Online Dial-a-Ride Problems: Minimizing the Completion Time

Transcription

1 Online Dial-a-Ride Problems: Minimizing the Completion Time Norbert Ascheuer, Sven O. Krumke, and Jörg Rambau Konrad-Zuse-Zentrum für Informationstechnik Berlin, Department Optimization, Takustr. 7, D Berlin-Dahlem, Germany, Abstract. We consider the following online dial-a-ride problem (OlDarp): Objects are to be transported between points in a metric space. Transportation requests arrive online, specifying the objects to be transported and the corresponding source and destination. These requests are to be handled by a server which starts its work at a designated origin and which picks up and drops objects at their sources and destinations. The server can move at constant unit speed. After the end of its service the server returns to its start in the origin. The goal of OlDarp is to come up with a transportation schedule for the server which finishes as early as possible, i.e., which minimizes the makespan. We analyze several competitive algorithms for OlDarp and establish tight competitiveness results. The first two algorithms, REPLAN and IGNORE are very simple and natural: REPLAN completely discards its (preliminary) schedule and recomputes a new one when a new request arrives. IGNORE always runs a (locally optimal) schedule for a set of known requests and ignores all new requests until this schedule is completed. We show that both strategies, REPLAN and IGNORE, are 5/2-competitive. We then present a somewhat less natural strategy SMARTSTART, which in contrast to the other two strategies may leave the server idle from time to time although unserved requests are known. The SMARTSTARTalgorithm has an improved competitive ratio of 2, which matches our lower bound. 1 Introduction Transportation problems where objects are to be transported between given sources and destinations in a metric space are classical problems in combinatorial optimization. In the classical setting, one assumes that the complete input for an instance is available for an algorithm to compute a solution. In many cases this offline optimization does not reflect the real-world situation appropriately. For instance, the transportation requests in an elevator system are hardly known in advance. Decisions have to be made online without the knowledge of future requests. Research supported by the German Science Foundation (grant 883/5-2) H. Reichel and S. Tison (Eds.): STACS 2000, LNCS 1770, pp , c Springer-Verlag Berlin Heidelberg 2000

2 640 Norbert Ascheuer, Sven O. Krumke, and Jörg Rambau Online algorithms are tailored to cope with such situations. Whereas offline algorithms work on the complete input sequence, online algorithms only get to see the requests released so far and thus have to account for future requests that may or may not arise at a later time. A common way to evaluate the quality of online algorithms is competitive analysis [6]: An algorithm ALG is called c- competitive if its cost on any input sequence is at most c-times the optimum offline cost. In this paper we consider the following online dial-a-ride problem (OlDarp): Objects are to be transported between the points of a metric space. A request consists of the objects to be transported and the corresponding source and destination of the transportation request. The requests arrive online and must be handled by a server which starts and ends its work at a distinguished origin. The server picks up and drops objects at their sources and destinations, respectively. The goal of OlDarp is to come up with a transportation schedule for the server which finishes as early as possible. Related Work. We do not claim originality for the two online-algorithms IG- NORE and REPLAN; instead, we show how to analyze them for OlDarp and how ideas from both strategies can be used to construct a new online strategy SMARTSTART with better competitive ratio. The first to the best of our knowledge occurrence of the strategy IGNORE can be found in the paper by Shmoys, Wein, and Williamson [12]: They show a general result about obtaining competitive algorithms for minimizing the total completion time (also called the makespan) in machine scheduling problems when the jobs arrive over time: If there is a ρ-approximation algorithm for the offline version, then this implies a 2ρ-competitive algorithm for the online-version, which is essentially the IGNORE strategy. The results from [12] show that IG- NORE-type strategies are 2-competitive for a number of online-scheduling problems. The strategy REPLAN is probably folklore; it can be found also under different names like reopt or optimal. The difference of OlDarp studied here to the machine scheduling problems treated in [12] are as follows: For OlDarp, the execution time of jobs depends on their execution order. OlDarp can be viewed as a generalized scheduling problem with setup costs and order dependent execution times. It should be stressed that in this paper we do not allow additive constants in the definition of the competitive ratio. If one allowed an additive constant equal to the diameter of the metric space, then the algorithms REPLAN and IGNORE wouldinfactbe also 2-competitive. In [5,4,3] the authors studied the Online Traveling Salesman Problem (OlTsp) which is obtained as a special case of OlDarp treated in this paper, when for each request its source and destination coincide. It is shown in [5] that there is a metric space (the boundary of the unit square) where any deterministic algorithm for the OlTsp has a competitive ratio of at least 2. For the case that the metric space is the real line, a lower bound of is given in [4]. A 2-competitive algorithm that works in an arbitrary (symmetric) metric space and a 7/4-competitive algorithm for the real line are presented in [5].

