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

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

Margin-based Ranking and an Equivalence between AdaBoost and RankBoost

We study boosting algorithms for learning to rank. We give a general margin-based bound for ranking based on covering numbers for the hypothesis space. Our bound suggests that algorithms that maximize the ranking margin will generalize well. We then describe a new algorithm, smooth margin ranking, that precisely converges to a maximum ranking-margin solution. The algorithm is a modification of RankBoost, analogous to “approximate coordinate ascent boosting.” Finally, we prove that AdaBoost and RankBoost are equally good for the problems of bipartite ranking and classification in terms of their asymptotic behavior on the training set. Under natural conditions, AdaBoost achieves an area under the ROC curve that is equally as good as RankBoost’s; furthermore, RankBoost, when given a specific intercept, achieves a misclassification error that is as good as AdaBoost’s. This may help to explain the empirical observations made by Cortes andMohri, and Caruana and Niculescu-Mizil, about the excellent performance of AdaBoost as a bipartite ranking algorithm, as measured by the area under the ROC curve

Innovative approaches to the organization of social service of elderly people in a management of the social work : regional experience

The article is devoted to a problem of formation of new system of social service, relevant for Russia, covering a general population. In these conditions the new tasks are set for management of social work. They are connected with creation and introduction in practice of social work of new technologies of work with elderly people who will be directed not only to the solution of the existing social problems, but promoting involvement of pensioners in a "active" old age. At the description of regional experiment of Krasnoyarsk Region on introduction of innovative technologies the analysis and synthesis, logical and complex approaches are used to assessment of level of social service of the elderly. In work, it is shown that the faces of the senior generation are the unprotected social group of the population of Russia. By the way it is added also the financial problems. As a solution of this problem, innovative approach is offered to the organization of social service of the elderly in management of social work of Krasnoyarsk Region. The regional experience will promote the choice of optimal model of management of social work.peer-reviewe

The Complexity of Distributed Approximation of Packing and Covering Integer Linear Programs

In this paper, we present a low-diameter decomposition algorithm in the LOCAL model of distributed computing that succeeds with probability $1 - 1/poly(n)$. Specifically, we show how to compute an $\left(\epsilon, O\left(\frac{\log n}{\epsilon}\right)\right)$ low-diameter decomposition in $O\left(\frac{\log^3(1/\epsilon)\log n}{\epsilon}\right)$ round Further developing our techniques, we show new distributed algorithms for approximating general packing and covering integer linear programs in the LOCAL model. For packing problems, our algorithm finds an $(1-\epsilon)$-approximate solution in $O\left(\frac{\log^3 (1/\epsilon) \log n}{\epsilon}\right)$ rounds with probability $1 - 1/poly(n)$. For covering problems, our algorithm finds an $(1+\epsilon)$-approximate solution in $O\left(\frac{\left(\log \log n + \log (1/\epsilon)\right)^3 \log n}{\epsilon}\right)$ rounds with probability $1 - 1/poly(n)$. These results improve upon the previous $O\left(\frac{\log^3 n}{\epsilon}\right)$-round algorithm by Ghaffari, Kuhn, and Maus [STOC 2017] which is based on network decompositions. Our algorithms are near-optimal for many fundamental combinatorial graph optimization problems in the LOCAL model, such as minimum vertex cover and minimum dominating set, as their $(1\pm \epsilon)$-approximate solutions require $\Omega\left(\frac{\log n}{\epsilon}\right)$ rounds to compute.Comment: To appear in PODC 202

