Using oracles for the design of efficient approximation algorithms

International audienceWe are interested here in oracle techniques for the design of approximation algo- rithms. Following the classical definition, an oracle is a black box capable of answering correctly and instantaneously any question. Several classical approximation scheme de- sign techniques (typically PTAS) can be revisited using oracle. Our objective in this work is to show that, conversely, oracle techniques are not limited to the design of PTAS. In particular, interactivity (using queries to oracle) may also lead to parameterized algorithm (whose complexity is exponential in a parameter, supposed to be "small"), that can be more practical than classical P T AS. Moreover, we aim at showing how it is possible to "degenerate" questions asked to the oracle to derive fast implementations of these interactive algorithms. These ideas will be illustrated on the classical makespan minimization on uniform machines problem (QCmax )

A (2+ɛ)-approximation for scheduling parallel jobs in platforms

We consider the problem of \textsc{Scheduling parallel Jobs in heterogeneous Platforms}: We are given a set $\mathcal{J}=\{1,\ldots,n\}$ of $n$ jobs, where a job $j\in\mathcal{J}$ is described by a pair $(p_j,q_j)$ of a processing time $p_j\in\mathbb{Q}_{>0}$ and the number of processors $q_j\in\mathbb{N}$ that are required to execute $j$. We are also given a set $\mathcal{B}$ of $N$ heterogeneous platforms $P_1,\ldots,P_N$, where each $P_i$ contains $m_{i}$ processors for $i\in\{1,\ldots, N\}.$ The objective is to find a schedule for the jobs in the platforms minimizing the makespan, i.e. the latest finishing time of a job. Unless $\mathcal{P}=\mathcal{NP}$ there is no approximation algorithm with absolute ratio strictly better than $2$ for the problem. We give a $(2+\epsilon)$-approximation for the problem improving the previously best known absolute approximation ratio $3$

Multi-Objective Group Discovery on the Social Web (Technical Report)

Les rapports de recherche du LIG - ISSN: 2105-0422We are interested in discovering user groups from collabo-rative rating datasets of the form i, u, s, where i ∈ I, u ∈ U, and s is the integer rating that user u has assigned to item i. Each user has a set of attributes that help find labeled groups such as young computer scientists in France and American female designers. We formalize the problem of finding user groups whose quality is optimized in multiple dimensions and show that it is NP-Complete. We develop α-MOMRI, an α-approximation algorithm, and h-MOMRI, a heuristic-based algorithm , for multi-objective optimization to find high quality groups. Our extensive experiments on real datasets from the social Web examine the performance of our algorithms and report cases where α-MOMRI and h-MOMRI are useful

Multiple Strip Packing and Scheduling Parallel Jobs in Platforms

We consider two strongly related problems, multiple strip packing and scheduling parallel jobs in platforms. In the first one we are given a list of n rectangles with heights and widths bounded by one and N strips of unit width and infinite height. The objective is to find a non-overlapping orthogonal packing without rotations of all rectangles into the strips minimizing the maximum height used. In the scheduling problem we consider jobs instead of rectangles, i.e. we are allowed to cut the rectangles vertically and we may have target areas of different size, called platforms. A platform $P_\ell$ is a collection of $m_\ell$ processors running at speed $s_\ell$ and the objective is to minimize the makespan, i.e. the latest finishing time of a job

Scheduling with Storage Constraints

International audienceWe are interested in this paper to study scheduling problems in systems where many users compete to perform their respective jobs on shared parallel resources. Each user has specific needs or wishes for computing his/her jobs expressed as a function to optimize (among maximum completion time, sum of completion times and sum of weighted completion times). Such problems have been mainly studied through Game Theory. In this work, we focus on solving the problem by optimizing simultaneously each user's objective function independently using classical combinatorial optimization techniques. Some results have already been proposed for two users on a single computing resource. However, no generic combinatorial method is known for many objectives. The analysis proposed in this paper concerns an arbitrarily fixed number of users and is not restricted to a single resource. We first derive inapproximability bounds; then we analyze several greedy heuristics whose approximation ratios are close to these bounds. However, they remain high since they are linear in the number of users. We provide a deeper analysis which shows that a slightly modified version of the algorithm is a constant approximation of a Pareto-optimal solution

Models for scheduling on large scale platforms: which policy for which application?

