1 ( l q=i u 1 +1 L 0l p +1 holds l s.t. t l i u ; 2b.2. i u 1 > L 0i u 1 : Chec whether L 0l p +1 holds l, s.t. t l i u. 2c. Chec whether L 0i j p +1 holds j, s.t. u < j M. l v=q eqv ) p +1 and 5
Condition (3): Chec whether L 0j +1 u 1 h=1 p i h holds u s.t. 1 < u M and j s.t. i u 1 < j i u. Condition (3) will ensure that there is always enough epty space to fit the straddlers in the final schedule (Lea 8). Conditions (1a) (and (2a)) are satisfied when there is enough space to fit J +1 as tardy (or early) in a non-preeptive schedule. Since we will prove (Lea 6) that Conditions (2b), (2c) are enough to guarantee (with a soe shuffling around) that early jobs can always be inserted non-preeptively in a preeptive schedule, and Lea 7 will show that even if Condition (1a) is not satisfied, we are able to insert tardy jobs preeptively in a preeptive schedule if Conditions (1b), (1c) hold, Conditions (1a),(2a) are redundant if we are looing for a preeptive schedule. But we will use the fact that Conditions (1a),(2a),(3) are enough for the construction of an optial DP algorith which produces an optial non-preeptive schedule in the analysis of our FPTAS (Sections 4, 5). There is a ore concise way of expressing Condition (2), as shown in the following Lea 4 Condition (2b) can be replaced by the following: 2b. Chec whether d l d iu 1 ( l l q=i u 1 +1 v=q eqv + ax{ i u 1 l s.t. t l i u. L 0i u 1,0}) p +1 holds The new state ( + 1,Z +1,...) after the (feasible) insertion of the ( + 1)-th job J +1 of class C t in superinterval G iu 1 i u is coputed as follows: J +1 is early: Set Z +1 = Z, y iuj +1 = yiuj, W iuj +1 = W iuj for all 1 u M, 1 j i u. J +1 is tardy: Set Z +1 = Z + w +1 ( i u 1 + p +1 + d iu 1 d t ), y i u 1t +1 = y i u 1t + p +1, W i u 1t +1 = W i u 1t + w +1. Note that we reject the insertion if Z +1 > Z ub, and if at soe point we deterine that this inequality is true for all possible insertions of J +1 then we reject Z ub, we replace it with a new Z ub := 2Z ub and start the algorith fro scratch. We need to show that the assignent of jobs to the superintervals eets the definition of the abstract schedule. First we elucidate the relation of the L values with the actual epty space. Lea 5 Let u M, 1. If L 0i j 0, j s.t. 1 j u, then there is enough actual epty space to pac preeptively the tardy jobs that have so far been assigned to the first u superintervals. Proof: Note that these tardy jobs ust each be scheduled so that they coplete in their respective superinterval. Their processing can tae place anywhere before their copletion tie. For a superinterval G ij 1 i j, define L i (j 1)i j := L 0i j L0i (j 1). By Lea 3 this quantity equals the epty space in [d ij 1,d ij ) plus the space potentially needed in [d ij 1,d ij ) by pieces of preepted tardy jobs with copletion tie after d ij. Clearly L 0iu = u j=1 L(i j 1)i j. Each of the ters in the su can be negative or nonnegative. A negative ter corresponds to a superinterval with an excess portion of tardy jobs which needs to be oved (preepted) towards the past. A nonnegative ter corresponds to a superinterval with an excess of space which can be used to accoodate preepted parts of jobs that coplete in future superintervals. Therefore, if L 0i h 0, h s.t. 1 h j, the su L 0i j 0, is the net epty space available for accoodating preeptions fro jobs that coplete after d ij once all tardy jobs assigned in [d 0,d ij ) have been paced. We establish that the early jobs are feasibly paced. Lea 6 Assue state (2) corresponds to an abstract -schedule. Conditions (2) and (3) iply that job J +1 is paced non-preeptively as early in the intervals I iu 1 +1,...,I iu, so that we obtain an abstract ( + 1)-schedule. Moreover all early jobs coplete as close to their due date as possible. 6
