We consider robust variants of the bin packing problem with uncertain item sizes. Specifically we consider two uncertainty sets previously studied in the literature. The first is budgeted uncertainty (the $U^\Gamma$ model), in which at most $\Gamma$ items deviate, each reaching its peak value, while other items assume their nominal values. The second uncertainty set, the $U^\Omega$ model, bounds the total amount of deviation in each scenario. We show that a variant of the Next-cover algorithm is a $2$ approximation for the $U^\Omega$ model, and another variant of this algorithm is a $2\Gamma$ approximation for the $U^\Gamma$ model. Unlike the classical bin packing problem, it is shown that (unless $\mathcal{P}=\mathcal{NP}$) no asymptotic approximation scheme exists for the $U^\Gamma$ model, for $\Gamma=1$. This motivates the question of the existence of a constant approximation factor algorithm for the $U^\Gamma$ model. Our main result is to answer this question by proving a (polynomial-time) $4.5$ approximation algorithm, based on a dynamic-programming approach.

We describe a learning procedure enhancing L1-penalized regression by adding dynamically generated rules describing multidimensional “box” sets. Our rule-adding procedure is based on the classical column generation method for high-dimensional linear programming. The pricing problem for our column generation procedure reduces to the NP-hard rectangular maximum agreement (RMA) problem of finding a box that best discriminates between two weighted datasets. We solve this problem exactly using a parallel branch-and-bound procedure. The resulting rule-enhanced regression procedure is computation-intensive, but has promising prediction performance.

MOTIVATION: Elementary (flux) modes (EMs) have served as a valuable tool for investigating structural and functional properties of metabolic networks. Identification of the full set of EMs in genome-scale networks remains challenging due to combinatorial explosion of EMs in complex networks. It is often, however, that only a small subset of relevant EMs needs to be known, for which optimization-based sequential computation is a useful alternative. Most of the currently available methods along this line are based on the iterative use of mixed integer linear programming (MILP), the effectiveness of which significantly deteriorates as the number of iterations builds up. To alleviate the computational burden associated with the MILP implementation, we here present a novel optimization algorithm termed alternate integer linear programming (AILP). RESULTS: Our algorithm was designed to iteratively solve a pair of integer programming (IP) and linear programming (LP) to compute EMs in a sequential manner. In each step, the IP identifies a minimal subset of reactions, the deletion of which disables all previously identified EMs. Thus, a subsequent LP solution subject to this reaction deletion constraint becomes a distinct EM. In cases where no feasible LP solution is available, IP-derived reaction deletion sets represent minimal cut sets (MCSs). Despite the additional computation of MCSs, AILP achieved significant time reduction in computing EMs by orders of magnitude. The proposed AILP algorithm not only offers a computational advantage in the EM analysis of genome-scale networks, but also improves the understanding of the linkage between EMs and MCSs. AVAILABILITY AND IMPLEMENTATION: The software is implemented in Matlab, and is provided as supplementary information . CONTACT: [email protected]. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

This article introduces novel formulations for optimally responding to epidemics and cyber attacks in networks. In our models, at a given time period, network nodes (e.g., users or computing resources) are associated with probabilities of being infected, and each network edge is associated with some probability of propagating the infection. A decision maker would like to maximize the network's utility; keeping as many nodes open as possible, while satisfying given bounds on the probabilities of nodes being infected in the next time period. The model's relation to previous deterministic optimization models and to both probabilistic and deterministic asymptotic models is explored. Initially, maintaining the stochastic independence assumption of previous work, we formulate a nonlinear integer program with high-order multilinear terms. We then propose a quadratic formulation that provides a lower bound and feasible solution to the original problem. Further motivation for the quadratic model is given by showing that it alleviates the assumption of stochastic independence. The quadratic formulation is then linearized in order to be solved by standard integer programming solvers. We develop valid inequalities for the resulting formulations. © 2015 Wiley Periodicals, Inc.NETWORKS, Vol. 66(2), 145–158 2015

This paper proposes a convex relaxation of a sparse support vector machine (SVM) based on the perspective relaxation of mixed-integer nonlinear programs. We seek to minimize the zero-norm of the hyperplane normal vector with a standard SVM hinge-loss penalty and extend our approach to a zero-one loss penalty. The relaxation that we propose is a second-order cone formulation that can be efficiently solved by standard conic optimization solvers. We compare the optimization properties and classification performance of the second-order cone formulation with previous sparse SVM formulations suggested in the literature.

The 𝒩𝒫-hard maximum monomial agreement problem consists of finding a single logical conjunction that is most consistent with or “best fits” a weighted data set of “positive” and “negative” binary vectors. Computing weighted voting classifiers using boosting methods involves a maximum agreement subproblem at each iteration, although such subproblems are typically solved in practice by heuristic methods. Here, we describe an exact branch-and-bound method for maximum agreement over Boolean monomials, improving on the earlier work of Goldberg and Shan [Goldberg, N., C. Shan. 2007. Boosting optimal logical patterns. Proc. 7th SIAM Internat. Conf. Data Mining, SIAM, Philadelphia, 228–236]. Specifically, we develop a tighter upper bounding function and an improved branching procedure that exploits knowledge of the bound and the particular data set, while having a lower branching factor. Experimental results show that the new method is able to solve larger problem instances and runs faster within a linear programming boosting procedure applied to medium-sized data sets from the UCI Machine Learning Repository. The new algorithm also runs much faster than applying a commercial mixed-integer programming solver, which uses linear programming relaxation-based bounds, to an integer linear programming formulation of the problem.

We consider the supervised learning of a binary classifier from noisy observations. We use smooth boosting to linearly combine abstaining hypotheses, each of which maps a subcube of the attribute space to one of the two classes. We introduce a new branch-and-bound weak learner to maximize the agreement rate of each hypothesis. Dobkin et al. give an algorithm for maximizing agreement with real-valued attributes [9]. Our algorithm improves on the time complexity of Dobkin et al.'s as long as the data can be binarized so that the number of binary attributes is o(log of the number of observations × number of real-valued attributes). Furthermore, we have fine-tuned our branch-and-bound algorithm with a queuing discipline and optimality gap to make it fast in practice. Finally, since logical patterns in Hammer et al.'s Logical Analysis of Data (LAD) framework [8, 6] are equivalent to abstaining monomial hypotheses, any boosting algorithm can be combined with our proposed weak learner to construct LAD models. On various data sets, our method outperforms state-of-the-art methods that use suboptimal or heuristic weak learners, such as SLIPPER. It is competitive with other optimizing classifiers that combine monomials, such as LAD. Compared to LAD, our method eliminates many free parameters that restrict the hypothesis space and require extensive fine-tuning by cross-validation.