3 Online Dial-a-Ride Problems: Minimizing the Completion Time 641 Recently, in an independent effort Feuerstein and Stougie [9] analyzed the algorithm IGNORE (which they call dlt for Don t listen while traveling ) and established the same competitive ratio as in this paper. Our Contribution. We provide a number of competitive algorithms for the basic version of the problem OlDarp. The best algorithm, SMARTSTART, has competitive ratio 2 which improves the result of 5/2 established in [9]. This competitiveness result closes the gap to the lower bound. Our algorithm SMARTSTART induces an alternative 2-competitive algorithm for the OlTsp. Moreover,SMARTSTART canbeusedtoobtaina competitive polynomial time algorithm for the OlTsp. Since , this improves the result of [5,4], where a 3-competitive polynomial time algorithm for the OlTsp was presented. IGNORE and SMARTSTART can be applied to the generalization of OlDarp (and OlTsp) where there are k-servers with arbitrary capacities C 1,...,C k IN. Their competitive ratios of 5/2 of 2, respectively, hold as well in the generalized case. As a corollary, we obtain 2-competitive algorithms for a number of machine scheduling problems with setup-costs which generalizes a result from [12]. Paper Outline. This paper is organized as follows: In Section 2 we formally define the problem OlDarp and introduce notation. In Section 3 we establish lower bounds for the competitive ratio of deterministic online algorithms for OlDarp. In Section 4 we analyze two simple strategies, REPLAN and IGNORE. Section 5 contains our improved algorithm SMARTSTART. The competitive ratios of our three algorithms are summarized in Table 1. Table 1. Competitive ratios of algorithms in this paper. Algorithm REPLAN IGNORE SMARTSTART Lower bound Closed Schedules 5/2 5/ Preliminaries An instance of the basic online dial-a-ride problem OlDarp consists of a metric space M =(X, d) with a distinguished origin o X and a sequence σ = r 1,...,r m of requests. It is assumed that for all pairs (x, y) ofpointsfromm, there is a path p :[0, 1] X in X with p(0) = x and p(1) = y of length d(x, y) (see [5] for a thorough discussion of the model). Examples of a metric spaces that satisfy the above condition are the Euclidean space IR p and a metric space induced by an undirected edge-weighted graph. Each request is a triple r i =(t i,a i,b i ), where t i is a real number, the time where request r i is released (becomes known), and a i V and b i V are

4 642 Norbert Ascheuer, Sven O. Krumke, and Jörg Rambau the source and destination, respectively, between which the new object is to be transported. We assume that the sequence σ = r 1,...,r m of requests is given in order of non-decreasing release times. A server is located at the origin o X at time 0 and can move at constant unit speed. In the basic version the server has unit-capacity, i.e., it can carry at most one object at a time. (Extensions to the case of more servers with arbitrary capacities are also considered in this paper.) We do not allow preemption: once the server has picked up an object, it is not allowed to drop it at any other place than its destination. An online algorithm for OlDarp does neither have information about the release time of the last request nor about the total number of requests. The online algorithm must determine the behavior of the server at a certain moment t of time as a function depending on all the requests released up to time t and on the current time t. In contrast, the offline algorithm has information about all requests in the whole sequence σ already at time 0. Given a sequence σ of requests, a valid transportation schedule for σ is a sequence of moves of the server such that the following conditions are satisfied: (a) The server starts its movement in the origin vertex o, (b) each transportation request in σ is served, but starting not earlier than the time it becomes known, and (c) the server returns to the origin vertex after having served the last request. The objective function of the OlDarp is the (total) completion time (also called the makespan in scheduling) of the server, that is, the time when the server has served all requests and returned to the origin. Let ALG(σ) denote the completion time of the server moved by algorithm ALG on the sequence σ of requests. We use OPT to denote an optimal offline algorithm. An online algorithm ALG for OlDarp is c-competitive, ifthereexists a constant c such that for any request sequence σ the inequality ALG(σ) c OPT(σ) holds true. It should be mentioned that in sometimes literature the definition of the competitive ratio allows an additive constant. In that case a c- competitive algorithm with additive constant zero is then referred to as strictly c-competitive. In this paper we do not allow additive constants. 3 Lower Bounds We first address the question how well an online algorithm can perform compared to the optimal offline algorithm. Since OlDarp generalizes the OlTsp we obtain the following result from the lower bound established in [5]: Theorem 1. If ALG is a deterministic c-competitive algorithm for OlDarp, then c 2. For the case that the metric space is the real line, a lower bound of was given in [4,3]. We provide an improved lower bound for OlDarp below: Theorem 2. If ALG is a deterministic c-competitive algorithm for OlDarp on the real line, then c 1+ 2/