Proof: If Condition (2a) holds, there is at least p +1 epty space in the superinterval G iu 1 i u although (i) it ay not be contiguous (ii) it ay not occur in its entirety before d t (iii) part of it ay be earared to accoodate preeptions fro tardy jobs assigned after d iu. If Condition (2b) holds, one has in addition to ove parts of tardy jobs fro G iu 1 i u towards the past in order to create the epty space of (2a). If neither of the holds, it is ipossible to pac J +1 as early within this superinterval. We establish that assigning J +1 under Conditions (2a) or (2b) has no ill effect on the first jobs. Then we consider how the possibly fragented epty space can be used to feasibly pac J +1. After assigning J +1 to G iu 1 i u, L 0i j +1 = L0i j +1, j, s.t. 1 j i u 1. By Lea 5, the feasible assignent of jobs to intervals before d iu 1 is not affected. Space for straddlers is preserved because of Condition (3). Early jobs assigned after d iu are not affected either. We only have to worry about tardy jobs assigned after d iu 1 and early jobs in the superinterval G iu 1 i u. The forer can afford to lose soe of their coveted space because of Lea 5 and Conditions (2a) (or (2b)) and (2c). The latter are paced according to the schee that follows. Since our reasoning applies regardless of whether Condition (2a) or (2b) holds let L i u 1l denote L 0l L0i u 1 in the forer case and d l d iu 1 ( l l q=i u 1 +1 v=q eqv + ax{ i u 1 L 0i u 1,0}) in the latter (fro Lea 4). Recall that i u 1 < t i u. We have L i u 1i u p +1, and that L i u 1(i u 1) is the space that is either epty or contains (parts of) jobs fro class C iu (due to Lea 1) in [d iu 1,d iu 1). We can greedily push all the latter jobs as close to d iu as possible using the epty space closest to d iu. After we are done, the epty space between d iu 1 and d iu 1 ust be at least p +1 : If all C iu jobs fit in the epty space of I iu, then L i u 1(i u 1) (2) L i u 1(i u 1) represents actual epty space and by Condition p +1 ; otherwise, there ust be no epty space left in I iu, which eans that the whole epty space in G iu 1 i u, which we now to be at least p +1, is concentrated in [d iu 1,d iu 1). Continuing to successively push jobs of classes C iu 1,C iu 2,...,C t+1 as close as possible to due dates d iu 1,d iu 2,...,d t+1 respectively, at the end we will have at least p +1 units of epty space in [d iu 1,d t ], and in this epty space we can insert (early) job J +1, without disturbing the previous jobs just as the stateent of the lea specifies. It reains to argue about the pacing of tardy jobs (proof in the appendix): Lea 7 Assue state (2) corresponds to an abstract -schedule. Conditions (1) and (3) iply that one can assign job J +1 to coplete as tardy in the superinterval G iu 1 i u, so that we obtain an abstract ( + 1)-schedule. 4 Producing an optial schedule The abstract schedule produced so far by the dynaic prograing algorith has placed the early jobs in their superintervals non-preeptively and as close to their due date as possible (as shown by Lea 6). It has also placed the copletion ties of the tardy jobs in their superintervals 3. But we have not specified how the (preepted) tardy jobs are arranged, since Condition (1) only ensures that there is enough epty space to fit each tardy job, possibly broen in pieces. Now we describe the procedure that allocates the tardy jobs on the tie horizon: 1. The (tardy) jobs in the last interval I K,K+1 are scheduled in that interval non-preeptively in WSPT order. 3 In fact, we now the specific interval of each copletion tie, since only the first interval of every superinterval can be used for the copletion of tardy jobs. 7
