The Lovasz Local Lemma due to Erdos and Lovasz is a powerful tool in proving the existence of rare events. We present an extension of this lemma, which works well when the event to be shown to exist is a conjunction of individual events, each of which asserts that a random variable does not deviate much from its mean. As applications, we consider two classes of NP-hard integer programs: minimax and covering integer programs. A key technique, randomized rounding of linear relaxations, was developed by Raghavan and Thompson to derive good approximation algorithms for such problems. We use our extension of the Local Lemma to prove that randomized rounding produces, with non-zero probability, much better feasible solutions than known before, if the constraint matrices of these integer programs are column-sparse (e.g., routing using short paths, problems on hypergraphs with small dimension degree). This complements certain well-known results from discrepancy theory. We also generalize the method of pessimistic estimators due to Raghavan, to obtain constructive (algorithmic) versions of our results for covering integer programs.
Next, there is growing interest in optimization, since different participating individuals and or organizations may have different objective functions in a given problem instance; see, e.g., @cite_32 . Motivated by this, we study multi-criteria optimization in the setting of covering problems:
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We study problems in multiobjective optimization, in which solutions to a combinatorial optimization problem are evaluated with respect to several cost criteria, and we are interested in the trade-off between these objectives (the so-called Pareto curve). We point out that, under very general conditions, there is a polynomially succinct curve that spl epsiv -approximates the Pareto curve, for any spl epsiv >0. We give a necessary and sufficient condition under which this approximate Pareto curve can be constructed in time polynomial in the size of the instance and 1 spl epsiv . In the case of multiple linear objectives, we distinguish between two cases: when the underlying feasible region is convex, then we show that approximating the multi-objective problem is equivalent to approximating the single-objective problem. If however the feasible region is discrete, then we point out that the question reduces to an old and recurrent one: how does the complexity of a combinatorial optimization problem change when its feasible region is intersected with a hyperplane with small coefficients; we report some interesting new findings in this domain. Finally, we apply these concepts and techniques to formulate and solve approximately a cost-time-quality trade-off for optimizing access to the World-Wide Web, in a model first studied by (1996) (which was actually the original motivation for this work). This paper addresses the solution of bound-constrained optimization problems using algorithms that require only the availability of objective function values but no derivative information. We refer to these algorithms as derivative-free algorithms. Fueled by a growing number of applications in science and engineering, the development of derivative-free optimization algorithms has long been studied, and it has found renewed interest in recent time. Along with many derivative-free algorithms, many software implementations have also appeared. The paper presents a review of derivative-free algorithms, followed by a systematic comparison of 22 related implementations using a test set of 502 problems. The test bed includes convex and nonconvex problems, smooth as well as nonsmooth problems. The algorithms were tested under the same conditions and ranked under several criteria, including their ability to find near-global solutions for nonconvex problems, improve a given starting point, and refine a near-optimal solution. A total of 112,448 problem instances were solved. We find that the ability of all these solvers to obtain good solutions diminishes with increasing problem size. For the problems used in this study, TOMLAB MULTIMIN, TOMLAB GLCCLUSTER, MCS and TOMLAB LGO are better, on average, than other derivative-free solvers in terms of solution quality within 2,500 function evaluations. These global solvers outperform local solvers even for convex problems. Finally, TOMLAB OQNLP, NEWUOA, and TOMLAB MULTIMIN show superior performance in terms of refining a near-optimal solution. Recent work on Bayesian optimization has shown its effectiveness in global optimization of difficult black-box objective functions. Many real-world optimization problems of interest also have constraints which are unknown a priori. In this paper, we study Bayesian optimization for constrained problems in the general case that noise may be present in the constraint functions, and the objective and constraints may be evaluated independently. We provide motivating practical examples, and present a general framework to solve such problems. We demonstrate the effectiveness of our approach on optimizing the performance of online latent Dirichlet allocation subject to topic sparsity constraints, tuning a neural network given test-time memory constraints, and optimizing Hamiltonian Monte Carlo to achieve maximal effectiveness in a fixed time, subject to passing standard convergence diagnostics. Robust optimization has traditionally focused on uncertainty in data and costs in optimization problems to formulate models whose solutions will be optimal in the worst-case among the various uncertain scenarios in the model. While these approaches may be thought of defining data- or cost-robust problems, we formulate a new "demand-robust" model motivated by recent work on two-stage stochastic optimization problems. We propose this in the framework of general covering problems and prove a general structural lemma about special types of first-stage solutions for