Steven S. Seiden

On the online bin packing problem

A new framework for analyzing online bin packing algorithms is presented. This framework presents a unified way of explaining the performance of algorithms based on the Harmonic approach. Within this framework, it is shown that a new algorithm, Harmonic++, has asymptotic performance ratio at most 1.58889. It is also shown that the analysis of Harmonic+1 presented in Richey [1991] is incorrect; this is a fundamental logical flaw, not an error in calculation or an omitted case. The asymptotic performance ratio of Harmonic+1 is at least 1.59217. Thus, Harmonic++ provides the best upper bound for the online bin packing problem to date.

Semi-online scheduling with decreasing job sizes

We investigate the problem of semi-online scheduling jobs on m identical parallel machines where the jobs arrive in order of decreasing sizes. We present a complete solution for the preemptive variant of semi-online scheduling with decreasing job sizes. We give matching lower and upper bounds on the competitive ratio for any fixed number m of machines; these bounds tend to (1+3)/2~1.36603, as the number of machines goes to infinity. Our results are also the best possible for randomized algorithms. For the non-preemptive variant of semi-online scheduling with decreasing job sizes, a result of Graham (SIAM J. Appl. Math. 17 (1969) 263-269) yields a (4/3-1/(3m))-competitive deterministic non-preemptive algorithm. For m=2 machines, we prove that this algorithm is the best possible (it is 7/6-competitive). For m=3 machines we give a lower bound of (1+37)/6~1.18046. Finally, we present a randomized non-preemptive 8/7-competitive algorithm for m=2 machines and prove that this is optimal.

Preemptive multiprocessor scheduling with rejection

Abstract The problem of online multiprocessor scheduling with rejection was introduced by Bartal et al. (SIAM J. Discrete Math. 13(1) (2000) 64–78). They show that for this problem the competitive ratio is 1+φ≈2.61803 , where φ is the golden ratio. A modified model of multiprocessor scheduling with rejection is presented where preemption is allowed. For this model, it is shown that better performance is possible. An online algorithm which is (4+ 10 )/3 -competitive is presented. We prove that the competitive ratio of any online algorithm is at least 2.12457. We say that an algorithm schedules obliviously if the accepted jobs are scheduled without knowledge of the rejection penalties. We also show a lower bound of 2.33246 on the competitive ratio of any online algorithm which schedules obliviously. As a subroutine in our algorithm, we use a new optimal online algorithm for preemptive scheduling without rejection. This algorithm never achieves a larger makespan than that of the previously known algorithm of Chen et al. (Oper. Res. Lett. 18(3) (1995) 127–131), and achieves a smaller makespan for some inputs.

New Bounds for Variable-Sized Online Bin Packing

In the variable-sized online bin packing problem, one has to assign items to bins one by one. The bins are drawn from some fixed set of sizes, and the goal is to minimize the sum of the sizes of the bins used. We present new algorithms for this problem and show upper bounds for them which improve on the best previous upper bounds. We also show the first general lower bounds for this problem. The case in which bins of two sizes, 1 and

Randomized algorithms for metrical task systems

\alpha \in (0,1)

On the Online Bin Packing Problem

, are used is studied in detail. This investigation leads us to the discovery of several interesting fractal-like curves.

An Optimal Online Algorithm for Bounded Space Variable-Sized Bin Packing

Borodin et al. (1992) introduce a general model for online systems in [3] called task systems and show a deterministic algorithm which achieves a competitive ratio of 2n − 1 for any metrical task system with n states. We present a randomized algorithm which achieves a competitive ratio of e/(e − 1)n − 1/(e − 1) ≈ 1.5820n − 0.5820 for this same problem. For the uniform metric space, Borodin et al. present an algorithm which achieves a competitive ratio of 2Hn, and they show a lower bound of Hn, for any randomized algorithm. We improve their upper bound for the uniform metric space by showing a randomized algorithm which is (Hn + O(√log n))-competitive.

New results for online page replication

A new framework for analyzing online bin packing algorithms is presented. This framework presents a unified way of explaining the performance of algorithms based on the Harmonic approach [3, 5, 8, 10, 11, 12]. Within this framework, it is shown that a new algorithm, Harmonic++, has asymptotic performance ratio at most 1.58889. It is also shown that the analysis of Harmonic+1 presented in [11] is incorrect; this is a fundamental logical flaw, not an error in calculation or an omitted case. The asymptotic performance ratio of Harmonic+1 is at least 1.59217. Thus Harmonic++ provides the best upper bound for the online bin packing problem to date.

A heuristic storage for minimizing access time of arbitrary data patterns

An online algorithm for variable-sized bin packing, based on the Harmonic algorithm of Lee and Lee,[J. ACM, 32 (1985), pp. 562--572], is investigated. This algorithm was proposed by Csirik, [Acta Inform., 26 (1989), pp. 697--709], who proved that for all sets of bin sizes, 1.69103 upper bounds its performance ratio. The upper bound is improved in the sense that we give a method of calculating the performance ratio to any accuracy for any set of bin sizes. Further, it is shown that the algorithm is optimal among those which use bounded space. An interesting feature of the analysis is that, although it is shown that our algorithm achieves a performance ratio arbitrarily close to the optimum value, it is not known precisely what that value is. The case where bins of capacity 1 and

Linear time approximation schemes for vehicle scheduling problems

\alpha \in (0,1)