2. For u = M,M 1,...,1 loo at the tardy jobs with copletion ties in G iu 1 i u, i.e., in interval I iu 1,i u 1 +1 in WSPT order. While there is epty space in this interval, fit in it as uch processing tie as possible of the job currently under consideration. If at soe point there is no ore epty space, the rest of the processing ties of these tardy jobs will becoe preepted pieces to be fitted soewhere in [d 0,d iu 1 ). Then, we fill as uch of the reaining epty space in G iu 1 i u as possible using preepted pieces belonging to preepted tardy jobs in [d iu,d K ] in WSPT order (although the particular order doesn t atter). When we run out of either epty space or preepted pieces, we ove to the next u := u 1. We note that the above process does not change the quantities L 0j, j = 1,2,...,K, and therefore Condition (3) continues to hold. The placeent of the tardy straddlers will coplete the schedule the algorith will output. The following lea shows how we will place the straddlers preeptively so that two properties are aintained: (a) straddler S iu copletes at or after d iu and before d iu+1, for all u = 1,2,...,M 1, and (b) the prefix of the schedule that contains all straddlers processing tie is contiguous, i.e., there are no holes of epty space in it. We will need property (b) in the calculation of the total tardiness of the final schedule below and in our FPTAS. We ephasize that (b) ay force us to preept straddlers: for exaple, suppose that the epty space in [d 0,d 1 ) is uch bigger than M ; then our schedule will use M units at the beginning of that epty space to process S j1,...,s jm, while setting their copletion ties at d j1,...,d im respectively. Lea 8 The placeent of the tardy straddlers can be done so that properties (a),(b) above are aintained. Given that Z ub is large enough, the dynaic prograing will ultiately produce a set of states with their first coordinate equal to, i.e., states that correspond to partial schedules of all nonstraddling jobs. Since these states satisfy Condition (3), Lea 8 iplies that we can re-insert the straddlers at their correct position without affecting the earliness of the early or the placeent in intervals of the tardy non-straddling jobs, thus creating a nuber of candidate full schedules. Let {T iu } M be the tardiness of the M tardy straddlers. Also, note that due to property (b) in Lea 8, x iu := ax{0, u p Sil L 0iu } (4) is the part of S iu beyond due date d iu. Then, if S iu C t (with t i u ), we have T iu = x iu + d iu d t, u = 1,...,M, and the total weighted tardiness of a candidate schedule is Z = Z + w iu T iu + ( i u W iul )x i u. (5) The algorith outputs a schedule with iniu Z by tracing bac the feasible transitions, starting fro the state that has the Z which produced the iniu Z. It should be obvious how to extend the description of the algorith above to include the case of a straddler being the sae for ore than one (consecutive) due dates. The following is also fairly easy to prove (proof in the appendix): Theore 1 The dynaic prograing algorith above produces an optial schedule. Note that in the proof of Theore 1 we didn t need to chec Conditions (1b),(2b). If, in addition, we require that the algorith is non-preeptive, then the proof goes through without checing for Conditions (1c),(2c), since they are satisfied trivially by the optial non-preeptive schedule. Hence we have the following 8