such problems: there exists a first-stage solution that is a minimal feasible solution for the union of the demands for some subset of the scenarios and its objective function value is no more than twice the optimal. We then provide approximation algorithms for a variety of standard discrete covering problems in this setting, including minimum cut, minimum multi-cut, shortest paths, Steiner trees, vertex cover and un-capacitated facility location. While many of our results draw from rounding approaches recently developed for stochastic programming problems, we also show new applications of old metric rounding techniques for cut problems in this demand-robust setting. Abstract We consider the NP-hard problem of scheduling jobs on a single machine against common due dates with respect to earliness and tardiness penalties. The paper covers two aspects: Firstly, we develop a problem generator and solve 280 instances with two new heuristics to obtain upper bounds on the optimal objective function value. Secondly, we demonstrate computationally that our heuristics are efficient in obtaining near-optimal solutions for small problem instances. The generated problem instances in combination with the upper bounds can be used as benchmarks for future approaches in the field of common due-date scheduling. Scope and purpose In connection with just-in-time production and delivery, earliness as well as tardiness penalties are of interest. Thus scheduling against common due dates has received growing attention during the last decade. Many algorithms have been developed to solve the different variants of this problem. But whenever a new algorithm for scheduling against common due dates is proposed, its quality is assessed only on a few self-generated examples. Hence it is difficult to evaluate the various approaches, particularly in comparison with each other. Therefore the goal of this paper is to present numerous benchmark problems together with some upper bounds on the optimal objective function value. The problems of obesity and overweight are commonly cited as the motivation behind recent efforts to develop technology that promotes physical activity. Prompted by the social nature of many of the emerging applications, this paper presents our investigation of the sociality of weight management as experienced by a broad demographic of individuals. Our findings highlight the broad scope of peer involvement, and provide insight into the context and mechanics of related interaction that may prove valuable in informing the next generation of peer-based weight management technology for use in everyday life. Given a collection @math of weighted subsets of a ground set @math , the set cover problem is to find a minimum weight subset of @math which covers all elements of @math . We study a strategic game defined upon this classical optimization problem. Every element of @math is a player which chooses one set of @math where it appears. Following a public tax function, every player is charged a fraction of the weight of the set that it has selected. Our motivation is to design a tax function having the following features: it can be implemented in a distributed manner, existence of an equilibrium is guaranteed and the social cost for these equilibria is minimized. Dynamic Multi-objective Optimization is a challenging research topic since the objective functions, constraints, and problem parameters may change over time. Although dynamic optimization and multi-objective optimization have separately obtained a great interest among many researchers, there are only few studies that have been developed to solve Dynamic Multi-objective Optimisation Problems (DMOPs). Moreover, applying Evolutionary Algorithms (EAs) to solve this category of problems is not yet highly explored although this kind of problems is of significant importance in practice. This paper is devoted to briefly survey EAs that were proposed in the literature to handle DMOPs. In addition, an overview of the most commonly used test functions, performance measures and statistical tests is presented. Actual challenges and future research directions are also discussed. Applications in complex systems such as the Internet have spawned a recent interest in studying situations involving multiple agents with their individual cost or utility functions. In this paper, we introduce an algorithmic framework for studying combinatorial optimization problems in the presence of multiple agents with submodular cost functions. We study several fundamental covering problems in this framework and establish upper and lower bounds on their approximability. In crowdsourcing systems, the interests of contributing participants and system stakeholders are often not fully aligned. Participants seek to learn, be entertained, and perform easy tasks, which offer them instant gratification; system stakeholders want users to complete more difficult tasks, which bring higher value to the crowdsourced application. We directly address this problem by presenting techniques that optimize the crowdsourcing process by jointly maximizing the user longevity in the system and the true value that the system derives from user participation. We first present models that predict the "survival probability" of a user at any given moment, that is, the probability that a user will proceed to the next task offered by the system. We then leverage this survival model to dynamically decide what task to assign and what motivating goals to present to the user. This allows us to jointly optimize for the short term (getting difficult tasks done) and for the long term (keeping users engaged for longer periods of time). We show that dynamically assigning tasks significantly increases the value of a crowdsourcing system. In an extensive empirical evaluation, we observed that our task allocation strategy increases the amount of information collected by up to 117.8 . We also explore