Driver scheduling problem modelling

The Drivers Scheduling Problem (DSP) consists of selecting a set of duties for vehicle drivers, for example buses, trains, plane or boat drivers or pilots, for the transportation of passengers or goods. This is a complex problem because it involves several constraints related to labour and company rules and can also present different evaluation criteria and objectives. Being able to develop an adequate model for this problem that can represent the real problem as close as possible is an important research area.The main objective of this research work is to present new mathematical models to the DSP problem that represent all the complexity of the drivers scheduling problem, and also demonstrate that the solutions of these models can be easily implemented in real situations. This issue has been recognized by several authors and as important problem in Public Transportation. The most well-known and general formulation for the DSP is a Set Partition/Set Covering Model (SPP/SCP). However, to a large extend these models simplify some of the specific business aspects and issues of real problems. This makes it difficult to use these models as automatic planning systems because the schedules obtained must be modified manually to be implemented in real situations. Based on extensive passenger transportation experience in bus companies in Portugal, we propose new alternative models to formulate the DSP problem. These models are also based on Set Partitioning/Covering Models; however, they take into account the bus operator issues and the perspective opinions and environment of the user.We follow the steps of the Operations Research Methodology which consist of: Identify the Problem; Understand the System; Formulate a Mathematical Model; Verify the Model; Select the Best Alternative; Present the Results of the Analysis and Implement and Evaluate. All the processes are done with close participation and involvement of the final users from different transportation companies. The planner‘s opinion and main criticisms are used to improve the proposed model in a continuous enrichment process. The final objective is to have a model that can be incorporated into an information system to be used as an automatic tool to produce driver schedules. Therefore, the criteria for evaluating the models is the capacity to generate real and useful schedules that can be implemented without many manual adjustments or modifications. We have considered the following as measures of the quality of the model: simplicity, solution quality and applicability. We tested the alternative models with a set of real data obtained from several different transportation companies and analyzed the optimal schedules obtained with respect to the applicability of the solution to the real situation. To do this, the schedules were analyzed by the planners to determine their quality and applicability. The main result of this work is the proposition of new mathematical models for the DSP that better represent the realities of the passenger transportation operators and lead to better schedules that can be implemented directly in real situations.Drivers Scheduling Problem, Duties, Modelling

Simultaneous column-and-row generation for solving large-scale linear programs with column-dependent-rows

In this thesis, we handle a general class of large-scale linear programming problems. These problems typically arise in the context of linear programming formulations with exponentially many variables. The defining property for these formulations is a set of linking constraints, which are either too many to be included in the formulation directly, or the full set of linking constraints can only be identified, if all variables are generated explicitly. Due to this dependence between columns and rows, we refer to this class of linear programs as problems with column-dependent-rows. To solve these problems, we need to be able to generate both columns and rows on-the-fly within a new solution method. The proposed approach in this thesis is called simultaneous column-and-row generation. We first characterize the underlying assumptions for the proposed column-and-row generation algorithm. These assumptions are general enough and cover all problems with column-dependent-rows studied in the literature up until now. We then introduce, in detail, a set of pricing subproblems, which are used within the proposed column-and-row generation algorithm. This is followed by a formal discussion on the optimality of the algorithm. Additionally, this generic algorithm is combined with Lagrangian relaxation approach, which provides a different angle to deal with simultaneous column-and-row generation. This observation then leads to another method to solve problems with column-dependent-rows. Throughout the thesis, the proposed solution methods are applied to solve different problems, namely, the multi-stage cutting stock problem, the time-constrained routing problem and the quadratic set covering problem. We also conduct computational experiments to evaluate the performance of the proposed approaches

Nonconvex and mixed integer multiobjective optimization with an application to decision uncertainty