Corollary 1 The non-preeptive DP algorith with feasible transitions restricted to only those that satisfy Conditions (1a), (2a) and (3) still produces an optial (non-preeptive) schedule. Corollary 1 will be iportant for the proof of the approxiation ratio guarantee below, since we will copare the solution produced by our FPTAS to the optial schedule of the corollary. 5 The FPTAS The transforation of the pseudopolynoial algorith described in Sections 3, 4 into an FPTAS follows closely the FPTAS (Algorith Eps) in [4]. Since the running tie of the dynaic prograing part doinates the total running tie, in what follows we use the ter DP to refer to the entire process. Let ε > 0. Recall that we have guessed Z ub such that Z ub /2 OPT Z ub, and let Z lb := Z ub /2. Define δ = εz lb 4. Consider a state (,Z,yi 11,W i 11,y i 21,W i 21,,y i MK,W i MK ) of the exact dynaic prograing. Fro this state, we will deduce the states (,Z,y i 11,W i 11,y i 21,W i 21,,y i MK,W i MK ) used by the FPTAS dynaic prograing as follows: We round variable Z to the next ultiple of δ (hence Z taes at ost Zub δ = O( n ε ) distinct values). For every 1 u M, we round W iuj to the nearest power of (1 + ε/2) 1/ (hence W iuj taes O(n log W) values, where W is the total weight of the n jobs). After ordering the non-straddling jobs in WSPT order, let w π(1) > w π(2) > > w π(n) be the N distinct weight values of the non-straddlers in decreasing order. The rounding of y iuj, 1 u M is ore coplicated. Define a division of tie interval [0, Z ub w π(n) ] into subintervals {H i subintervals {Ĥi j (i)}xi i j =1 of length δ i = Z ub Zub w π(i w ) π(i 1) := [ Z ub w π(i 1), Z ub w π(i ) ]} N i =1. In turn, divide each H i into δ iw π(i ) for all 1 and 1 i K, where x i i = δ i is the nuber of such subintervals (note that the length of the last subinterval ay be less than δ i ). For each state (,Z,y i 11,,yi MK,W i MK ), the dynaic progra,w i 11 applies its O(K) transitions to generate new states ( + 1,Z +1,y i 11 +1,W i 11 +1,,yi MK +1,W i MK For the set of states which have the sae values of Z +1,W i 11 +1,,W i MK +1 +1 )., we round yiuj +1 in the following way: we group all the y iuj +1 values that fall into the sae subinterval Ĥ i j together, and eep only the sallest and the largest values in this group, say y iujax +1 and y iujin +1. We ephasize that these two values correspond to the actual processing ties of two sets of tardy jobs, and therefore none of these two values is greater than A j +1. Hence, fro the group of states generated by the DP transition, we produce and store states with at ost two values at position y ij +1, i.e., ( + 1,Z +1,y i 11 +1,W i 11 +1,,yiujax +1, y i MK +1,W i MK +1 ) and ( + 1,Z +1,y i 11 +1,W i 11 +1,,yiujin +1, y i MK +1,W i MK +1 ). Lea 9 The algorith runs in tie O((ε 1 n log W log P) Θ(K2) ). We focus on states (,Z,yi 11,W i 11,,y i MK,W i MK ), = 0,1,..., that are the sequence of transitions in the DP of Corollary 1 that produces an optial non-preeptive schedule. The following lea shows that despite the rounding used after every transition in our algorith, there is a sequence of states (,Z,y i 11,W i 11,,y i MK,W i MK ), = 0,1,..., whose transitions fro one state to the next atch exactly the job placeent decisions of the optial DP step-for-step. 9
The ey idea is that when our algorith overestiates the space needed by tardy jobs (i.e., the y s are rounded up), the space needed by the corresponding early jobs is decreased (rounded down), since the total space needed reains the sae, as (1) shows. The preeption of the tardy jobs allows us to treat the total space taen by the jobs in a class C t as a unified entity, because the overestiated processing tie of tardy jobs in this class can be placed (preepted) in the place of early jobs, whose processing tie is reduced by an equal aount. This is the basic otivation behind our introduction of tardy job preeption. Lea 10 For every = 1,2,...,, given the identical placeent of the first 1 jobs, if a placeent of job J is feasible for the optial DP, then the sae placeent is feasible for our DP. Proof: We use induction. Obviously the lea is true for = 1, since both DPs start fro the sae initial state (0,0,0,...,0). Assuing that it is true up to the placeent of job J, i.e., the optial and