5 Online Dial-a-Ride Problems: Minimizing the Completion Time 643 Proof. Suppose that ALG is a deterministic online algorithm with competitive ratio c 1+ 2/2. At time t = 0, the algorithm ALG is faced with two requests r 1 =(0,o,2) and r 2 =(0, 2,o).Theoptimumofflinecosttoservethesetwo requests is 4. The server operated by ALG must start serving request r 2 at some time 2 T 4c 2, because otherwise ALG could not be c-competitive. At time T the adversary issues another request r 3 =(T,T,2). Then OPT(r 1,r 2,r 3 )=2T. On the other hand, ALG(r 1,r 2,r 3 ) 3T +2. Thus, the competitive ratio c of ALG satisfies 3T +2 c 3 2T T c 2. The smallest value c 1 such that c 3/2+1/(4c 2) is c =1+ 2/2. 4 Two Simple Strategies In this section we present and analyze two very natural online-strategies for for OlDarp and prove both of them to be 5/2-competitive. Strategy REPLAN As soon as a new request arrives the server completes the current carrying move (if it is performing one), then the server stops and replans: it computes a new shortest schedule which starts at the current position of the server, takes care of all yet unserved requests, and returns to the origin. Strategy IGNORE The server remains idle until the point in time t when the first requests become known. The algorithm then serves the requests released at time t immediately, following a shortest schedule S. All requests that arrive during the time when the algorithm follows this schedule are temporarily ignored. After S has been completed and the server is back in the origin, the algorithm computes a shortest schedule for all unserved requests and follows this schedule. Again, all new requests that arrive during the time that the server is following the schedule are temporarily ignored. A schedule for the ignored requests is computed as soon as the server has completed its current schedule. The algorithm keeps on following schedules and temporarily ignoring requests this way. Both algorithms above repeatedly solve offline instances of OlDarp.These offline instances have the property that all release times are at least as large as the current time. Thus, the corresponding offline problem is the following: given a number of transportation requests (with release times all zero), find a shortest transportation for them. For a sequence σ of requests and a point x inthemetricspacem let L (t, x, σ) denote the length of a shortest schedule (i.e., the time difference between its completion time and the start time t) whichstartsinx at time t, servesall requests from σ (but not earlier than their release times), and ends in the origin. Clearly, for t t we have that L (t,x,σ) L (t, x, σ). Moreover, OPT(σ) = L (0,o,σ) and thus OPT(σ) L (t, o, σ) for any time t 0.

