Leah Epstein

Approximation Schemes for Scheduling on Uniformly Related and Identical Parallel Machines

We give a polynomial approximation scheme for the problem of scheduling on uniformly related parallel machines for a large class of objective functions that depend only on the machine completion times, including minimizing the lp norm of the vector of completion times. This generalizes and simplifies many previous results in this area.

On-line scheduling of unit time jobs with rejection: minimizing the total completion time

We consider on-line scheduling of unit time jobs on a single machine with job-dependent penalties. The jobs arrive on-line (one by one) and can be either accepted and scheduled, or be rejected at the cost of a penalty. The objective is to minimize the total completion time of the accepted jobs plus the sum of the penalties of the rejected jobs. We give an on-line algorithm for this problem with competitive ratio 12(2+3)~1.86602. Moreover, we prove that there does not exist an on-line algorithm with competitive ratio better than 1.63784.

Improved Approximation Guarantees for Weighted Matching in the Semi-streaming Model

We study the maximum weight matching problem in the semi-streaming model, and improve on the currently best one-pass algorithm due to Zelke [Proceedings of the 25th Annual Symposium on Theoretical Aspects of Computer Science, 2008, pp. 669–680] by devising a deterministic approach whose performance guarantee is 4.91+e. In addition, we study preemptive online algorithms, a class of algorithms related to one-pass semi-streaming algorithms, where we are allowed to maintain only a feasible matching in memory at any point in time. We provide a lower bound of 4.967 on the competitive ratio of any such deterministic algorithm, and hence show that future improvements will have to store in memory a set of edges that is not necessarily a feasible matching. We conclude by presenting an empirical study, conducted in order to compare the practical performance of our approach to that of previously suggested algorithms.

All-norm approximation algorithms

A major drawback in optimization problems and in particular in scheduling problems is that for every measure there may be a different optimal solution. In many cases the various measures are different lp norms. We address this problem by introducing the concept of an all-norm p-approximation algorithm, which supplies one solution that guarantees p-approximation to all lp norms simultaneously. Specifically, we consider the problem of scheduling in the restricted assignment model, where there are m machines and n jobs, each job is associated with a subset of the machines and should be assigned to one of them. Previous work considered approximation algorithms for each norm separately. Lenstra et al. [Math. Program. 46 (1990) 259-271] showed a 2-approximation algorithm for the problem with respect to the l∞ norm. For any fixed lp norm the previously known approximation algorithm has a performance of θ(p). We provide an all-norm 2-approximation polynomial algorithm for the restricted assignment problem. On the other hand, we show that for any given lp norm (p > 1) there is no PTAS unless P=NP by showing an APX-hardness result. We also show for any given lp norm a FPTAS for any fixed number of machines.

A lower bound for on-line scheduling on uniformly related machines

We consider the problem of on-line scheduling of jobs arriving one by one on uniformly related machines, with or without preemption. We prove a lower bound of 2, both with and without preemption, for randomized algorithms working for an arbitrary number of machines. For a constant number of machines we give new lower bounds for the preemptive case.

Bin stretching revisited

Abstract. We study three on-line models of bin stretching on two machines. For the case where the machines are identical and the jobs arrive sorted by non-increasing sizes, we show a tight bound of 10/9 on the competitive ratio. For two related machines, we show a preemptive algorithm with competitive ratio 1 for any speed ratio, and two new non-preemptive algorithms. We prove that the upper bound on the competitive ratio achieved by the non-preemptive algorithms is optimal for almost any speed ratio, and close to optimal for all other speed ratios.

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

Better bounds for minimizing SONET ADMs

\alpha \in (0,1)

On Bin Packing with Conflicts

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

Online Bin Packing with Cardinality Constraints

SONET add-drop multiplexers (ADMs) are the dominant cost factor in SONET /WDM rings. The number of SONET ADMs required by a set of traffic streams is determined by the routing and wavelength assignment of the traffic streams. Following previous work, we consider the problem where the route of each traffic stream is given as input, and we need to assign wavelengths so as to minimize the total number of used SONET ADMs. This problem is known to be NP-hard, and the best known approximation algorithm for this problem has a performance guarantee of