the utility of motivating users with goals. We demonstrate that setting specific, static goals can be highly detrimental to the long-term user participation, as the completion of a goal (e.g., earning a badge) is also a common drop-off point for many users. We show that setting the goals dynamically, in conjunction with judicious allocation of tasks, increases the amount of information collected by the crowdsourcing system by up to 249 , compared to the existing baselines that use fixed objectives. In recent years scholars have built maps of science by connecting the academic fields that cite each other, are cited together, or that cite a similar literature. But since scholars cannot always publish in the fields they cite, or that cite them, these science maps are only rough proxies for the potential of a scholar, organization, or country, to enter a new academic field. Here we use a large dataset of scholarly publications disambiguated at the individual level to create a map of science-or research space-where links connect pairs of fields based on the probability that an individual has published in both of them. We find that the research space is a significantly more accurate predictor of the fields that individuals and organizations will enter in the future than citation based science maps. At the country level, however, the research space and citations based science maps are equally accurate. These findings show that data on career trajectories-the set of fields that individuals have previously published in-provide more accurate predictors of future research output for more focalized units-such as individuals or organizations-than citation based science maps. Decomposing a complex shape into geometrically simple primitives is a fundamental problem in geometry processing. We are interested in a shape decomposition problem where the simple primitives sought are generalized cylinders, which are ubiquitous in both organic forms and man-made artifacts. We introduce a quantitative measure of cylindricity for a shape part and develop a cylindricity-driven optimization algorithm, with a global objective function, for generalized cylinder decomposition. As a measure of geometric simplicity and following the minimum description length principle, cylindricity is defined as the cost of representing a cylinder through skelet al and cross-section profile curves. Our decomposition algorithm progressively builds local to non-local cylinders, which form over-complete covers of the input shape. The over-completeness of the cylinder covers ensures a conservative buildup of the cylindrical parts, leaving the final decision on decomposition to global optimization. We solve the global optimization by finding an exact cover, which optimizes the global objective function. We demonstrate results of our optimal decomposition algorithm on numerous examples and compare with other alternatives. This paper focuses on kernel methods for multi-instance learning. Existing methods require the prediction of the bag to be identical to the maximum of those of its individual instances. However, this is too restrictive as only the sign is important in classification. In this paper, we provide a more complete regularization framework for MI learning by allowing the use of different loss functions between the outputs of a bag and its associated instances. This is especially important as we generalize this for multi-instance regression. Moreover, both bag and instance information can now be directly used in the optimization. Instead of using heuristics to solve the resultant non-linear optimization problem, we use the constrained concave-convex procedure which has well-studied convergence properties. Experiments on both classification and regression data sets show that the proposed method leads to improved performance. In a document network such as a citation network of scientific documents, web-logs, etc., the content produced by authors exhibits their interest in certain topics. In addition some authors influence other authors' interests. In this work, we propose to model the influence of cited authors along with the interests of citing authors. Moreover, we hypothesize that apart from the citations present in documents, the context surrounding the citation mention provides extra topical information about the cited authors. However, associating terms in the context to the cited authors remains an open problem. We propose novel document generation schemes that incorporate the context while simultaneously modeling the interests of citing authors and influence of the cited authors. Our experiments show significant improvements over baseline models for various evaluation criteria such as link prediction between document and cited author, and quantitatively explaining unseen text. Scheduling theory has received a growing interest since its origins in the second half of the 20th century. Developed initially for the study of scheduling problems with a single objective, the theory has been recently extended to problems involving multiple criteria. However, this extension has still left a gap between the classical multi-criteria approaches and some real-life problems in which not all jobs contribute to the evaluation of each criterion. In this book, we close this gap by presenting and developing multi-agent scheduling models in which subsets of jobs sharing the same resources are evaluated by different criteria. Several scenarios are introduced, depending on the definition and the intersection structure of the job subsets. Complexity results, approximation schemes, heuristics and exact algorithms are discussed for single-machine and parallel-machine scheduling environments. Definitions and algorithms are illustrated with the help of examples and figures. We motivate and describe a new parameterized approximation paradigm which studies the interaction between performance ratio and running time for any