Scheduling on Hybrid Platforms: Improved Approximability Window

Modern platforms are using accelerators in conjunction with standard processing units in order to reduce the running time of specific operations, such as matrix operations, and improve their performance. Scheduling on such hybrid platforms is a challenging problem since the algorithms used for the case of homogeneous resources do not adapt well. In this paper we consider the problem of scheduling a set of tasks subject to precedence constraints on hybrid platforms, composed of two types of processing units. We propose a $(3+2\sqrt{2})$-approximation algorithm and a conditional lower bound of 3 on the approximation ratio. These results improve upon the 6-approximation algorithm proposed by Kedad-Sidhoum et al. as well as the lower bound of 2 due to Svensson for identical machines. Our algorithm is inspired by the former one and distinguishes the allocation and the scheduling phases. However, we propose a different allocation procedure which, although is less efficient for the allocation sub-problem, leads to an improved approximation ratio for the whole scheduling problem. This approximation ratio actually decreases when the number of processing units of each type is close and matches the conditional lower bound when they are equal

Scheduling to Minimize Total Weighted Completion Time via Time-Indexed Linear Programming Relaxations

We study approximation algorithms for scheduling problems with the objective of minimizing total weighted completion time, under identical and related machine models with job precedence constraints. We give algorithms that improve upon many previous 15 to 20-year-old state-of-art results. A major theme in these results is the use of time-indexed linear programming relaxations. These are natural relaxations for their respective problems, but surprisingly are not studied in the literature. We also consider the scheduling problem of minimizing total weighted completion time on unrelated machines. The recent breakthrough result of [Bansal-Srinivasan-Svensson, STOC 2016] gave a $(1.5-c)$-approximation for the problem, based on some lift-and-project SDP relaxation. Our main result is that a $(1.5 - c)$-approximation can also be achieved using a natural and considerably simpler time-indexed LP relaxation for the problem. We hope this relaxation can provide new insights into the problem

Communication-Aware Scheduling of Precedence-Constrained Tasks on Related Machines

Scheduling precedence-constrained tasks is a classical problem that has been studied for more than fifty years. However, little progress has been made in the setting where there are communication delays between tasks. Results for the case of identical machines were derived nearly thirty years ago, and yet no results for related machines have followed. In this work, we propose a new scheduler, Generalized Earliest Time First (GETF), and provide the first provable, worst-case approximation guarantees for the goals of minimizing both the makespan and total weighted completion time of tasks with precedence constraints on related machines with machine-dependent communication times

Algorithms For Clustering Problems:Theoretical Guarantees and Empirical Evaluations

Clustering is a classic topic in combinatorial optimization and plays a central role in many areas, including data science and machine learning. In this thesis, we first focus on the dynamic facility location problem (i.e., the facility location problem in evolving metrics). We present a new LP-rounding algorithm for facility location problems, which yields the first constant factor approximation algorithm for the dynamic facility location problem. Our algorithm installs competing exponential clocks on clients and facilities, and connects every client by the path that repeatedly follows the smallest clock in the neighborhood. The use of exponential clocks gives rise to several properties that distinguish our approach from previous LP-roundings for facility location problems. In particular, we use \emph{no clustering} and we enable clients to connect through paths of \emph{arbitrary lengths}. In fact, the clustering-free nature of our algorithm is crucial for applying our LP-rounding approach to the dynamic problem. Furthermore, we present both empirical and theoretical aspects of the $k$-means problem. The best known algorithm for $k$-means with a provable guarantee is a simple local-search heuristic that yields an approximation guarantee of $9+\epsilon$, a ratio that is known to be tight with respect to such methods. We overcome this barrier by presenting a new primal-dual approach that enables us (1) to exploit the geometric structure of $k$-means and (2) to satisfy the hard constraint that at most $k$ clusters are selected without deteriorating the approximation guarantee. Our main result is a $6.357$-approximation algorithm with respect to the standard LP relaxation. Our techniques are quite general and we also show improved guarantees for the general version of $k$-means where the underlying metric is not required to be Euclidean and for $k$-median in Euclidean metrics. We also improve the running time of our algorithm to almost linear running time and still maintain a provable guarantee. We compare our algorithm with {\sc K-Means++} (a widely studied algorithm) and show that we obtain better accuracy with comparable and even better running time