our partial schedules have identical placeents of jobs J 1,J 2,...,J in superintervals, we loo at the placeent of job J +1. In what follows, starred quantities refer to the optial schedule, and non-starred ones to ours. Let J +1 C t, and suppose that the optial placeent is in superinterval G iu 1 i u. Throughout the proof, we will use the fact that L 0l L0l l, s.t. 1 l K, due to the identical placeent of the first jobs, Eq. (3), and the fact that the y s are always rounded up. The rest of the technical details are in the Appendix. In the rest of the paper, we wor with these two special sequences and their transitions. We observe that L 0j u 1 j,u s.t. 1 < u M and i u 1 < j i u fro Condition (3), which is satisfied by the optial DP. Moreover, L 0l L 0l l s.t. 1 l K (cf. Lea 10). Hence L 0j u 1 j,u s.t. 1 < u M and i u 1 < j i u, i.e., Condition (3) is satisfied by the last state produced by our algorith in the sequence of transitions we study, and therefore we can feasibly coplete the schedule produced in this way with the insertion of the tardy straddlers. Theore 2 proves the approxiation ratio guarantee for the schedule produced by our algorith, by proving this guarantee when the special transition sequence above is followed, and with the use of Lea 11. We ephasize that our algorith ay not output the schedule corresponding to that sequence, since its approxiate estiation of the total tardiness ay lead it to picing another one, with a saller estiate of the total tardiness. For every 1 and 1 u M, let Bi u () := ax{w h h,y iuj h 0,1 j i u }, and if no job is tardy in superinterval G iui u+1, set Bi u () := 0. Lea 11 For every 1, 1 u M, and 1 j i u K we have Z Z + 2δ (6) 0 i ubi u ()(y iuj y iuj ) δ (7) Theore 2 If Z is the total tardiness of the schedule returned by the algorith and Z is the optial, we have that Z (1 + ε)z. General straddler placeent: Till now we have assued that each one of the (guessed) tardy straddlers straddles only one due date. Fro the above, it is easy to see how the algoriths can be odified to wor for the general case of straddlers spanning over ore than one due date. Acnowledgent We than George Steiner for several enlightening discussions. 10
References [1] T. S. Abdul-Razaq, C. N. Potts, and L. N. Van Wassenhove. A survey of algoriths for the single achine total weighted tardiness scheduling proble. Discrete Applied Matheatics, Vol. 26, pp. 235 253, 1990. [2] T. C. E. Cheng, C. T. Ng, J. J.Yuan, Z. H. Liu. Single achine scheduling to iniize total weighted tardiness. European J. Oper. Res., Vol. 165, pp. 423 443, 2005 [3] J. Du and J. Y.-T. Leung. Miniizing total tardiness on one achine is NP-hard. Matheatics of Operations Research, Vol. 15, pp. 483 495, 1990. [4] H. Kellerer and V. A. Strusevich. A Fully Polynoial Approxiation Schee for the Single Machine Weighted Total Tardiness Proble With a Coon Due Date. Theoretical Coputer Science, Vol. 369, pp. 230 238, 2006. [5] S. G. Kolliopoulos and G. Steiner. Approxiation Algoriths for Miniizing the Total Weighted Tardiness on a Single Machine. Theoretical Coputer Science, Vol. 355(3), pp. 261 273, 2006. [6] E. L. Lawler. A pseudopolynoial algorith for sequencing jobs to iniize total tardiness. Ann. Discrete Math., Vol. 1, pp. 331 342, 1977. [7] E.L. Lawler. A fully polynoial approxiation schee for the total tardiness proble. Operations Research Letters, Vol. 1, pp. 207 208, 1982. [8] J. K. Lenstra, A. H. G. Rinnooy Kan, P.Brucer. Coplexity of achine scheduling probles. Ann. Discrete Math., Vol. 1, pp. 343 362, 1977. [9] R. McNaughton. Scheduling with due dates and loss functions, Manageent Sci. Vol. 6, pp. 1 12, 1959. [10] T. Sen, J. M. Sule, and P. Dileepan. Static scheduling research to iniize weighted and unweighted tardiness: a state-of-the-art