Algorithms for Hierarchical and Semi-Partitioned Parallel Scheduling

We propose a model for scheduling jobs in a parallel machine setting that takes into account the cost of migrations by assuming that the processing time of a job may depend on the specific set of machines among which the job is migrated. For the makespan minimization objective, the model generalizes classical scheduling problems such as unrelated parallel machine scheduling, as well as novel ones such as semi-partitioned and clustered scheduling. In the case of a hierarchical family of machines, we derive a compact integer linear programming formulation of the problem and leverage its fractional relaxation to obtain a polynomial-time 2-approximation algorithm. Extensions that incorporate memory capacity constraints are also discussed

ILP-based approaches to partitioning recurrent workloads upon heterogeneous multiprocessors

The problem of partitioning systems of independent constrained-deadline sporadic tasks upon heterogeneous multiprocessor platforms is considered. Several different integer linear program (ILP) formulations of this problem, offering different tradeoffs between effectiveness (as quantified by speedup bound) and running time efficiency, are presented

Algorithms for Constructing Overlay Networks For Live Streaming

We present a polynomial time approximation algorithm for constructing an overlay multicast network for streaming live media events over the Internet. The class of overlay networks constructed by our algorithm include networks used by Akamai Technologies to deliver live media events to a global audience with high fidelity. We construct networks consisting of three stages of nodes. The nodes in the first stage are the entry points that act as sources for the live streams. Each source forwards each of its streams to one or more nodes in the second stage that are called reflectors. A reflector can split an incoming stream into multiple identical outgoing streams, which are then sent on to nodes in the third and final stage that act as sinks and are located in edge networks near end-users. As the packets in a stream travel from one stage to the next, some of them may be lost. A sink combines the packets from multiple instances of the same stream (by reordering packets and discarding duplicates) to form a single instance of the stream with minimal loss. Our primary contribution is an algorithm that constructs an overlay network that provably satisfies capacity and reliability constraints to within a constant factor of optimal, and minimizes cost to within a logarithmic factor of optimal. Further in the common case where only the transmission costs are minimized, we show that our algorithm produces a solution that has cost within a factor of 2 of optimal. We also implement our algorithm and evaluate it on realistic traces derived from Akamai's live streaming network. Our empirical results show that our algorithm can be used to efficiently construct large-scale overlay networks in practice with near-optimal cost

On Derandomized Approximation Algorithms

With the design of powerful randomized algorithms the transformation of a randomized algorithm or probabilistic existence result for combinatorial problems into an efficient deterministic algorithm (called derandomization) became an important issue in algorithmic discrete mathematics. In the last years several interesting examples of derandomization have been published, like discrepancy in hypergraph colouring, packing integer programs and an algorithmic version of the Lovász-Local-Lemma. In this paper the derandomization method of conditional probabilities of Raghavan/Spencer is extended using discrete martingales. As a main result pessimistic estimators are constructed for combinatorial approximation problems involving non-linear objective functions with bounded martingale differences. The theory gives polynomial-time algorithms for the linear and quadratic lattice approximation problem and a quadratic variant of the matrix balancing problem extending results of Spencer, Beck/Fiala and Raghavan. Finally a probabilistic existence result of Erdös on the average graph bisection is transformed into a deterministic algorithm

Dependent randomized rounding for clustering and partition systems with knapsack constraints

Clustering problems are fundamental to unsupervised learning. There is an increased emphasis on fairness in machine learning and AI; one representative notion of fairness is that no single demographic group should be over-represented among the cluster-centers. This, and much more general clustering problems, can be formulated with "knapsack" and "partition" constraints. We develop new randomized algorithms targeting such problems, and study two in particular: multi-knapsack median and multi-knapsack center. Our rounding algorithms give new approximation and pseudo-approximation algorithms for these problems. One key technical tool, which may be of independent interest, is a new tail bound analogous to Feige (2006) for sums of random variables with unbounded variances. Such bounds are very useful in inferring properties of large networks using few samples

Real-Time Message Routing and Scheduling

Exchanging messages between nodes of a network (e.g., embedded computers) is a fundamental issue in real-time systems involving critical routing and scheduling decisions. In order for messages to meet their deadlines, one has to determine a suitable (short) origin-destination path for each message and resolve conflicts between messages whose paths share a communication link of the network. With this paper we contribute to the theoretic foundations of real-time systems. On the one hand, we provide efficient routing strategies yielding origin-destination paths of bounded dilation and congestion. In particular, we can give good a priori guarantees on the time required to send a given set of messages which, under certain reasonable conditions, implies that all messages can be scheduled to reach their destination on time. Finally, for message routing along a directed path (which is already NP-hard), we identify a natural class of instances for which a simple scheduling heuristic yields provably optimal solutions

The Moser-Tardos Framework with Partial Resampling

The resampling algorithm of Moser \& Tardos is a powerful approach to develop constructive versions of the Lov\'{a}sz Local Lemma (LLL). We generalize this to partial resampling: when a bad event holds, we resample an appropriately-random subset of the variables that define this event, rather than the entire set as in Moser & Tardos. This is particularly useful when the bad events are determined by sums of random variables. This leads to several improved algorithmic applications in scheduling, graph transversals, packet routing etc. For instance, we settle a conjecture of Szab\'{o} & Tardos (2006) on graph transversals asymptotically, and obtain improved approximation ratios for a packet routing problem of Leighton, Maggs, & Rao (1994)

An automated routing method for VLSI with three interconnection layers

Recently, to the extent allowed by the fabricating technology, approaches have been made to develop an automated router for the multi-layer IC layout design. In this thesis, we examine the VLSI routing problem where three layers are available for interconnection;We investigate the routing problem in three stages: global routing, power/ground routing, and channel routing. The global routing for three-interconnection layer model is not much different from that of two-layer madel. We study the global routing problem for two cases: gate array and general cell layout. In our three-layer grid model, power/ground wires keep the direction-per-layer scheme as signal net wires. However, the power/ground routing is further constrained by the width of wires and the layers they are laid on;The channel routing stage of our router is based on directional model where overlaps of horizontal wire segments are allowed. We improve the dogleg method so that it is applicable to the three-layer model and it can handle multi-terminal nets more efficiently. Applying the extensive dogleg method and the three-layer merge algorithm, we not only remove the cyclic vertical constraints graph but also eliminate the effect of the height of long vertical constraints tree to the channel width and thus we reduce the lower bound of the channel width to half of the density of the channel. We expand the applicability of channel router by eliminating some of the limitations assumed in channel routing problems by some existing algorithms. Routability conditions are examined for various cases of channel routing problem;The major result presented in this dissertation is an algorithm for a channel routing problem. Given a rectangular channel with terminals on top and bottom sides, the algorithm will find a three-layer channel routing which minimizes the channel width and the wire length. Experimental results show that our router is close to optimal