Bounding Search Space Size via (Hyper)tree Decompositions

This paper develops a measure for bounding the performance of AND/OR search algorithms for solving a variety of queries over graphical models. We show how drawing a connection to the recent notion of hypertree decompositions allows to exploit determinism in the problem specification and produce tighter bounds. We demonstrate on a variety of practical problem instances that we are often able to improve upon existing bounds by several orders of magnitude.Comment: Appears in Proceedings of the Twenty-Fourth Conference on Uncertainty in Artificial Intelligence (UAI2008

Positive-Instance Driven Dynamic Programming for Graph Searching

Research on the similarity of a graph to being a tree - called the treewidth of the graph - has seen an enormous rise within the last decade, but a practically fast algorithm for this task has been discovered only recently by Tamaki (ESA 2017). It is based on dynamic programming and makes use of the fact that the number of positive subinstances is typically substantially smaller than the number of all subinstances. Algorithms producing only such subinstances are called positive-instance driven (PID). We give an alternative and intuitive view on this algorithm from the perspective of the corresponding configuration graphs in certain two-player games. This allows us to develop PID-algorithms for a wide range of important graph parameters such as treewidth, pathwidth, and treedepth. We analyse the worst case behaviour of the approach on some well-known graph classes and perform an experimental evaluation on real world and random graphs.Comment: WADS 201

HYPERGRAPH PARTITIONING FOR EFFICIENT BOOLEAN SATISFIABILITY

This paper presents hypergraph partitioning based constraint decomposition procedures to guide Boolean Satisfiability search. Variable-constraint relationships are modeled on a hypergraph and partitioning based techniques are employed to decompose the constraints. Subsequently, the decomposition is analyzed to solve the CNF-SAT problem efficiently. An important aspect of CNF-SAT search procedures is to derive an ordering of variables to guide constraint resolution. Most conventional SAT solvers [1] [2] [3] employ variable-activity based branching heuristics to resolve the constraints. Recently, tree-decomposition techniques, borrowed from constraint satisfaction problems, have been employed to derive variable orderings to guide SAT diagnosis. Even though these techniques provide good variable orders for some SAT instances, their computational complexity makes them impractical for solving large and hard CNF-SAT problems. To overcome this limitation, this research advocates the use of hypergraph partitioning methods to decompose the constraints. This decomposition suggests a good variable order for SAT search. The contributions of this research are two-fold: 1) to engineer a constraint decomposition technique using hypergraph partitioning; 2) to engineer a constraint resolution method based on this decomposition. Preliminary experiments show that our approach is fast, scalable and can significantly increase the performance (often orders of magnitude) of the SAT engine

AND/OR Multi-Valued Decision Diagrams (AOMDDs) for Graphical Models

Inspired by the recently introduced framework of AND/OR search spaces for graphical models, we propose to augment Multi-Valued Decision Diagrams (MDD) with AND nodes, in order to capture function decomposition structure and to extend these compiled data structures to general weighted graphical models (e.g., probabilistic models). We present the AND/OR Multi-Valued Decision Diagram (AOMDD) which compiles a graphical model into a canonical form that supports polynomial (e.g., solution counting, belief updating) or constant time (e.g. equivalence of graphical models) queries. We provide two algorithms for compiling the AOMDD of a graphical model. The first is search-based, and works by applying reduction rules to the trace of the memory intensive AND/OR search algorithm. The second is inference-based and uses a Bucket Elimination schedule to combine the AOMDDs of the input functions via the the APPLY operator. For both algorithms, the compilation time and the size of the AOMDD are, in the worst case, exponential in the treewidth of the graphical model, rather than pathwidth as is known for ordered binary decision diagrams (OBDDs). We introduce the concept of semantic treewidth, which helps explain why the size of a decision diagram is often much smaller than the worst case bound. We provide an experimental evaluation that demonstrates the potential of AOMDDs

Towards automated restructuring of object oriented systems

The work introduces a method for diagnosing design flaws in object oriented systems, and finding meaningful refactorings to remove such flaws. The method is based on pairing up a structural pattern that is considered pathological (e.g. a code smell or anti-pattern) with a so called design context. The design context describes the design semantics associated to the pathological structure, and the desired strategic closure for that fragment. The process is tool supported and largely automated

Solution Counting for CSP and SAT with Large Tree-Width

Рассмотрена проблема подсчета количества решений задачи совместимости ограничений (Constraint Satisfaction Problem). Для ее решения был адаптирован метод обратного прослеживания с ацикличным представлением графа ограничений (Backtracking with Tree-Decomposition). Предложен точный алгоритм, сложность которого экспоненциально зависит от ширины дерева, и приближенный алгоритм, экспоненциально зависящий от размера максимальной клики.The problem of counting the number of solutions of a CSP is considered. For solving the problem the Backtracking with a Tree-Decomposition method was adapted. The exact algorithm is suggested which has the worst-time complexity exponential in a tree width, as well as iterative algorithm that has computational complexity exponential in a maximum clique size.Розглянуто проблему підрахунку кількості розв’язків задачі сумісності обмежень (Constraint Satisfaction Problem). Для її розв’язку було адаптовано метод зворотного простеження з ациклічним поданням графа обмежень (Backtracking with Tree-Decomposition). Запропоновано точний алгоритм, складність якого експоненційно залежить від ширини дерева, і наближений алгоритм, експоненційно залежний від розміру максимальної кліки

Exploiting Structure in Backtracking Algorithms for Propositional and Probabilistic Reasoning

Boolean propositional satisfiability (SAT) and probabilistic reasoning represent two core problems in AI. Backtracking based algorithms have been applied in both problems. In this thesis, I investigate structure-based techniques for solving real world SAT and Bayesian networks, such as software testing and medical diagnosis instances. When solving a SAT instance using backtracking search, a sequence of decisions must be made as to which variable to branch on or instantiate next. Real world problems are often amenable to a divide-and-conquer strategy where the original instance is decomposed into independent sub-problems. Existing decomposition techniques are based on pre-processing the static structure of the original problem. I propose a dynamic decomposition method based on hypergraph separators. Integrating this dynamic separator decomposition into the variable ordering of a modern SAT solver leads to speedups on large real world SAT problems. Encoding a Bayesian network into a CNF formula and then performing weighted model counting is an effective method for exact probabilistic inference. I present two encodings for improving this approach with noisy-OR and noisy-MAX relations. In our experiments, our new encodings are more space efficient and can speed up the previous best approaches over two orders of magnitude. The ability to solve similar problems incrementally is critical for many probabilistic reasoning problems. My aim is to exploit the similarity of these instances by forwarding structural knowledge learned during the analysis of one instance to the next instance in the sequence. I propose dynamic model counting and extend the dynamic decomposition and caching technique to multiple runs on a series of problems with similar structure. This allows us to perform Bayesian inference incrementally as the evidence, parameter, and structure of the network change. Experimental results show that my approach yields significant improvements over previous model counting approaches on multiple challenging Bayesian network instances

Une approche basée sur la décomposition arborescente pour la résolution d'instances SAT structurées

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