Cardinality constraints or, more generally, weight constraints are well recognized as an important extension of answer-set programming. Clearly, all common algorithmic tasks related to programs with cardinality or weight constraints - like checking the consistency of a program - are intractable. Many intractable problems in the area of knowledge representation and reasoning have been shown to become linear time tractable if the treewidth of the programs or formulas under consideration is bounded by some constant. The goal of this paper is to apply the notion of treewidth to programs with cardinality or weight constraints and to identify tractable fragments. It will turn out that the straightforward application of treewidth to such class of programs does not suffice to obtain tractability. However, by imposing further restrictions, tractability can be achieved.

This paper addresses a mixed integer programming (MIP) formulation for the multi-item uncapacitated lot-sizing problem that is inspired from the trailer manufacturer. The proposed MIP model has been utilized to find out the optimum order quantity, optimum order time, and the minimum total cost of purchasing, ordering, and holding over the predefined planning horizon. This problem is known as NP-hard problem. The model was presented in an optimal software form using LINGO 13.0.

High throughput genetic sequencing arrays with thousands of measurements per sample and a great amount of related censored clinical data have increased demanding need for better measurement specific model selection. In this paper we establish strong oracle properties of nonconcave penalized methods for nonpolynomial (NP) dimensional data with censoring in the framework of Cox's proportional hazards model. A class of folded-concave penalties are employed and both LASSO and SCAD are discussed specifically. We unveil the question under which dimensionality and correlation restrictions can an oracle estimator be constructed and grasped. It is demonstrated that nonconcave penalties lead to significant reduction of the "irrepresentable condition" needed for LASSO model selection consistency. The large deviation result for martingales, bearing interests of its own, is developed for characterizing the strong oracle property. Moreover, the nonconcave regularized estimator, is shown to achieve asymptotically the information bound of the oracle estimator. A coordinate-wise algorithm is developed for finding the grid of solution paths for penalized hazard regression problems, and its performance is evaluated on simulated and gene association study examples.

This paper presents theory for Normalized Random Measures (NRMs), Normalized Generalized Gammas (NGGs), a particular kind of NRM, and Dependent Hierarchical NRMs which allow networks of dependent NRMs to be analysed. These have been used, for instance, for time-dependent topic modelling. In this paper, we first introduce some mathematical background of completely random measures (CRMs) and their construction from Poisson processes, and then introduce NRMs and NGGs. Slice sampling is also introduced for posterior inference. The dependency operators in Poisson processes and for the corresponding CRMs and NRMs is then introduced and Posterior inference for the NGG presented. Finally, we give dependency and composition results when applying these operators to NRMs so they can be used in a network with hierarchical and dependent relations.

The sum-product or belief propagation (BP) algorithm is a widely-used message-passing algorithm for computing marginal distributions in graphical models with discrete variables. At the core of the BP message updates, when applied to a graphical model with pairwise interactions, lies a matrix-vector product with complexity that is quadratic in the state dimension $d$, and requires transmission of a $(d-1)$-dimensional vector of real numbers (messages) to its neighbors. Since various applications involve very large state dimensions, such computation and communication complexities can be prohibitively complex. In this paper, we propose a low-complexity variant of BP, referred to as stochastic belief propagation (SBP). As suggested by the name, it is an adaptively randomized version of the BP message updates in which each node passes randomly chosen information to each of its neighbors. The SBP message updates reduce the computational complexity (per iteration) from quadratic to linear in $d$, without assuming any particular structure of the potentials, and also reduce the communication complexity significantly, requiring only $\log{d}$ bits transmission per edge. Moreover, we establish a number of theoretical guarantees for the performance of SBP, showing that it converges almost surely to the BP fixed point for any tree-structured graph, and for graphs with cycles satisfying a contractivity condition. In addition, for these graphical models, we provide non-asymptotic upper bounds on the convergence rate, showing that the $\ell_{\infty}$ norm of the error vector decays no slower than $O(1/\sqrt{t})$ with the number of iterations $t$ on trees and the mean square error decays as $O(1/t)$ for general graphs. These analysis show that SBP can provably yield reductions in computational and communication complexities for various classes of graphical models.

Compact representations of objects is a common concept in computer science. Automated planning can be viewed as a case of this concept: a planning instance is a compact implicit representation of a graph and the problem is to find a path (a plan) in this graph. While the graphs themselves are represented compactly as planning instances, the paths are usually represented explicitly as sequences of actions. Some cases are known where the plans always have compact representations, for example, using macros. We show that these results do not extend to the general case, by proving a number of bounds for compact representations of plans under various criteria, like efficient sequential or random access of actions. In addition to this, we show that our results have consequences for what can be gained from reformulating planning into some other problem. As a contrast to this we also prove a number of positive results, demonstrating restricted cases where plans do have useful compact representations, as well as proving that macro plans have favourable access properties. Our results are finally discussed in relation to other relevant contexts.

Probabilistic graphs model uncertainty by means of probabilistic edges whose value quantifies the likelihood of the edge existence or the strength of the link it represents. One of the main issues in probabilistic graphs is how to compute the connectivity of the network. The network reliability problem [4] is a generalization of the pairwise reachability, in which the goal is to determine the probability that all pairs of nodes are reachable from one another. Unlike a deterministic graph in which the reachability function is a binary value function indicating whether or not there is a path connecting two nodes, in the case of probabilistic graphs the function assumes probabilistic values. The concept of reachability in probabilistic graphs is used, along with its specialization, as a tool to compute how two nodes in the graph are likely to be connected. Reachability plays an important role in wide range of applications, such as in peer-to-peer networks [3, 18], for probabilistic-routing problem [2, 10], in road network [11], and in trust analysis in social networks [22].As adopted in these works, reachability is quite similar to the general concept of link prediction [9], whose task may be formalized as follows. Given a networked structure (V,E) made up of a set of data instances V and set of observed links E among some nodes in V, the task corresponds to predict how likely should exist an unobserved link between two nodes in the network. The extension to probabilistic graphs adds an important ingredient that should be adequately exploited.

We present a new algorithm for exactly solving decision making problems represented as influence diagrams. We do not require the usual assumptions of no forgetting and regularity; this allows us to solve problems with simultaneous decisions and limited information. The algorithm is empirically shown to outperform a state-of-the-art algorithm on randomly generated problems of up to 150 variables and 10^64 solutions. We show that these problems are NP-hard even if the underlying graph structure of the problem has low treewidth and the variables take on a bounded number of states, and that they admit no provably good approximation if variables can take on an arbitrary number of states.

We develop the Latent Multi-group Membership Graph (LMMG) model, a model of networks with rich node feature structure. In the LMMG model, each node belongs to multiple groups and each latent group models the occurrence of links as well as the node feature structure. The LMMG can be used to summarize the network structure, to predict links between the nodes, and to predict missing features of a node. We derive efficient inference and learning algorithms and evaluate the predictive performance of the LMMG on several social and document network datasets.

Large Markov decision processes (MDPs) are common in reinforcement learning and operations research and are often solved by approximate dynamic programming (ADP). Many ADP algorithms have been developed and studied, often with impressive empirical performance. However, because many ADP methods must be carefully tuned to work well and offer insufficient theoretical guarantees, it is important to develop new methods that have both good theoretical guarantees and empirical performance. Approximate linear programming (ALP)--an ADP method--has been developed with the goal of achieving convergence and good theoretical guarantees (de Farias & van Roy, 2003). Approximate bilinear programming (ABP) improves on the theoretical properties of ALP at the cost of additional computational complexity (Petrik & Zilberstein, 2009, 2011).