6 644 Norbert Ascheuer, Sven O. Krumke, and Jörg Rambau Since the optimum offline server OPT cannot serve the last request r m = (t m,a m,b m )fromσ before this request is released we get that OPT(σ) max{l (t, o, σ),t m + d(a m,b m )+d(b m,o)} for any t 0. (1) Lemma 1. Let σ = r 1,...,r m be a sequence of requests. Then for any t t m and any request r i =(t i,a i,b i ) from σ L (t, b i,σ\r i ) L (t, o, σ) d(a i,b i )+d(a i,o). Here σ\r i denotes the sequence obtained from σ by deleting the request r i. Proof. Consider an optimum schedule S which starts at the origin o at time t, serves all requests in σ and has length L (t, o, σ). It suffices to construct another schedule S which starts in b i no earlier than time t, serves all requests in σ\r i and has length at most L (t, o, σ) d(a i,b i )+d(a i,o). Let S serve the requests in the order r j1,...,r jm and let r i = r jk.notice that if we start in b at time t and serve the requests in the order r jk+1,...,r jm,r j1,...,r jk 1 and move back to the origin we obtain a schedule S with the desired properties. We are now ready to prove the result about the performance of REPLAN: Theorem 3. Algorithm REPLAN is 5/2-competitive. Proof. Let σ = r 1,...,r m be any sequence of requests. We distinguish between two cases depending on the current load of the REPLAN-server at the time t m (i.e., the time when the last request is released). If the server is currently empty it recomputes an optimal schedule which starts at its current position, denoted by s(t m ), serves all unserved requests, and returns to the origin. This schedule has length at most L (t m,s(t m ),σ) d(o, s(t m )) + L (t m,o,σ). Thus, REPLAN(σ) t m + d(o, s(t m )) + L (t m,o,σ) (1) t m + d(o, s(t m )) + OPT(σ) (2) New now consider the second case, when the server is currently serving a request r = (t, a, b). The time needed to complete this move is d(s(t m ),b). Then a shortest schedule starting at b serving all unserved requests is computed which has length at most L (t m,b,σ\r). Thus in the second case REPLAN(σ) t m + d(s(t m ),b)+l (t m,b,σ\r) t m + d(s(t m ),b)+l (t m,o,σ) d(a, b)+d(a, o) by Lemma 1 t m + OPT(σ) d(a, b)+d(s(t m ),b)+d(a, s(t m )) +d(s(t m ),o) }{{} =d(a,b) = t m + d(o, s(t m )) + OPT(σ).

7 Online Dial-a-Ride Problems: Minimizing the Completion Time 645 This means that inequality (2) holds in both cases. Since the REPLAN server has traveled to position s(t m )attimet m, there must be a request r j =(t j,a j,b j ) in σ where either d(o, a i ) d(o, s(t m )) or d(o, b i ) d(o, s(t m )). By the triangle inequality this implies that the optimal offline server will have to travel at least twice the distance d(o, s(t m )) during its schedule. Thus, d(o, s(t m )) OPT(σ)/2. Plugging this result into inequality (2) we get that the total time the REPLAN server needs is no more than 5/2 OPT(σ). We are now going to analyze the competitiveness of the second simple strategy IGNORE. Theorem 4. Algorithm IGNORE is 5/2-competitive. Proof. Consider again the point in time t m when the last request r m becomes known. If the IGNORE server is currently idle at the origin o, then its completes its last schedule no later than t m + L (t m,o,σ =tm ), where σ =tm is the set of requests released at time t m.sincel (t m,o,σ =tm ) OPT(σ) andopt(σ) t m, it follows that in this case IGNORE completes no later than time 2 OPT(σ). It remains the case that at time t m the IGNORE-server is currently working on a schedule S forasubsetσ S of the requests. Let t S denote the starting time of this schedule. Thus, the IGNORE-server will complete S at time t S +L (t S,o,σ S ). Denote by σ ts the set of requests presented after the IGNORE-server started with S at time t S.Noticethatσ ts is exactly the set of requests that are served by IGNORE in its last schedule. The IGNORE-server will complete its total service no later than time t S + L (t S,o,σ S )+L (t m,o,σ ts ). Let r f σ ts be the first request from σ ts served by OPT. Thus OPT(σ) t f + L (t f,a f,σ ts ) t S + L (t m,a f,σ ts ). (3) Now, L (t m,o,σ ts ) d(o, a f )+L (t m,a f,σ ts )andl (t S,o,σ S ) OPT(σ). Therefore, IGNORE(σ) t S + OPT(σ)+d(o, a f )+L (t m,a f,σ ts ) (3) 2 OPT(σ)+d(o, a f ) 5 2 OPT(σ). This completes the proof. The following instance shows that the competitive ratio of 5/2 proved for IGNORE is asymptotically tight even for the case when the metric space is the real line. At time 0 there is a request r 1 =(0, 1, 0). The next requests are r 2 =(ε, 2, 3) and r 3 =(2+ε, 2, 1). It is easy to see that IGNORE(r 1,r 2,r 3 )=10+4ε, while OPT(r 1,r 2,r 3 )=4+2ε. Thus,theratioIGNORE(r 1,r 2,r 3 )/OPT(r 1,r 2,r 3 )can be made arbitrarily close to 5/2 bychoosingε>0 small enough.