Multiobjective optimization problems commonly arise in different fields like economics or engineering. In general, when dealing with several conflicting objective functions, there is an infinite number of optimal solutions which cannot usually be determined analytically. This thesis presents new branch-and-bound-based approaches for computing the globally optimal solutions of multiobjective optimization problems of various types. New algorithms are proposed for smooth multiobjective nonconvex optimization problems with convex constraints as well as for multiobjective mixed integer convex optimization problems. Both algorithms guarantee a certain accuracy of the computed solutions, and belong to the first deterministic algorithms within their class of optimization problems. Additionally, a new approach to compute a covering of the optimal solution set of multiobjective optimization problems with decision uncertainty is presented. The three new algorithms are tested numerically. The results are evaluated in this thesis as well. The branch-and-bound based algorithms deal with box partitions and use selection rules, discarding tests and termination criteria. The discarding tests are the most important aspect, as they give criteria whether a box can be discarded as it does not contain any optimal solution. We present discarding tests which combine techniques from global single objective optimization with outer approximation techniques from multiobjective convex optimization and with the concept of local upper bounds from multiobjective combinatorial optimization. The new discarding tests aim to find appropriate lower bounds of subsets of the image set in order to compare them with known upper bounds numerically.Multikriterielle Optimierungprobleme sind in diversen Anwendungsgebieten wie beispielsweise in den Wirtschafts- oder Ingenieurwissenschaften zu finden. Da hierbei mehrere konkurrierende Zielfunktionen auftreten, ist die Lösungsmenge eines derartigen Optimierungsproblems im Allgemeinen unendlich groß und kann meist nicht in analytischer Form berechnet werden. In dieser Dissertation werden neue Branch-and-Bound basierte Algorithmen zur Lösung verschiedener Klassen von multikriteriellen Optimierungsproblemen entwickelt und vorgestellt. Der Branch-and-Bound Ansatz ist eine typische Methode der globalen Optimierung. Einer der neuen Algorithmen löst glatte multikriterielle nichtkonvexe Optimierungsprobleme mit konvexen Nebenbedingungen, während ein zweiter zur Lösung multikriterieller gemischt-ganzzahliger konvexer Optimierungsprobleme dient. Beide Algorithmen garantieren eine gewisse Genauigkeit der berechneten Lösungen und gehören damit zu den ersten deterministischen Algorithmen ihrer Art. Zusätzlich wird ein Algorithmus zur Berechnung einer Überdeckung der Lösungsmenge multikriterieller Optimierungsprobleme mit Entscheidungsunsicherheit vorgestellt. Alle drei Algorithmen wurden numerisch getestet. Die Ergebnisse werden ebenfalls in dieser Arbeit ausgewertet. Die neuen Algorithmen arbeiten alle mit Boxunterteilungen und nutzen Auswahlregeln, sowie Verwerfungs- und Terminierungskriterien. Dabei spielen gute Verwerfungskriterien eine zentrale Rolle. Diese entscheiden, ob eine Box verworfen werden kann, da diese sicher keine Optimallösung enthält. Die neuen Verwerfungskriterien nutzen Methoden aus der globalen skalarwertigen Optimierung, Approximationstechniken aus der multikriteriellen konvexen Optimierung sowie ein Konzept aus der kombinatorischen Optimierung. Dabei werden stets untere Schranken der Bildmengen konstruiert, die mit bisher berechneten oberen Schranken numerisch verglichen werden können