survey. Int. J. Production Econo., Vol. 83, pp. 1 12, 2003. [11] J. Yuan. The NP-hardness of the single achine coon due date weighted tardiness proble. Systes Sci. Math. Sci., Vol. 5, pp. 328 333, 1992. Appendix Proof of Lea 1: Suppose that for soe 1 ˆl i and K û i+1, eˆlû > 0. This iplies that there are soe Cû jobs which are early in interval Iˆl. Therefore, by exchanging soe of the tardy part of S i with soe part of these Cû jobs will reduce the total tardiness, since the tardiness of S i is reduced and the Cû jobs used in the exchange are still early. This is a contradiction of optiality. Proof of Lea 2: Suppose there exists 2 i K such that S i 1 is early, while y (i 1)u > 0. Then there are soe C u jobs (1 u i 1) which are tardy in I i. Then exchanging part of S i 1 with soe or part of these C u jobs will reduce their total tardiness, and S i 1 is still early. This is a contradiction of optiality. 11
Proof of Lea 4: We give the proof of deriving 2b.2 (the rest of the proof is obvious): if iu 1 L 0i u 1, then inequality d l d iu 1 ( l l q=i u 1 +1 v=q eqv + ax{ i u 1 L 0i u 1,0}) p +1 is equivalent to d l d iu 1 ( l l q=i u 1 +1 v=q eqv + iu 1 L 0i u 1 ) p +1. By exchanging the positions of the ters we have L 0i u 1 L 0l l v=q eqv l q=i u 1 +1 v=q eqv + d l d iu 1 ( l q=i u 1 +1 + i u 1 ) p +1. By (3), we now that L0i u 1 = d l d iu 1 ( l + i u 1 ) for all t l i u. An interpretation of these inequalities is that for all the class l jobs which are early in the superinterval G iu 1 i u are still early after inserting the new early job J +1. Proof of Lea 7: The proof is very siilar to the proof of Lea 6. We argue first that Conditions (1a) (or (1b)), (1c) and (3) do not affect the assignent of the first jobs. If Condition (1a) holds, there is at least p +1 epty space in the superinterval G iu 1 i u although (i) it ay not be contiguous (ii) part of it ay be earared to accoodate preeptions fro tardy jobs assigned after d iu. Condition (1b) corresponds to the assignent of job J +1 as floating in the prefix [d 0,d iu ). In both cases, we ay need to shift the early jobs of the superinterval as in the previous proof. Proof of Lea 8: First we note that the quantity L 0K is the actual epty space in [d 0,d K ), and since Condition (3) is true, for u = M,j = i M we have L 0i M M 1, i.e., we have enough actual epty space in [d 0,d K ) for all tardy straddlers other than S im. If L 0i M M 1 p SM, then use the extra epty space (L 0i M M 1 ) to fit (L 0i M M 1 ) units of p SM, and fit the rest p SM (L 0i M M 1 ) units right after d im (shifting the jobs in I K,K+1 towards the future by an equal aount of units). Otherwise (i.e., L 0i M p SM ), set the copletion tie of S im at d im, leave L 0i M to d im epty, and fit S im right before this epty space. M 1 > M units of epty space closest After the placeent of S im as described, we are left with exactly M 1 units of actual epty space before S im, and in this space we fit exactly S i1,...,s im 1 in this order. As a result, property (b) is fulfilled. To prove property (a), we use arguents siilar to the shifting schee in Lea 6. We loo at each due date d iu for u = M 1,M 2,...,1: For u = M 1, note that L 0,i M 1+1 still contains in it at least M 1 units of epty space, because of Lea 6, M 1, and the fact that there are no preepted pieces Condition (3) that states L 0,i M 1+1 counted in L 0,i M 1+1 yet. Therefore we now that S i1,...,s im 1 will be fitted in [d 0,d im 1 +1), which iplies that S im 1 is placed correctly. Now we consider two cases (as in Lea 6): if interval I im 1,i M 1 +1 contains any epty space, then there are no preepted pieces of tardy jobs copleting after d im 1 in [d 0,d im 1 ), since these pieces could only have coe fro the tardy jobs in G im 1 i M, but then it is