8 646 Norbert Ascheuer, Sven O. Krumke, and Jörg Rambau 4.1 Application to Machine Scheduling Problems We close this section by showing how the IGNORE strategy can be applied to a generalization of the basic problem OlDarp and how this generalization can be used to model certain machine scheduling problems where there are setup costs for different job types. Consider the generalization k-oldarp of OlDarp when there are k IN servers with arbitrary capacities C 1,...,C k IN. Capacity C j means that server j can carry at most C items at a time. The objective function for k-oldarp is the time when the last server has returned to the origin (after all requests have been served). For k-oldarp the IGNORE strategy always plans schedules for its servers such that the length of the longest schedule is minimized. All schedules are constructed in such a way that they start and end in the origin. New requests are ignored until the last of the servers has returned to the origin. It is not too hard to see that the proof of Theorem 4 remains valid even for k-oldarp: Theorem 5. Algorithm IGNORE is 5/2-competitive even for the extension k- OlDarp of OlDarp where there are k IN servers with arbitrary capacities C 1,...,C k IN. Suppose there are k uniform machines where jobs can be run and assume that there are l types of jobs. For each job type j setup cost of s j is given: if a job of type k j has been processed immediately before on a machine, then an additional cost of s j is incurred to start processing a job of type j on this machine. This setup cost models for instance the situation where special auxiliary device must be installed at the machine to perform a certain job type. The problem of minimizing the makespan in this machine scheduling problem when jobs arrive over time can be modeled as k-oldarp (with k unit-capacity servers) on a star shaped metric space with center o: For each of the p job types there is one ray in the metric space emanating from p. OnrayR j there is a special point x j at distance s j /2fromo. Ajoboftypej with processing time p is modeled by a transportation request from the point x j to the point x j + p/2. 5 The SMARTSTART Strategy In this section we present and analyze our algorithm SMARTSTART which achieves a best-possible competitive ratio of 2 (cf. the lower bound given in Theorem 1). The idea of the algorithm is basically to emulate the IGNOREstrategy but to make sure that each sub-schedule is completed not too late : if a sub-schedule would take too long to complete then the algorithm waits for a specified amount of time. Intuitively this construction tries to avoid the worstcase situation for IGNORE where right after the algorithm started a schedule a new request becomes known. The algorithm SMARTSTART has a fixed waiting scaling parameter θ> 1. From time to time the algorithm consults its work-or-sleep routine: this subroutine computes an (approximately) shortest schedule S for all unserved

9 Online Dial-a-Ride Problems: Minimizing the Completion Time 647 requests starting and ending in the origin. If this schedule can be completed no later than time θt (where t is the current time) the subroutine returns (S, work), otherwise it returns (S, sleep). In the sequel it will be convenient to assume that the work-or-sleep subroutine uses a ρ-approximation algorithm for computing a schedule: the approximation algorithm always finds a schedule of length at most ρ times the optimum one. While in online computation one is usually not interested in time complexity (and thus in view of competitive analysis we can assume that ρ =1),employing a polynomial time approximation algorithm will enable us to get a practical algorithm (and in particular to improve the result of [5,4] for the OlTsp). The server of algorithm SMARTSTART can assume three states: idle In this case the server has served all known requests, is sitting in the origin and waiting for new requests to occur. sleeping In this case the server knows of some unserved requests but also knows that they take too long to serve (what too long means will be formalized in the algorithm below). working In this state the algorithm (or rather the server operated by it) is following a computed schedule. We now formalize the behavior of the algorithm by specifying how it reacts in each of the three states. Strategy SMARTSTART If the algorithm is idle at time T and new requests arrive, calls work-or-sleep. If the result is (S, work), the algorithm enters the working state where it follows S. Otherwise the algorithm enters the sleeping state with wakeup time t,wheret T is the earliest time such that t + l(s) θt,wherel(s) is the length of the just computed schedule S, i.e., t =min{ t T : t + l(s) θt }. In the sleeping state the algorithm simply does nothing until its wakeup time t. At this time the algorithm reconsults the work-or-sleep subroutine. If the result is (S, work), then the algorithm enters the working state and follows S. Otherwise the algorithm continues to sleep with new wakeup time min{ t t : t + l(s) θt }. In the working state, i.e, while the server is following a schedule all new requests are (temporarily) ignored. As soon as the current schedule is completed the server either enters the idle-state (if there are no unserved requests) or it reconsults the work-or-sleep subroutine which determines the next state (sleeping or working). Theorem 6. For all θ ρ, θ>1, Algorithm SMARTSTART is c-competitive with { ( c =max θ, ρ 1+ 1 ), θ } θ ρ. ( Moreover, the best possible choice of θ is 1 ) ( ρ and yields a competitive ratio of 1 ) 4 4ρ ρ.

