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Register a new userOn The Power of Tree Projections: Structural Tractability of Enumerating CSP Solutions
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Francesco Scarcello
Publication date
2010
fields
Informatics Engineering
and research's language is
English
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The problem of deciding whether CSP instances admit solutions has been deeply studied in the literature, and several structural tractability results have been derived so far. However, constraint satisfaction comes in practice as a computation problem where the focus is either on finding one solution, or on enumerating all solutions, possibly projected to some given set of output variables. The paper investigates the structural tractability of the problem of enumerating (possibly projected) solutions, where tractability means here computable with polynomial delay (WPD), since in general exponentially many solutions may be computed. A general framework based on the notion of tree projection of hypergraphs is considered, which generalizes all known decomposition methods. Tractability results have been obtained both for classes of structures where output variables are part of their specification, and for classes of structures where computability WPD must be ensured for any possible set of output variables. These results are shown to be tight, by exhibiting dichotomies for classes of structures having bounded arity and where the tree decomposition method is considered.
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Evaluating conjunctive queries and solving constraint satisfaction problems are fundamental problems in database theory and artificial intelligence, respectively. These problems are NP-hard, so that several research efforts have been made in the literature for identifying tractable classes, known as islands of tractability, as well as for devising clever heuristics for solving efficiently real-world instances. Many heuristic approaches are based on enforcing on the given instance a property called local consistency, where (in database terms) each tuple in every query atom matches at least one tuple in every other query atom. Interestingly, it turns out that, for many well-known classes of queries, such as for the acyclic queries, enforcing local consistency is even sufficient to solve the given instance correctly. However, the precise power of such a procedure was unclear, but for some very restricted cases. The paper provides full answers to the long-standing questions about the precise power of algorithms based on enforcing local consistency. The classes of instances where enforcing local consistency turns out to be a correct query-answering procedure are however not efficiently recognizable. In fact, the paper finally focuses on certain subclasses defined in terms of the novel notion of greedy tree projections. These latter classes are shown to be efficiently recognizable and strictly larger than most islands of tractability known so far, both in the general case of tree projections and for specific structural decomposition methods.
Tree projections provide a unifying framework to deal with most structural decomposition methods of constraint satisfaction problems (CSPs). Within this framework, a CSP instance is decomposed into a number of sub-problems, called views, whose solutions are either already available or can be computed efficiently. The goal is to arrange portions of these views in a tree-like structure, called tree projection, which determines an efficiently solvable CSP instance equivalent to the original one. Deciding whether a tree projection exists is NP-hard. Solution methods have therefore been proposed in the literature that do not require a tree projection to be given, and that either correctly decide whether the given CSP instance is satisfiable, or return that a tree projection actually does not exist. These approaches had not been generalized so far on CSP extensions for optimization problems, where the goal is to compute a solution of maximum value/minimum cost. The paper fills the gap, by exhibiting a fixed-parameter polynomial-time algorithm that either disproves the existence of tree projections or computes an optimal solution, with the parameter being the size of the expression of the objective function to be optimized over all possible solutions (and not the size of the whole constraint formula, used in related works). Tractability results are also established for the problem of returning the best K solutions. Finally, parallel algorithms for such optimization problems are proposed and analyzed. Given that the classes of acyclic hypergraphs, hypergraphs of bounded treewidth, and hypergraphs of bounded generalized hypertree width are all covered as special cases of the tree projection framework, the results in this paper directly apply to these classes. These classes are extensively considered in the CSP setting, as well as in conjunctive database query evaluation and optimization.
SHAP explanations are a popular feature-attribution mechanism for explainable AI. They use game-theoretic notions to measure the influence of individual features on the prediction of a machine learning model. Despite a lot of recent interest from both academia and industry, it is not known whether SHAP explanations of common machine learning models can be computed efficiently. In this paper, we establish the complexity of computing the SHAP explanation in three important settings. First, we consider fully-factorized data distributions, and show that the complexity of computing the SHAP explanation is the same as the complexity of computing the expected value of the model. This fully-factorized setting is often used to simplify the SHAP computation, yet our results show that the computation can be intractable for commonly used models such as logistic regression. Going beyond fully-factorized distributions, we show that computing SHAP explanations is already intractable for a very simple setting: computing SHAP explanations of trivial classifiers over naive Bayes distributions. Finally, we show that even computing SHAP over the empirical distribution is #P-hard.
Accurate retrieval of the power equipment information plays an important role in guiding the full-lifecycle management of power system assets. Because of data duplication, database decentralization, weak data relations, and sluggish data updates, the power asset management system eager to adopt a new strategy to avoid the information losses, bias, and improve the data storage efficiency and extraction process. Knowledge graph has been widely developed in large part owing to its schema-less nature. It enables the knowledge graph to grow seamlessly and allows new relations addition and entities insertion when needed. This study proposes an approach for constructing power equipment knowledge graph by merging existing multi-source heterogeneous power equipment related data. A graph-search method to illustrate exhaustive results to the desired information based on the constructed knowledge graph is proposed. A case of a 500 kV station example is then demonstrated to show relevant search results and to explain that the knowledge graph can improve the efficiency of power equipment management.
Tree projections provide a mathematical framework that encompasses all the various (purely) structural decomposition methods that have been proposed in the literature to single out classes of nearly-acyclic (hyper)graphs, such as the tree decomposition method, which is the most powerful decomposition method on graphs, and the (generalized) hypertree decomposition method, which is its natural counterpart on arbitrary hypergraphs. The paper analyzes this framework, by focusing in particular on minimal tree projections, that is, on tree projections without useless redundancies. First, it is shown that minimal tree projections enjoy a number of properties that are usually required for normal form decompositions in various structural decomposition methods. In particular, they enjoy the same kind of connection properties as (minimal) tree decompositions of graphs, with the result being tight in the light of the negative answer that is provided to the open question about whether they enjoy a slightly stronger notion of connection property, defined to speed-up the computation of hypertree decompositions. Second, it is shown that tree projections admit a natural game-theoretic characterization in terms of the Captain and Robber game. In this game, as for the Robber and Cops game characterizing tree decompositions, the existence of winning strategies implies the existence of monotone ones. As a special case, the Captain and Robber game can be used to characterize the generalized hypertree decomposition method, where such a game-theoretic characterization was missing and asked for. Besides their theoretical interest, these results have immediate algorithmic applications both for the general setting and for structural decomposition methods that can be recast in terms of tree projections.
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