Calculus inference graphs and analog circuit synthesis

Abstract In this article, we consider a calculus system as a set of variables linked by calculus operations. These operations can be algebric equations, differential equations, algorithmic functions, boolean equations... We interest in finding a particular set of variables, called parameters, such that the values of the resting variables can be infered by using the operations. This problem reduces to a combinatorial optimization problem, called the calculus inference subgraph problem, for which we propose both exact and approximative solving methods. We focus on applying these results to analog circuit synthesis in order to compute the component sizing variable values from the expected values of both input and output circuit parameters

Scheduling data flow program in xkaapi: A new affinity based Algorithm for Heterogeneous Architectures

Efficient implementations of parallel applications on heterogeneous hybrid architectures require a careful balance between computations and communications with accelerator devices. Even if most of the communication time can be overlapped by computations, it is essential to reduce the total volume of communicated data. The literature therefore abounds with ad-hoc methods to reach that balance, but that are architecture and application dependent. We propose here a generic mechanism to automatically optimize the scheduling between CPUs and GPUs, and compare two strategies within this mechanism: the classical Heterogeneous Earliest Finish Time (HEFT) algorithm and our new, parametrized, Distributed Affinity Dual Approximation algorithm (DADA), which consists in grouping the tasks by affinity before running a fast dual approximation. We ran experiments on a heterogeneous parallel machine with six CPU cores and eight NVIDIA Fermi GPUs. Three standard dense linear algebra kernels from the PLASMA library have been ported on top of the Xkaapi runtime. We report their performances. It results that HEFT and DADA perform well for various experimental conditions, but that DADA performs better for larger systems and number of GPUs, and, in most cases, generates much lower data transfers than HEFT to achieve the same performance

A linear programming-based method for job shop scheduling

We present a decomposition heuristic for a large class of job shop scheduling problems. This heuristic utilizes information from the linear programming formulation of the associated optimal timing problem to solve subproblems, can be used for any objective function whose associated optimal timing problem can be expressed as a linear program (LP), and is particularly effective for objectives that include a component that is a function of individual operation completion times. Using the proposed heuristic framework, we address job shop scheduling problems with a variety of objectives where intermediate holding costs need to be explicitly considered. In computational testing, we demonstrate the performance of our proposed solution approach

Two-level lot-sizing with inventory bounds

We study a two-level uncapacitated lot-sizing problem with inventory bounds that occurs in a supply chain composed of a supplier and a retailer. The first level with the demands is the retailer level and the second one is the supplier level. The aim is to minimize the cost of the supply chain so as to satisfy the demands when the quantity of item that can be held in inventory at each period is limited. The inventory bounds can be imposed at the retailer level, at the supplier level or at both levels. We propose a polynomial dynamic programming algorithm to solve this problem when the inventory bounds are set on the retailer level. When the inventory bounds are set on the supplier level, we show that the problem is NP-hard. We give a pseudo-polynomial algorithm which solves this problem when there are inventory bounds on both levels. In the case where demand lot-splitting is not allowed, i.e. each demand has to be satisfied by a single order, we prove that the uncapacitated lot-sizing problem with inventory bounds is strongly NP-hard. This implies that the two-level lot-sizing problems with inventory bounds are also strongly NP-hard when demand lot-splitting is considered

MATHEMATICAL MODELS AND LAGRANGIAN HEURISTICS FOR A TWO-LEVEL LOT-SIZING PROBLEM WITH BOUNDED INVENTORY

We consider a two-level lot-sizing problem where the first level consists of N end products competing for a single type of raw material (second level), which is supposed to be critical. In particular, the storage capacity of raw materials is limited and must be carefully managed. The goal is to simultaneously determine an optimal replenishment plan for the raw material and optimal production plans for the end products on a horizon of T periods. The problem is modeled as an integer linear program and solved using both a Lagrangian relaxation-based heuristic and a commercial optimization software. The results obtained using the Lagrangian heuristic are promising and new ideas are generated to further improve the quality of the solution

Scheduling on Two Unbounded Resources with Communication Costs

International audienceHeterogeneous computing systems are popular and powerful platforms, containing several heterogeneous computing elements (e.g. CPU+GPU). In this work, we consider a platform with two types of machines , each containing an unbounded number of elements. We want to execute an application represented as a Directed Acyclic Graph (DAG) on this platform. Each task of the application has two possible execution times, depending on the type of machine it is executed on. In addition we consider a cost to transfer data from one platform to the other between successive tasks. We aim at minimizing the execution time of the DAG (also called makespan). We show that the problem is NP-complete for graphs of depth at least three but polynomial for graphs of depth at most two. In addition, we provide polynomial-time algorithms for some usual classes of graphs (trees, series-parallel graphs)

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