10 648 Norbert Ascheuer, Sven O. Krumke, and Jörg Rambau Proof. Let σ =tm be the set of requests released at time t m,wheret m denotes again the point in time when the last requests becomes known. We distinguish between different cases depending on the state of the SMARTSTART-server at time t m : Case 1: The server is idle. In this case the algorithm consults its work-or-sleep routine which computes an approximately shortest schedule S for the requests in σ =tm.thesmart- START-server will start its work at time t =min{ t t m : t+l(s) θt }, where l(s) =L (t m,o,σ =tm ) denotes the length of the schedule S. If t = t m, then by construction the algorithm completes no later than time θt m θ OPT(σ). Otherwise t >t m and it follows that t + l(s) =θt. By the performance guarantee of ρ of the approximation algorithm employed in work-or-sleep we have that OPT(σ) l(s)/ρ = θ 1 ρ t. Thus, it follows that SMARTSTART(σ) =t + l(s) θt θ ρ OPT(σ) ( = ρ 1+ 1 ) OPT(σ). θ 1 θ 1 Case 2: The server is sleeping. Note that the wakeup time of the server is no later than min{ t t m : t + l(s) θt }, wheres is now a shortest schedule for all the requests in σ not yet served by SMARTSTART at time t m, and we can proceed as in Case 1. Case 3: The algorithm is working. If after completion of the current schedule the server enters the sleeping state then the arguments given above establish ( that ) the completion time of the SMARTSTART-server does not exceed ρ 1+ 1 θ 1 OPT(σ). The remaining case is that the SMARTSTART-server starts its final schedule S immediately after having completed S. Lett S be the time when the server started S and denote by σ ts the set of requests presented after the server started S at time t S.Noticethatσ ts is exactly the set of requests that are served by SMARTSTART in its last schedule S. SMARTSTART(σ) =t S + l(s)+l(s ). (4) Here l(s) andl(s ) ρl (t m,o,σ ts ) denotes the length of the schedule S and S, respectively. We have that t S + l(s) θt S, (5) since the SMARTSTART only starts a schedule at some time t if it can complete it not later than time θt. Letr f σ ts be the first request from σ ts served by OPT. Using the arguments given in the proof of Theorem 4 we conclude that OPT(σ) t S + L (t m,a f,σ ts ). (6) Moreover, since the tour of length L (t m,a f,σ ts )startsina f and returns to the origin it follows from the triangle inequality that L (t m,a f,σ ts ) d(o, a f ). Thus, from (6) we get OPT(σ) t S + d(o, a f ). (7)

