The paper introduces mixed networks, a new framework for expressing and reasoning with probabilistic and deterministic information. The framework combines belief networks with constraint networks, defining the semantics and graphical representation. We also introduce the AND/OR search space for graphical models, and develop a new linear space search algorithm. This provides the basis for understanding the benefits of processing the constraint information separately, resulting in the pruning of the search space. When the constraint part is tractable or has a small number of solutions, using the mixed representation can be exponentially more effective than using pure belief networks which model constraints as conditional probability tables.

We introduce a probabilistic formalism subsuming Markov random fields of bounded tree width and probabilistic context free grammars. Our models are based on a representation of Boolean formulas that we call case-factor diagrams (CFDs). CFDs are similar to binary decision diagrams (BDDs) but are more concise than BDDs for circuits of bounded tree width and can concisely represent the set of parse trees over a given string under a given context free grammar (unlike BDDs). A probabilistic model consists of a CFD defining a feasible set of Boolean assignments and a weight (or cost) for each individual Boolean variable. We give an inside-outside algorithm for simultaneously computing the marginal of each Boolean variable, and a Viterbi algorithm for finding the minimum cost variable assignment. Both algorithms run in time proportional to the size of the CFD. 1 1

In the efficient social choice problem, the goal is to assign values, subject to side constraints, to a set of variables to maximize the total utility across a population of agents, where each agent has private information about its utility function. In this paper we model the social choice problem as a distributed constraint optimization problem (DCOP), in which each agent can communicate with other agents that share an interest in one or more variables. Whereas existing DCOP algorithms can be easily manipulated by an agent, either by misreporting private information or deviating from the algorithm, we introduce M-DPOP, the first DCOP algorithm that provides a faithful distributed implementation for efficient social choice. This provides a concrete example of how the methods of mechanism design can be unified with those of distributed optimization. Faithfulness ensures that no agent can benefit by unilaterally deviating from any aspect of the protocol, neither informationrevelation, computation, nor communication, and whatever the private information of other agents. We allow for payments by agents to a central bank, which is the only central authority that we require. To achieve faithfulness, we carefully integrate the Vickrey-Clarke-Groves (VCG) mechanism with the DPOP algorithm, such that each agent is only asked to perform computation, report

Inspired by AND/OR search spaces for graphical models recently introduced, we propose to augment Ordered Decision Diagrams with AND nodes, in order to capture function decomposition structure. This yields AND/OR multivalued decision diagram (AOMDD) which compiles a constraint network into a canonical form that supports polynomial time queries such as solution counting, solution enumeration or equivalence of constraint networks. We provide a compilation algorithm based on Variable Elimination for assembling an AOMDD for a constraint network starting from the AOMDDs for its constraints. The algorithm uses the APPLY operator which combines two AOMDDs by a given operation. This guarantees the complexity upper bound for the compilation time and the size of the AOMDD to be exponential in the treewidth of the constraint graph, rather than pathwidth as is known for ordered binary decision diagrams (OBDDs).

We introduce a Probabilistic Grammar-Markov Model (PGMM) which couples probabilistic context free grammars and Markov Random Fields. These PGMMs are generative models defined over attributed features and are used to detect and classify objects in natural images. PGMMs are designed so that they can perform rapid inference, parameter learning, and the more difficult task of structure induction. PGMMs can deal with unknown 2D pose (position, orientation, and scale) in both inference and learning, different appearances, or aspects, of the model. The PGMMs can be learnt in an unsupervised manner where the image can contain one of an unknown number of objects of different categories or even be pure background. We first study the weakly supervised case, where each image contains an example of the (single) object of interest, and then generalize to less supervised cases. The goal of this paper is theoretical but, to provide proof of concept, we demonstrate results from this approach on a subset of the Caltech dataset (learning on a training set and evaluating on a testing set). Our results are generally comparable with the current state of the art, and our inference is performed in less than five seconds.

Optimal planners in the classical setting are built around two notions: branching and pruning. SAT-based planners for example branch by trying the values of a selected variable, and prune by propagating constraints and checking consistency. In the conformant setting, a similar branching scheme can be used if restricted to action variables, but the pruning scheme must be modified. Indeed, pruning branches that encode inconsistent partial plans is not sufficient since a partial plan may be consistent and complete (covering all the action variables) and still fail to be a conformant plan. This happens indeed when the plan does not conform to some possible initial state or transition. A remedy to this problem is to use a criterion stronger than consistency for pruning. This is actually what we do in this paper where the consistency-based

Abstract. This paper describes an approach to computation in a semiringbased system, which includes semiring-based CSPs (in particular weighted CSPs, fuzzy CSPs and standard CSPs) as well as Bayesian networks. The approach to computation is based on what we call semiring-labelled decision diagrams (SLDDs), which are strongly related to binary decision diagrams and finite-state automata. SLDDs can be generated in a similar way to a standard search tree (decision tree) for solving a CSP, but some nodes are merged, creating a more compact representation; for certain classes of CSPs, the number of nodes in the resulting network will be a tiny fraction of the number of nodes in the corresponding search tree. A method is given for generating an SLDD that represents e.g., a particular instance of a semiring-based CSP; it is shown how this can be used to perform various computations of interest, such as computing solutions, determining the possible values of each variable, finding optimal solutions, counting solutions and random generation of solutions of a CSP. 1

We introduce a new generation of depth-first Branch-and-Bound algorithms that explore the AND/OR search tree using static and dynamic variable orderings for solving general constraint optimization problems. The virtue of the AND/OR representation of the search space is that its size may be far smaller than that of a traditional OR representation, which can translate into significant time savings for search algorithms. The focus of this paper is on linear space search which explores the AND/OR search tree rather than the search graph and therefore make no attempt to cache information. We investigate the power of the minibucket heuristics within the AND/OR search space, in both static and dynamic setups. We focus on two most common optimization problems in graphical models: finding the Most Probable Explanation (MPE) in Bayesian networks and solving Weighted CSPs (WCSP). In extensive empirical evaluations we demonstrate that the new AND/OR Branch-and-Bound approach improves considerably over the traditional OR search strategy and show how various variable ordering schemes impact the performance of the AND/OR search scheme.

In this paper we formulate a novel AND/OR graph representation capable of describing the different configurations of deformable articulated objects such as horses. The representation makes use of the summarization principle so that lower level nodes in the graph only pass on summary statistics to the higher level nodes. The probability distributions are invariant to position, orientation, and scale. We develop a novel inference algorithm that combined a bottom-up process for proposing configurations for horses together with a top-down process for refining and validating these proposals. The strategy of surround suppression is applied to ensure that the inference time is polynomial in the size of input data. The algorithm was applied to the tasks of detecting, segmenting and parsing horses. We demonstrate that the algorithm is fast and comparable with the state of the art approaches. 1

Abstract. AND/OR search spaces have recently been introduced as a unifying paradigm for advanced algorithmic schemes for graphical models. The main virtue of this representation is its sensitivity to the structure of the model, which can translate into exponential time savings for search algorithms. In this paper we extend the recently introduced AND/OR Branch-and-Bound algorithm (AOBB) [1] for solving pure 0/1 Integer Linear Programs [2]. Since the variable selection can have a dramatic impact on search performance, we introduce a new dynamic AND/OR Branch-and-Bound algorithm able to accommodate variable ordering heuristics. The effectiveness of the dynamic AND/OR approach is demonstrated on a variety of benchmarks for pure 0/1 integer programming, including instances from the MIPLIB library, real-world combinatorial auctions and random uncapacitated warehouse location problems. 1