Essays on Approximation Algorithms for Robust Linear Optimization Problems

Solving optimization problems under uncertainty has been an important topic since the appearance of mathematical optimization in the mid 19th century. George Dantzig’s 1955 paper, “Linear Programming under Uncertainty” is considered one of the ten most influential papers in Management Science [26]. The methodology introduced in Dantzig’s paper is named stochastic programming, since it assumes an underlying probability distribution of the uncertain input parameters. However, stochastic programming suffers from the “curse of dimensionality”, and knowing the exact distribution of the input parameter may not be realistic. On the other hand, robust optimization models the uncertainty using a deterministic uncertainty set. The goal is to optimize the worst-case scenario from the uncertainty set. In recent years, many studies in robust optimization have been conducted and we refer the reader to Ben-Tal and Nemirovski [4–6], El Ghaoui and Lebret [19], Bertsimas and Sim [15, 16], Goldfarb and Iyengar [23], Bertsimas et al. [8] for a review of robust optimization. Computing an optimal adjustable (or dynamic) solution to a robust optimization problem is generally hard. This motivates us to study the hardness of approximation of the problem and provide efficient approximation algorithms. In this dissertation, we consider adjustable robust linear optimization problems with packing and covering formulations and their approximation algorithms. In particular, we study the performances of static solution and affine solution as approximations for the adjustable robust problem. Chapter 2 and 3 consider two-stage adjustable robust linear packing problem with uncertain second-stage constraint coefficients. For general convex, compact and down-monotone uncertainty sets, the problem is often intractable since it requires to compute a solution for all possible realizations of uncertain parameters [22]. In particular, for a fairly general class of uncertainty sets, we show that the two-stage adjustable robust problem is NP-hard to approximate within a factor that is better than Ω(logn), where n is the number of columns of the uncertain coefficient matrix. On the other hand, a static solution is a single (here and now) solution that is feasible for all possible realizations of the uncertain parameters and can be computed efficiently. We study the performance of static solutions an approximation for the adjustable robust problem and relate its optimality to a transformation of the uncertain set. With this transformation, we show that for a fairly general class of uncertainty sets, static solution is optimal for the adjustable robust problem. This is surprising since the static solution is widely perceived as highly conservative. Moreover, when the static solution is not optimal, we provide an instance-based tight approximation bound that is related to a measure of non-convexity of the transformation of the uncertain set. We also show that for two-stage problems, our bound is at least as good (and in many case significantly better) as the bound given by the symmetry of the uncertainty set [11, 12]. Moreover, our results can be generalized to the case where the objective coefficients and right-hand-side are also uncertainty. In Chapter 3, we focus on the two-stage problems with a family of column-wise and constraint-wise uncertainty sets where any constraint describing the set involves entries of only a single column or a single row. This is a fairly general class of uncertainty sets to model constraint coefficient uncertainty. Moreover, it is the family of uncertainty sets that gives the previous hardness result. On the positive side, we show that a static solution is an O(\log n · min(\log \Gamma, \log(m+n))-approximation for the two-stage adjustable robust problem where m and n denote the numbers of rows and columns of the constraint matrix and \Gamma is the maximum possible ratio of upper bounds of the uncertain constraint coefficients. Therefore, for constant \Gamma, surprisingly the performance bound for static solutions matches the hardness of approximation for the adjustable problem. Furthermore, in general the static solution provides nearly the best efficient approximation for the two-stage adjustable robust problem. In Chapter 4, we extend our result in Chapter 2 to a multi-stage adjustable robust linear optimization problem. In particular, we consider the case where the choice of the uncertain constraint coefficient matrix for each stage is independent of the others. In real world applications, decision problems are often of multiple stages and a iterative implementation of two-stage solution may result in a suboptimal solution for multi-stage problem. We consider the static solution for the adjustable robust problem and the transformation of the uncertainty sets introduced in Chapter 2. We show that the static solution is optimal for the adjustable robust problem when the transformation of the uncertainty set for each stage is convex. Chapters 5 considers a two-stage adjustable robust linear covering problem with uncertain right-hand-side parameter. As mentioned earlier, such problems are often intractable due to astronomically many extreme points of the uncertainty set. We introduce a new approximation framework where we consider a “simple” set that is “close” to the original uncertainty set. Moreover, the adjustable robust problem can be solved efficiently over the extended set. We show that the approximation bound is related to a geometric factor that represents the Banach-Mazur distance between the two sets. Using this framework, we provide approximation bounds that are better than the bounds given by an affine policy in [7] for a large class of interesting uncertainty sets. For instance, we provide an approximation solution that gives a m^{1/4}-approximation for the two-stage adjustable robust problem with hypersphere uncertainty set, while the affine policy has an approximation ratio of O(\sqrt{m}). Moreover, our bound for general p-norm ball is m^{\frac{p-1}{p^2}} as opposed to m^{1/p} as given by an affine policy