11 Online Dial-a-Ride Problems: Minimizing the Completion Time 649 On the other hand l(s ) ρ (d(o, a f )+L (t m,a f,σ ts )) (6) ρ (OPT(σ) t S + d(o, a f )). (8) Using (5) and (8) in (4) and the assumption that θ ρ, weobtain SMARTSTART(σ) θt S + l(s ) by (5) (θ ρ)t S + ρd(o, a f )+ρopt(σ) by (8) { θ OPT(σ)+(2ρ θ)d(o, a f ) by (7) θ OPT(σ)+(2ρ θ) OPT(σ) 2,ifθ 2ρ θ OPT(σ), if θ>2ρ max { θ 2 + ρ, θ} OPT(σ) This completes the proof. For pure competitive analysis we may assume that each schedule S computed by work-or-sleep is in fact an optimal schedule, i.e., that ρ =1.The best competitive ratio for SMARTSTART is then achieved for that value of θ wherethethreetermsθ, 1+ 1 θ 1 and θ are equal. This is the case for θ =2 and yields a competitive ratio of 2. We thus obtain the following corollary. Corollary 1. For ρ =1, θ =2algorithm SMARTSTART is 2-competitive. The offline dial-a-ride problem is also known as the Stacker-Crane-Problem. In [10] the authors present a 9/5-approximationalgorithm. On paths the problem can be solved in polynomial time [2,11]. In [7] an approximation algorithm for the single server dial-a-ride problem with performance O( C log n log log n) was given, where C denotes the capacity of the server. For the special case of the Online Traveling Salesman Problem (OlTsp) Christofides algorithm [8] yields a polynomial time approximation algorithm with ρ = 3/2. For this value of ρ, the best competitive ratio of SMARTSTART is attained for θ = and equals Thus, for the OlTsp our algorithm SMARTSTART can be used to obtain a polynomial time competitive algorithm with competitive ratio approximately This improves the result of [5,4] where a 3-competitive polynomial time algorithm for OlTsp was given. We finally note that the SMARTSTART-strategy inherits some desirable properties from the IGNORE-strategy: The algorithm can also be used for k- OlDarp and provides the same competitive ratio. Theorem 7. Algorithm SMARTSTART is 2-competitive for the extension k- OlDarp of OlDarp where there are k IN servers with arbitrary capacities C 1,...,C k IN. The last result implies a 2-competitive algorithm for the machine scheduling problems with setup costs discussed in Section 4.1.

12 650 Norbert Ascheuer, Sven O. Krumke, and Jörg Rambau 6 Remarks Our investigations of the OlDarp were originally motivated by the performance analysis of a large distribution center of Herlitz AG, Berlin [1]. Its automatic pallet transportation system employs several vertical transportation systems (elevators) in order to move pallets between the various floors of the building. The pallets that have to be transported during one day of production are not known in advance. If the objective is chosen as minimizing the completion time (makespan) then this can be modeled by the OlDarp where the metric space is induced by a graph which is a simple path. References 1. N. Ascheuer, M. Grötschel, S. O. Krumke, and J. Rambau. Combinatorial online optimization. In Proceedings of the International Conference of Operations Research, pages Springer, M. J. Atallah and S. R. Kosaraju. Efficient solutions to some transportation problems with applications to minimizing robot arm travel. SIAM Journal on Computing, 17(5): , October G. Ausiello, E. Feuerstein, S. Leonardi, L. Stougie, and M. Talamo. Serving request with on-line routing. In Proceedings of the 4th Scandinavian Workshop on Algorithm Theory, volume 824 of Lecture Notes in Computer Science, pages 37 48, July G. Ausiello, E. Feuerstein, S. Leonardi, L. Stougie, and M. Talamo. Competitive algorithms for the traveling salesman. In Proceedings of the 4th Workshop on Algorithms and Data Structures, volume 955 of Lecture Notes in Computer Science, pages , August G. Ausiello, E. Feuerstein, S. Leonardi, L. Stougie, and M. Talamo. Algorithms for the on-line traveling salesman. Algorithmica, To appear. 6. A. Borodin and R. El-Yaniv. Online Computation and Competitive Analysis. Cambridge University Press, M. Charikar and B. Raghavachari. The finite capacity dial-a-ride problem. In Proceedings of the 39th Annual IEEE Symposium on the Foundations of Computer Science, N. Christofides. Worst-case analysis of a new heuristic for the traveling salesman problem. Technical report, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh, PA, E. Feuerstein and L. Stougie. On-line single server dial-a-ride problems. Theoretical Computer Science, special issue on on-line algorithms, to appear. 10. G. N. Frederickson, M. S. Hecht, and C. E. Kim. Approximation algorithms for some routing problems. SIAM Journal on Computing, 7(2): , May D. Hauptmeier, S. O. Krumke, J. Rambau, and H.-C. Wirth. Euler is standing in line. In Proceedings of the 25th International Workshop on Graph-Theoretic Concepts in Computer Science, Ascona, Switzerland, volume 1665 of Lecture Notes in Computer Science, pages 42 54, June D. B. Shmoys, J. Wein, and D. P. Williamson. Scheduling parallel machines on-line. SIAM Journal on Computing, 24(6): , December 1995.