Motivated by online advertisement and exchange settings, greedy randomized algorithms for the maximum matching problem have been studied, in which the algorithm makes (random) decisions that are essentially oblivious ...
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Motivated by online advertisement and exchange settings, greedy randomized algorithms for the maximum matching problem have been studied, in which the algorithm makes (random) decisions that are essentially oblivious to the input graph. Any greedy algorithm can achieve a performance ratio of 0.5, which is the expected number of matched nodes to the number of nodes in a maximum matching. Since Aronson, Dyer, Frieze, and Suen [Random Structures Algorithm, 6 (1991), pp. 29-46] proved that the modified randomized greedy algorithm achieves a performance ratio of 0:5 + epsilon (where epsilon = 1/400000) on arbitrary graphs in the midnineties, no further attempts in the literature have been made to improve this theoretical ratio for arbitrary graphs until two papers were published in FOCS 2012 [G. Goel and P. Tripathi, IEEE Computer Society, Los Alamitos, CA, 2012, pp. 718-727;M. Poloczek and M. Szegedy, IEEE Computer Society, Los Alamitos, CA, 2012, pp. 708-717]. In this paper, we revisit the ranking algorithm using the linearprogramming framework. Special care is given to analyze the structural properties of the ranking algorithm in order to derive the linearprogramming constraints, of which one known as the boundary constraint requires totally new analysis and is crucial to the success of our linear program (LP). We use continuous linear programming relaxation to analyze the limiting behavior as the finite LP grows. Of particular interest are new duality and complementary slackness characterizations that can handle the monotone and the boundary constraints in continuous linear programming. Improving previous work, this paper achieves a theoretical performance ratio of 2(5-root 7)/9 approximate to 0.523 on arbitrary graphs.
We consider continuouslinear programs over a continuous finite time horizon T, with a constant coefficient matrix, linear right hand side functions and linear cost coefficient functions. Specifically, we search for o...
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We consider continuouslinear programs over a continuous finite time horizon T, with a constant coefficient matrix, linear right hand side functions and linear cost coefficient functions. Specifically, we search for optimal solutions in the space of measures or of functions of bounded variation. These models generalize the separated continuous linear programming models and their various duals, as formulated in the past by Anderson, by Pullan, and by Weiss. In previous papers we formulated a symmetric dual and have shown strong duality. We also have presented a detailed description of optimal solutions and have defined a combinatorial analogue to basic solutions of standard LP. In this paper we present an algorithm which solves this class of problems in a finite bounded number of steps, using an analogue of the simplex method, in the space of measures.
The urban green areas represent a strategic resource for the contemporary city sustainable development. In addition to aesthetic and recreational functions, their presence contributes to increase the environmental qua...
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ISBN:
(纸本)9783319920993;9783319920986
The urban green areas represent a strategic resource for the contemporary city sustainable development. In addition to aesthetic and recreational functions, their presence contributes to increase the environmental quality level by improving the microclimate, preserving the biodiversity and promoting the territory economic growth. However, the interventions execution designed to provide the built areas of the so-called urban forests is rarely indicated as a priority action in the urban spaces planning because often a different allocation of available resources is preferred. In this work, starting from the definition of a indicators set useful for expressing the not only financial, but also social, cultural and environmental components of value of the projects for urban forestry, the aim is to build a multi-criteria economic analysis protocol purposeful at predicting the correct funds distribution between initiatives for realization of urban forests at a district scale. The characterization of the model is carried out through the logical-mathematical tools of the Operational Research developed according to continuous linear programming principles.
We consider continuouslinear programs over a continuous finite time horizon T, with linear cost coefficient functions, linear right-hand side functions, and a constant coefficient matrix, as well as their symmetric d...
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We consider continuouslinear programs over a continuous finite time horizon T, with linear cost coefficient functions, linear right-hand side functions, and a constant coefficient matrix, as well as their symmetric dual. We search for optimal solutions in the space of measures or of functions of bounded variation. These models generalize the separated continuous linear programming models and their various duals, as formulated in the past by Anderson, by Pullan, and by Weiss. In a recent paper, we have shown that under a Slater-type condition, these problems possess optimal strongly dual solutions. In this paper, we give a detailed description of optimal solutions and define a combinatorial analogue to basic solutions of standard LP. We also show that feasibility implies existence of strongly dual optimal solutions without requiring the Slater condition. We present several examples to illustrate the richness and complexity of these solutions.
We consider continuouslinear programs over a continuous finite time horizon T, with linear cost coefficient functions and linear right-hand side functions and a constant coefficient matrix, where we search for optima...
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We consider continuouslinear programs over a continuous finite time horizon T, with linear cost coefficient functions and linear right-hand side functions and a constant coefficient matrix, where we search for optimal solutions in the space of measures or of functions of bounded variation. These models generalize the separated continuous linear programming models and their various duals, as formulated in the past by Anderson, by Pullan, and by Weiss. We present simple necessary and sufficient conditions for feasibility. We formulate a symmetric dual and investigate strong duality by considering discrete time approximations. We prove that under a Slater type condition there is no duality gap and there exist optimal solutions which have impulse controls at 0 and T and have piecewise constant densities in (0, T). Moreover, we show that under nondegeneracy assumptions all optimal solutions are of this form, and are uniquely determined over (0, T).