ipossible for that epty space to exist. In this case, L 0,i M 1 in Condition (3) inequality L 0,i M 1 M 2 doesn t contain any preepted parts, and because of the way early jobs were pushed in Lea 6, we can conclude that [d 0,d im 1 ) contains at least M 2 units of epty space. if interval I im 1,i M 1 +1 contains no epty space, all the epty space of L 0,i M 1+1 at least M 1 M 2 ) actually is in [d 0,d im 1 ). (which is Both cases iply that [d 0,d im 1 ) contains at least M 2 units of epty space, and we can continue in the sae anner as before to show that S im 2 is correctly placed, then S im 3, etc. 12
Proof of Theore 1: Tae any optial non-preeptive schedule (which we already now that exists) and reove the straddlers. Consider also the sequence of partial schedules that result by reoving jobs {J 2,J 3,...,J }, {J 3,...,J },..., {J } respectively. We will show that these partial schedules can be produced by the algorith, i.e., Conditions (1)-(3) hold for every placeent of a job in the superinterval prescribed by the optial schedule. It is clear that Condition (3) is true for the whole sequence (since the straddlers were correctly placed in the schedule). Conditions (1a) and (2a) (depending on whether the (+1)-th job is tardy or early in its superinterval in the optial schedule) also hold. For exaple, for Condition (2a) (the arguent is the sae for Condition (1a)), assue that job J +1 C t is inserted early in G iu 1 i u and is the first for which Condition (2a) is not true, i.e., it holds that L 0l < p +1 for soe l s.t. L0i u 1 v q=i u 1+1 eqv + p +1 > d l d iu 1 (recall that t l i u. Then we have i u 1 + l v=i u 1 +1 jobs J 1,J 2,...,J have been inserted non-preeptively). This eans that the space in [d iu 1,d l ] is not enough to fit all tardy jobs and early jobs with due dates v l aong {J 1,J 2,...,J } that have been assigned to G iu 1 i u by the optial schedule. This contradicts the fact that all these jobs could be fitted there, as the optial schedule shows. Siilarly, we can show that Conditions (1c),(2c) also hold. Therefore there is a path in the DP transition diagra that corresponds to the placeent of jobs according to the given optial non-preeptive schedule, hence the final schedule produced by the algorith has optial tardiness. Proof of Lea 9: Assue the worst case M = K. For each one of the K(K + 1)/2 positions y iuj +1 we have at ost Z ub N w π(1) δ i + i=2 ( Zub w π(i) δ i Zub w π(i 1) ) = O( n2 ε ) distinct subintervals, or O(( n2 ε ) K(K+1) 2 ) cobinations of subintervals. When the cobination of subintervals is fixed, we have 2 K(K+1) 2 cobinations of possible values for the y iuj +1 s, since there are two choices for each of the. Therefore, for the sae values of Z +1,W i 11 +1,,W i KK +1, we have O(( n2 ε ) K(K+1) 2 2 K(K+1) 2 ) = O(ε K(K+1) 2 n K(K+1) ) states. Taing into account the rest of the state values and the initial guessing part (straddlers & Z ub ), overall the algorith runs in O((ε 1 n log W log P) Θ(K2) ) tie, where W,P are the total weight and total processing tie of all jobs. Rest of the proof of Lea 10: We distinguish two cases, according to the optial placeent of J +1 : Case 1: J +1 is early. Since J +1 is early, i u 1 < t i u and L 0l L 0i u 1 p +1 holds, l s.t. t l i u. Therefore we have L 0l L 0l L 0i u 1 p +1. In our algorith, if L 0l L0i u 1 p +1 holds l s.t. t l i u, Condition (2a) is satisfied. Otherwise, we exaine the two cases of Condition (2b): 1. i u 1 d l d iu 1 ( L 0i u 1 : Since l l q=i u 1 +1 v=q eqv has always been rounded down, we have q=i u 1 +1 v=q e qv ) d l d iu 1 ( q=i u 1 +1 v=q e qv ) L0l L 0i u 1 p +1 13
