Florian Diedrich

Approximation Algorithms for Scheduling with Reservations

AbstractWe study the problem of non-preemptively scheduling n independent sequential jobs on a system of m identical parallel machines in the presence of reservations, where m is constant. This setting is practically relevant because for various reasons, some machines may not be available during specified time intervals. The objective is to minimize the makespan Cmax, which is the maximum completion time.The general case of the problem is inapproximable unless

Approximation Algorithms for 3D Orthogonal Knapsack

\mathsf {P}=\mathsf {NP}

A Survey on Approximation Algorithms for Scheduling with Machine Unavailability

; hence, we study a suitable strongly

Approximation algorithms for scheduling with reservations

\mathsf {NP}

A framework for scheduling with online availability

-hard restriction, namely the case where at least one machine is always available. For this setting we contribute approximation schemes, complemented by inapproximability results. The approach is based on algorithms for multiple subset sum problems; our technique yields a PTAS which is best possible in the sense that an FPTAS is ruled out unless

Tight approximation algorithms for scheduling with fixed jobs and nonavailability

\mathsf {P}=\mathsf {NP}

Implementation of approximation algorithms for the max-min resource sharing problem

. The PTAS presented here is the first one for the problem under consideration; so far, not even for well-known special cases approximation schemes have been proposed. Furthermore we derive a low cost algorithm with a constant approximation ratio and discuss FPTASes for special cases as well as the complexity of the problem if m is part of the input.

Multi-objective problems in terms of relational algebra

We study non-overlapping axis-parallel packings of 3D boxes with profits into a dedicated bigger box where rotation is either forbidden or permitted, and we wish to maximize the total profit. Since this optimization problem is NP-hard, we focus on approximation algorithms. We obtain fast and simple algorithms for the non-rotational scenario with approximation ratios 9 + ϵ and 8 + ϵ, as well as an algorithm with approximation ratio 7 + ϵ that uses more sophisticated techniques; these are the smallest approximation ratios known for this problem. Furthermore, we show how the used techniques can be adapted to the case where rotation by 90° either around the z-axis or around all axes is permitted, where we obtain algorithms with approximation ratios 6 + ϵ and 5 + ϵ, respectively. Finally our methods yield a 3D generalization of a packability criterion and a strip packing algorithm with absolute approximation ratio 29/4, improving the previously best known result of 45/4.

An approximation algorithm for the general mixed packing and covering problem

In this chapter we present recent contributions in the field of sequential job scheduling on network machines which work in parallel; these are subject to temporary unavailability. This unavailability can be either unforeseeable (online models) or known a priori (offline models). For the online models we are mainly interested in preemptive schedules for problem formulations where the machine unavailability is given by a probabilistic model; objectives of interest here are the sum of completion times and the makespan. Here, the non-preemptive case is essentially intractable. For the offline models we are interested in non-preemptive schedules where we consider the makespan objective; we present approximation algorithms which are complemented by suitable inapproximability results. Here, the preemptive model is polynomial-time solvable for large classes of settings.

Approximation algorithms for 3D orthogonal Knapsack

We study the problem of scheduling n independent jobs on a system of m identical parallel machines in the presence of reservations. This constraint is practically important; for various reasons, some machines are not available during specified time intervals. The objective is to minimize the makespan. This problem is inapproximable in the general case unless P = NP which motivates the study of suitable restrictions. We use an approach based on algorithms for multiple subset sum problems; our technique yields a polynomial time approximation scheme (PTAS) which is best possible in the sense that the problem does not admit an FPTAS unless P = NP. The PTAS presented here is the first one for the problem under consideration; so far, not even for special cases approximation schemes have been proposed. We also derive a low cost algorithm with a constant approximation ratio and discuss additional FPTASes for special cases and complexity results.