In the recent years, there was a huge development of low cost large scale parallel systems. The design of efficient parallel algorithms has to be reconsidered by the influence of new parameters of such execution supports (namely, clusters of workstations, grid computing and global computing) which are characterized by a larger number of heterogeneous processors, often organized by hierarchical sub-systems. Alternative computational models have been designed in order to take into account new characteristics. Parallel Tasks model -- PT in short -- (i.e. tasks that require more than one processor for their execution) is a promising alternative for scheduling parallel applications, especially in the case of slow communication media. The basic idea is to consider the application at a rough level of granularity. Another way of looking at the problem (which is somehow a dual view) is the Divisible Load model (DL) where an application is considered as a collection of a large number of elementary -- sequential -- computing units that will be distributed among the available resources. As the main difficulty for scheduling in actual systems comes from handling efficiently the communications, these two new views of the problem allow us to consider them implicitly or to mask them, thus leading to more tractable problems. This paper aims first at presenting some examples of approximation algorithms for parallelizing applications for the PT model with a special emphasis on new execution supports. Then, we will show how to mix these results with the DLT model in order to integrate them into the previous model for managing the resources of an actual computational grid composed by more than 600 machines built in Grenoble (CiGri project)

Tight approximation for scheduling parallel jobs on identical clusters

International audienceWe consider the Multiple Cluster Scheduling Problem (MCSP), where the objective is to schedule n parallel rigid jobs on N identical clusters, minimizing the maximum completion time (makespan). MCSP is 2-inapproximable (unless P = NP), and several approximation algorithms have already been proposed. However, ratio 2 has only been reached by algorithms that use extremely costly and complex subroutines as "black boxes" which are polynomial and yet impractical due to prohibitive constants. Our objective within this work is to determine a reasonable restriction of MCSP where the inapproximability lower bound could be tightened in almost linear time. Thus, we consider a restriction of MCSP where jobs do not require strictly more than half of the processors of a cluster, and we provide a 2-approximation running in O(log(nhmax)n(N + log(n))), where hmax is the processing time of the longest job. This approximation is the best possible, as this restriction (and even simpler ones) remains 2-inapproximable

Investigations on path indexing for graph databases

Graph databases have become an increasingly popular choice for the management of the massive network data sets arising in many contemporary applications. We investigate the effectiveness of path indexing for accelerating query processing in graph database systems, using as an exemplar the widely used open-source Neo4j graph database. We present a novel path index design which supports efficient ordered access to paths in a graph dataset. Our index is fully persistent and designed for external memory storage and retrieval. We also describe a compression scheme that exploits the limited differences between consecutive keys in the index, as well as a workload-driven approach to indexing. We demonstrate empirically the speed-ups achieved by our implementation, showing that the path index yields query run-times from 2x up to 8000x faster than Neo4j. Empirical evaluation also shows that our scheme leads to smaller indexes than using general-purpose LZ4 compression. The complete stand-alone implementation of our index, as well as supporting tooling such as a bulk-loader, are provided as open source for further research and development

Master-slave Tasking on Heterogeneous Processors

In this paper, we consider the problem of scheduling independent identical tasks on heterogeneous processors where communication times and processing times are different. We assume that communication-computation overlap is possible for every processor, but only allow one send and one receive at a time. We propose an algorithm for chains of processors based on an iterative backward construction of the schedule, which is polynomial in the number of processors and in the number of tasks. The complexity is $O(np^2)$ where $n$ is the number of tasks and $p$ the number of processors. We prove this algorithm to be optimal with respect to the makespan. We extend this result to a special kind of tree called spider graphs. ) where n is the number of tasks and p the number of processors. We prove this algorithm to be optimal with respect to the makespan. We extend this result to a special kind of tree called spider graphs