parametrization of a given optimization problem. As a key tool, we introduce the concept of α-shrinking transformation, for α≥1. Applying such transformation to a parameterized problem instance decreases the parameter value, while preserving approximation ratio of α (or α-fidelity). For example, it is well-known that Vertex Cover cannot be approximated within any constant factor better than 2 [24] (under usual assumptions). Our parameterized α-approximation algorithm for k-Vertex Cover, parameterized by the solution size, has a running time of 1.273(2−α)k, where the running time of the best FPT algorithm is 1.273k [10]. Our algorithms define a continuous tradeoff between running times and approximation ratios, allowing practitioners to appropriately allocate computational resources. Moving even beyond the performance ratio, we call for a new type of approximative kernelization race. Our α-shrinking transformations can be used to obtain kernels which are smaller than the best known for a given problem. For the Vertex Cover problem we obtain a kernel size of 2(2−α)k. The smaller "α-fidelity" kernels allow us to solve exactly problem instances more efficiently, while obtaining an approximate solution for the original instance. We show that such transformations exist for several fundamental problems, including Vertex Cover, d-Hitting Set, Connected Vertex Cover and Steiner Tree. We note that most of our algorithms are easy to implement and are therefore practical in use. We consider the problem of next-best view selection for volumetric reconstruction of an object by a mobile robot equipped with a camera. Based on a probabilistic volumetric map that is built in real time, the robot can quantify the expected information gain from a set of discrete candidate views. We propose and evaluate several formulations to quantify this information gain for the volumetric reconstruction task, including visibility likelihood and the likelihood of seeing new parts of the object. These metrics are combined with the cost of robot movement in utility functions. The next best view is selected by optimizing these functions, aiming to maximize the likelihood of discovering new parts of the object. We evaluate the functions with simulated and real world experiments within a modular software system that is adaptable to other robotic platforms and reconstruction problems. We release our implementation open source. Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in Hk(Ω), and optimizing functionals arising from some problems in economics. In the continuous setting and assuming smoothness, the convexity constraints may be given locally by asking the Hessian matrix to be positive semidefinite, but in making discrete approximations two difficulties arise: the continuous solutions may be not smooth, and functions with positive semidefinite discrete Hessian need not be convex in a discrete sense. Previous work has concentrated on non-local descriptions of convexity, making the number of constraints to grow super-linearly with the number of nodes even in dimension 2, and these descriptions are very difficult to extend to higher dimensions. In this paper we propose a finite difference approximation using positive semidefinite programs and discrete Hessians, and prove convergence under very general conditions, even when the continuous solution is not smooth, working on any dimension, and requiring a linear number of constraints in the number of nodes. Using semidefinite programming codes, we show concrete examples of approximations to problems in two and three dimensions. Clustering on multiple views is witnessing increasing interests in both real-world application and machine learning community. A typical application is to discover communities of joint interests in social network, such as Facebook and Twitter. The network can be simply modeled as a graph in which the nodes are the people while the links show relationship between the people. There may exist many relationships between a pair of nodes, such as classmates, collaborators, playmates and so on. It is important to consider how to use these graphs together rather than a single graph if we want to understand the network and their participants effectively. Motivated by the fact, we present a clustering algorithm using spectral analysis in which multiple graphs are considered to get the clusters. Our study can also be considered as an instance of multi-views learning. The experimental results on UCI data set and Corel image data demonstrate the promising results that validate our proposed algorithm. This paper presents a hierarchical composition approach for multi-view object tracking. The key idea is to adaptively exploit multiple cues in both 2D and 3D, e.g., ground occupancy consistency, appearance similarity, motion coherence etc., which are mutually complementary while tracking the humans of interests over time. While feature online selection has been extensively studied in the past literature, it remains unclear how to effectively schedule these cues for the tracking purpose especially when encountering various challenges, e.g. occlusions, conjunctions, and appearance variations. To do so, we propose a hierarchical composition model and re-formulate multi-view multi-object tracking as a problem of compositional structure optimization. We setup a set of composition criteria, each of which corresponds to one particular cue. The hierarchical composition process is pursued by exploiting different criteria, which impose constraints between a graph node and its offsprings in the hierarchy. We learn the composition criteria using MLE on annotated data and efficiently construct the hierarchical graph by an iterative greedy pursuit algorithm. In the experiments, we demonstrate superior performance of our approach on three public datasets, one of which is newly created by us to test various challenges in multi-view multi-object tracking.