Approximation Algorithms for Scheduling with Resource and Precedence Constraints

We study non-preemptive scheduling problems on identical parallel machines and uniformly related machines under both resource constraints and general precedence constraints between jobs. Our first result is an O(logn)-approximation algorithm for the objective of minimizing the makespan on parallel identical machines under resource and general precedence constraints. We then use this result as a subroutine to obtain an O(logn)-approximation algorithm for the more general objective of minimizing the total weighted completion time on parallel identical machines under both constraints. Finally, we present an O(logm logn)-approximation algorithm for scheduling under these constraints on uniformly related machines. We show that these results can all be generalized to include the case where each job has a release time. This is the first upper bound on the approximability of this class of scheduling problems where both resource and general precedence constraints must be satisfied simultaneously

Scheduling MapReduce Jobs under Multi-Round Precedences

We consider non-preemptive scheduling of MapReduce jobs with multiple tasks in the practical scenario where each job requires several map-reduce rounds. We seek to minimize the average weighted completion time and consider scheduling on identical and unrelated parallel processors. For identical processors, we present LP-based O(1)-approximation algorithms. For unrelated processors, the approximation ratio naturally depends on the maximum number of rounds of any job. Since the number of rounds per job in typical MapReduce algorithms is a small constant, our scheduling algorithms achieve a small approximation ratio in practice. For the single-round case, we substantially improve on previously best known approximation guarantees for both identical and unrelated processors. Moreover, we conduct an experimental analysis and compare the performance of our algorithms against a fast heuristic and a lower bound on the optimal solution, thus demonstrating their promising practical performance

Approximation algorithms for min-max resource sharing and malleable tasks scheduling

This thesis deals with approximation algorithms for problems in mathematical programming, combinatorial optimization, and their applications. We first study the min-max resource-sharing problem (the packing problem as the linear case) with $M$ nonnegative convex constraints on a convex set $B$, which is a class of convex programming. In general block solvers are required for solving the problems. We generalize the algorithm by Grigoriadis et al. to the case with only weak approximate block solvers. In this way we present an approximation algorithm that needs at most $O(M(\ln M+\epsilon^{-2}\ln\epsilon^{-1}))$ calls to the block solver for any given relative accuracy $\epsilon\in(0,1)$. It is the first bound independent of the data and the approximation ratio of the block solver. As applications of the min-max resource-sharing problem, we study the multicast congestion problem in communication networks and the range assignment problem in arbitrary ad-hoc networks. We present improved approximation algorithms for these problems. We also study the problem of scheduling malleable tasks with precedence constraints. We are given $m$ identical processors and $n$ tasks. For each task the processing time is a discrete function of the number of processors allotted to it. In addition, the tasks must be processed according to the precedence constraints. The goal is to minimize the makespan (maximum completion time) of the resulting schedule. We improve the previous best approximation algorithm with a ratio $3+\sqrt{5}\approx 5.236$ to $100/43+100(\sqrt{4349}-7)/2451\approx 4.730598$. Finally, we propose a new model for malleable tasks and develop an approximation algorithm for the scheduling problem with a ratio $100/63+100(\sqrt{6469}+13)/5481\approx 3.291919$. We also show that our results are very close to the best asymptotic one

Parameterized complexity of machine scheduling: 15 open problems

Machine scheduling problems are a long-time key domain of algorithms and complexity research. A novel approach to machine scheduling problems are fixed-parameter algorithms. To stimulate this thriving research direction, we propose 15 open questions in this area whose resolution we expect to lead to the discovery of new approaches and techniques both in scheduling and parameterized complexity theory.Comment: Version accepted to Computers & Operations Researc