226,353 research outputs found

    Online Mixed Packing and Covering

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    Recent work has shown that the classical framework of solving optimization problems by obtaining a fractional solution to a linear program (LP) and rounding it to an integer solution can be extended to the online setting using primal-dual techniques. The success of this new framework for online optimization can be gauged from the fact that it has led to progress in several longstanding open questions. However, to the best of our knowledge, this framework has previously been applied to LPs containing only packing or only covering constraints, or minor variants of these. We extend this framework in a fundamental way by demonstrating that it can be used to solve mixed packing and covering LPs online, where packing constraints are given offline and covering constraints are received online. The objective is to minimize the maximum multiplicative factor by which any packing constraint is violated, while satisfying the covering constraints. Our results represent the first algorithm that obtains a polylogarithmic competitive ratio for solving mixed LPs online. We then consider two canonical examples of mixed LPs: unrelated machine scheduling with startup costs, and capacity constrained facility location. We use ideas generated from our result for mixed packing and covering to obtain polylogarithmic-competitive algorithms for these problems. We also give lower bounds to show that the competitive ratios of our algorithms are nearly tight

    The matching relaxation for a class of generalized set partitioning problems

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    This paper introduces a discrete relaxation for the class of combinatorial optimization problems which can be described by a set partitioning formulation under packing constraints. We present two combinatorial relaxations based on computing maximum weighted matchings in suitable graphs. Besides providing dual bounds, the relaxations are also used on a variable reduction technique and a matheuristic. We show how that general method can be tailored to sample applications, and also perform a successful computational evaluation with benchmark instances of a problem in maritime logistics.Comment: 33 pages. A preliminary (4-page) version of this paper was presented at CTW 2016 (Cologne-Twente Workshop on Graphs and Combinatorial Optimization), with proceedings on Electronic Notes in Discrete Mathematic

    Developing The Attitude And Creativity In Mathematics Education

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    The structures in a traditionally-organized classroom of mathematics teaching can usually be linked readily with the routine classroom activities of teacher-exposition and teacher-supervised desk work, teacher’s initiation, teacher’s direction and strongly teacher’s expectations of the outcome of student learning. If the teacher wants to develop appropriate attitude and creativities in mathematics teaching learning it needs for him to develop innovation in mathematics teaching. The teacher may face challenge to develop various style of teaching i.e. various and flexible method of teaching, discussion method, problem-based method, various style of classroom interaction, contextual and or realistic mathematics approach. To develop mathematical attitude and creativity in mathematics teaching learning processes, the teacher may understand the nature and have the highly skill of implementing the aspects of the following: mathematics teaching materials, teacher’s preparation, student’s motivation and apperception, various interactions, small-group discussions, student’s works sheet development, students’ presentations, teacher’s facilitations, students’ conclusions, and the scheme of cognitive development.In the broader sense of developing attitude and creativity of mathematics learning, the teacher may needs to in-depth understanding of the nature of school mathematics, the nature of students learn mathematics and the nature of constructivism in learning mathematics. Key Word: mathematical attitude, creativity in mathematics, innovation of mathematics teaching,school mathematics

    A Novel SAT-Based Approach to the Task Graph Cost-Optimal Scheduling Problem

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    The Task Graph Cost-Optimal Scheduling Problem consists in scheduling a certain number of interdependent tasks onto a set of heterogeneous processors (characterized by idle and running rates per time unit), minimizing the cost of the entire process. This paper provides a novel formulation for this scheduling puzzle, in which an optimal solution is computed through a sequence of Binate Covering Problems, hinged within a Bounded Model Checking paradigm. In this approach, each covering instance, providing a min-cost trace for a given schedule depth, can be solved with several strategies, resorting to Minimum-Cost Satisfiability solvers or Pseudo-Boolean Optimization tools. Unfortunately, all direct resolution methods show very low efficiency and scalability. As a consequence, we introduce a specialized method to solve the same sequence of problems, based on a traditional all-solution SAT solver. This approach follows the "circuit cofactoring" strategy, as it exploits a powerful technique to capture a large set of solutions for any new SAT counter-example. The overall method is completed with a branch-and-bound heuristic which evaluates lower and upper bounds of the schedule length, to reduce the state space that has to be visited. Our results show that the proposed strategy significantly improves the blind binate covering schema, and it outperforms general purpose state-of-the-art tool
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