There has been much research on network flows over time due to their important role in real world applications. This has led to many results, but the more challenging continuous time model still lacks some of the key ...
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There has been much research on network flows over time due to their important role in real world applications. This has led to many results, but the more challenging continuous time model still lacks some of the key concepts and techniques that are the cornerstones of static network flows. The aim of this paper is to advance the state of the art for dynamic network flows by developing the continuous time analogues of the theory for static network flows. Specifically, we make use of ideas from the static case to establish a reduced cost optimality condition, a negative cycle optimality condition, and a strong duality result for a very general class of network flows over time. (C) 2014 Elsevier B.V. All rights reserved.
Motivated by online advertisement and exchange settings, greedy randomized algorithms for the maximum matching problem have been studied, in which the algorithm makes (random) decisions that are essentially oblivious ...
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ISBN:
(纸本)9781611973389
Motivated by online advertisement and exchange settings, greedy randomized algorithms for the maximum matching problem have been studied, in which the algorithm makes (random) decisions that are essentially oblivious to the input graph. Any greedy algorithm can achieve performance ratio 0.5, which is the expected number of matched nodes to the number of nodes in a maximum matching. Since Aronson, Dyer, Frieze and Suen proved that the Modified Randomized Greedy algorithm achieves performance ratio 0.5+ε (where ε = 1/400000 ) on arbitrary graphs in the mid-nineties, no further attempts in the literature have been made to improve this theoretical ratio for arbitrary graphs until two papers were published in FOCS 2012. In this paper, we revisit the Ranking algorithm using the LP framework. Special care is given to analyze the structural properties of the Ranking algorithm in order to derive the LP constraints, of which one known as the boundary constraint requires totally new analysis and is crucial to the success of our LP. We use continuous LP relaxation to analyze the limiting behavior as the finite LP grows. Of particular interest are new duality and complementary slackness characterizations that can handle the monotone and the boundary constraints in continuous LP. Our work achieves the currently best the oretical performance ratio of 2(5-7~(1/2))/9≈0:523 on arbitrary graphs. Moreover, experiments suggest that Ranking cannot perform better than 0:724 in general.
continuouslinear programs have attracted considerable interest due to their potential for modeling manufacturing, scheduling, and routing problems. While efficient simplex-type algorithms have been developed for sepa...
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continuouslinear programs have attracted considerable interest due to their potential for modeling manufacturing, scheduling, and routing problems. While efficient simplex-type algorithms have been developed for separated continuouslinear programs, crude time discretization remains the method of choice for solving general (nonseparated) problem instances. In this paper we propose a more generic approximation scheme for nonseparated continuouslinear programs, where we approximate the functional decision variables (policies) by polynomial and piecewise polynomial decision rules. This restriction results in an upper bound on the original problem, which can be computed efficiently by solving a tractable semidefinite program. To estimate the approximation error, we also compute a lower bound by solving a dual continuouslinear program in (piecewise) polynomial decision rules. We establish the convergence of the primal and dual approximations under Slater-type constraint qualifications. We also highlight the potential of our method for optimizing large-scale multiclass queueing systems and dynamic Leontief models.
Temporal dynamics is a crucial feature of network flow problems occurring in many practical applications. Important characteristics of real-world networks such as arc capacities, transit times, transit and storage cos...
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Temporal dynamics is a crucial feature of network flow problems occurring in many practical applications. Important characteristics of real-world networks such as arc capacities, transit times, transit and storage costs, demands and supplies etc. are subject to fluctuations over time. Consequently, also flow on arcs can change over time which leads to so-called dynamic network flows. While time is a continuous entity by nature, discrete-time models are often used for modeling dynamic network flows as the resulting problems are in general much easier to handle computationally. In this paper, we study a general class of dynamic network flow problems in the continuous-time model, where the input functions are assumed to be piecewise linear or piecewise constant. We give two discrete approximations of the problem by dividing the considered time range into intervals where all parameters are constant or linear. We then present two algorithms that compute, or at least converge to optimum solutions. Finally, we give an empirical analysis of the performance of both algorithms.
This paper discusses a class of continuouslinear programs with fuzzy valued objective functions. A member of this class is called a fuzzy separated continuouslinear program (FSCLP). Such problems have applications i...
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This paper discusses a class of continuouslinear programs with fuzzy valued objective functions. A member of this class is called a fuzzy separated continuouslinear program (FSCLP). Such problems have applications in a number of domains, including, production and inventory systems, communication networks, and pipeline systems for transportation. The discretization approach is used to construct two ordinary fuzzy linearprogramming problems, which give a lower and an upper bound on the optimal value of FSCLP. It is then shown how to construct an improved feasible solution for FSCLP starting from a nonoptimal one. This leads to the development of a class of algorithms based on a sequence of discrete approximations to FSCLP. Numerical examples in the context of continuous-time networks are presented to show the applicability of the proposed method.
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