2. i u 1 L 0i j > L 0i u 1 : Since L 0l p +1 holds, we have L 0i j L0l, we have L0l Additionally, Condition (2c) is satisfied, because L 0i j Condition (2) is satisfied by state (,Z,y i 11 L 0i j p +1, j, s.t. u < j M.,W i 11 L 0i j,,y i MK p +1, t l i u. Also, since p +1, u < j M. Hence, ), and J +1 can be placed,w i MK (early) in superinterval G iu 1 i u by our algorith. Case 2: J +1 is tardy. Since J +1 is tardy, t i u 1. Also, L 0l L 0i u 1 s.t. t l i u. Therefore we have L 0l L 0l L 0i u 1 p +1. If L 0l p +1 holds, l L0i u 1 p +1 holds l s.t. t l i u, then Condition (1a) is satisfied. Otherwise, we exaine Condition (1b): We have L 0l L0l L 0l L 0i u 1 p +1 l, s.t. i u 1 < l i u. Additionally, Condition (1c) is satisfied, because L 0i j L 0i j p +1, j, s.t. u < j M. Hence, Condition (1) is satisfied by state (,Z,y i 11,W i 11,,y i MK,W i MK ), and J +1 can be placed (tardy) in superinterval G iu 1 i u by our algorith. Proof of Lea 11: The proof by induction is essentially the sae as the proof of Lea 1 in [4], but we include it here for copleteness. Assue that J +1 is fro C t where 1 t K, and to be inserted in superinterval G iui u+1. Then Z+1 = Z if J +1 is early, and Z+1 = Z + w +1 ( i u j=1 y iuj + p +1 + d iu d t ) if it is tardy. Define function φ iut(y iut ) = yiut + p +1 if J +1 is tardy in G iui u+1 and φ iut(y iut ) = yiut otherwise. Denote the rounded value of φ iut(y iut ) as yiut +1. For = 1, recall that the initial state is (0,0,...,0). It is easy to verify (6),(7) by the definition of the rounding. Now assue that these conditions hold for = s, where s <. We prove the lea for = s + 1. If, in the optial sequence, ys+1 iut = yiut s + p s+1 > 0, then J s+1 is tardy in G iui u+1 and hence Bi u (s+1) > 0. Assue that Bi u (s+1) = w v where v s+1. We have Bi u (s+1)ys+1 iut = w vys+1 iut w v yv iut Z ub since y iut is increasing with the increase of. Then we have ys+1 iut Zub Biu (s+1), and ys+1 iut belongs to soe subinterval of length at ost δ i ubiu (s+1). Now Lea 10 iplies that J s+1 is placed early or tardy by both the optial and our sequence, but in both cases we have 0 φ iut(ys iut ) ys+1 iut = yiut s ys iut δ δ i ubiu (s) i ubiu (s+1) since Bi u (s) Bi u (s + 1). Therefore φ iut(ys iut ) and ys+1 iut are either in the sae subinterval or in two consecutive subintervals. If the first case is true, the largest value in that interval is piced as the rounded value ys+1 iut ; if the second is true, the sallest value in the next subinterval is piced as the rounded value. Thus we have (7). Now we are bac to prove (6). If J s+1 is inserted early, the result is trivial. If it is tardy, we have Z s+1 Z s + w s+1 i u j=1 y iuj s + p s+1 + d iu d t + δ (6),(7) Zs + 2sδ + w i u s+1 (ys iuj δ + i j=1 u Bi u (s + 1) ) + p s+1 + d iu d t + δ i u = Zs + w s+1 ys iuj + p s+1 + d iu d t δ + i u w s+1 i u Bi u (s + 1) + δ + 2sδ j=1 Z s+1 + 2(s + 1)δ 14
where the first inequality taes into account the increase of Z s+1 by at ost δ due to its rounding, and the last inequality is due to the optial DP transition for J s+1. Proof of Theore 2: The proof is an extension of the proof of Lea 2 in [4], and we include it here for copleteness purposes. In exactly the sae way as in the proof of Lea 2 of [4], we can show that W iuj W iuj (1 + ε iuj )W 2, u,j s.t. 1 u M, 1 j i u. (8) Let Z be the total tardiness of the partial schedule coputed by the algorith before inserting the straddlers. Recall fro (4) that x iu := ax{0, u p S il L 0iu } (and respectively for x i u ). Since L 0iu is rounded up, x i u is rounded down (or becoes 0), i.e., x iu x iu. Then we have Z (5) = Z + w iu T iu + ( i u W iul )x iu = Z + w iu (x iu + d iu d t ) + ( Z + w iu (x i u + d iu d t ) + ( (6),(8) Z + 2δ + i u i u W iul )x iu W iul )x i u w iu (x i u + d iu d t ) + (1 + ε M 2 ) ( Z + ε 2 Z lb + w iu (x i u + d iu d t ) + ( (5) Z + ε 2 Z + ε 2 Z = (1 + ε)z i u i u W iul )x i u + ε 2 W iul )x i u ( i u W iul )x i u 15