Answer Set Solving with Bounded Treewidth Revisited

Parameterized algorithms are a way to solve hard problems more efficiently, given that a specific parameter of the input is small. In this paper, we apply this idea to the field of answer set programming (ASP). To this end, we propose two kinds of graph representations of programs to exploit their treewidth as a parameter. Treewidth roughly measures to which extent the internal structure of a program resembles a tree. Our main contribution is the design of parameterized dynamic programming algorithms, which run in linear time if the treewidth and weights of the given program are bounded. Compared to previous work, our algorithms handle the full syntax of ASP. Finally, we report on an empirical evaluation that shows good runtime behaviour for benchmark instances of low treewidth, especially for counting answer sets.Comment: This paper extends and updates a paper that has been presented on the workshop TAASP'16 (arXiv:1612.07601). We provide a higher detail level, full proofs and more example

Technical Communications of ICLP

Abstract Dynamic programming (DP) on tree decompositions is a well studied approach for solving hard problems efficiently. State-of-the-art implementations usually rely on tables for storing information, and algorithms specify how the tuples are manipulated during traversal of the decomposition. However, a major bottleneck of such table-based algorithms is relatively high memory consumption. The goal of the doctoral thesis herein discussed is to mitigate performance and memory shortcomings of such algorithms. The idea is to replace tables with an efficient data structure that no longer requires to enumerate intermediate results explicitly during the computation. To this end, Binary Decision Diagrams (BDDs) and related concepts are studied with respect to their applicability in this setting. Besides native support for efficient storage, from a conceptual point of view BDDs give rise to an alternative approach of how DP algorithms are specified. Instead of tuple-based manipulation operations, the algorithms are specified on a logical level, where sets of models can be conjointly updated. The goal of the thesis is to provide a general tool-set for problems that can be solved efficiently via DP on tree decompositions

A Greedy Algorithm for Constructing a Low-Width Generalized Hypertree Decomposition

ABSTRACT We propose a greedy algorithm which, given a hypergraph H and a positive integer k, produces a hypertree decomposition of width less than or equal to 3k − 1, or determines that H does not have a generalized hypertree-width less than k. The running time of this algorithm is O(m k+2 n), where m is the number of hyperedges and n is the number of vertices. If k is a constant, it is polynomial. The concepts of (generalized) hypertree decomposition and (generalized) hypertree-width were introduced by Gottlob et al. Many important NP-complete problems in database theory or artificial intelligence are polynomially solvable for classes of instances associated with hypergraphs of bounded hypertree-width. Gottlob et al. also developed a polynomial time algorithm det-k-decomp which, given a hypergraph H and a constant k, computes a hypertree decomposition of width less than or equal to k if the hypertree-width of H is less than or equal to k. The running time of det-k-decomp is O(m 2k n 2 ) in the worst case, where m and n are the number of hyperedges and the number of vertices, respectively. The proposed algorithm is faster than this. The key step of our algorithm is checking whether a set of hyperedges is an obstacle to a hypergraph having low generalized hypertree-width. We call such a local hypergraph structure a k-hyperconnected set. If a hypergraph contains a k-hyperconnected set with a size of at least 2k, it has hypertreewidth of at least k. Adler et al. propose another obstacle called a k-hyperlinked set. We discuss the difference between the two concepts with examples

Complexity of Secure Sets

A secure set $S$ in a graph is defined as a set of vertices such that for any $X\subseteq S$ the majority of vertices in the neighborhood of $X$ belongs to $S$. It is known that deciding whether a set $S$ is secure in a graph is co-NP-complete. However, it is still open how this result contributes to the actual complexity of deciding whether for a given graph $G$ and integer $k$, a non-empty secure set for $G$ of size at most $k$ exists. In this work, we pinpoint the complexity of this problem by showing that it is $\Sigma^P_2$-complete. Furthermore, the problem has so far not been subject to a parameterized complexity analysis that considers structural parameters. In the present work, we prove that the problem is $W[1]$-hard when parameterized by treewidth. This is surprising since the problem is known to be FPT when parameterized by solution size and "subset problems" that satisfy this property usually tend to be FPT for bounded treewidth as well. Finally, we give an upper bound by showing membership in XP, and we provide a positive result in the form of an FPT algorithm for checking whether a given set is secure on graphs of bounded treewidth.Comment: 28 pages, 9 figures, short version accepted at WG 201

Diagnostic distribué de systèmes respectant la confidentialité

Dans cette thèse, nous nous intéressons à diagnostiquer des systèmes intrinsèquement distribués (comme les systèmes pairs-à-pairs) où chaque pair n'a accès qu'à une sous partie de la description d'un système global. De plus, en raison d'une politique d'accès trop restrictive, il sera pourra qu'aucun pair ne puisse expliquer le comportement du système global. Dans ce contexte, le challenge du diagnostic distribué est le suivant: expliquer le comportement global d'un système distribué par un ensemble de pairs ayant chacun une vision limitée, tout comme l'aurait fait un unique pair diagnostiqueur ayant, lui, une vision globale du système.D'un point de vue théorique, nous montrons que tout nouveau système, logiquement équivalent au système pair-à-pairs initialement observé, garantit que tout diagnostic local d'un pair pourra être prolongé par un diagnostic global (dans ce cas, le nouveau système est dit correct pour le diagnostic distribué).Nous montrons aussi que si ce nouveau système est structuré (c-à-d: il contient un arbre couvrant pour lequel tous les pairs contenant une même variable forme un graphe connecté) alors il garantit que tout diagnostic global pourra être retrouvé à travers un ensemble de diagnostics locaux des pairs (dans ce cas le nouveau système est dit complet pour le diagnostic distribué).Dans un souci de représentation succincte et afin de respecter la politique de confidentialité du vocabulaire de chacun des pairs, nous présentons un nouvel algorithme Token Elimination (TE), qui décompose le système de pairs initial vers un système structuré.Nous montrons expérimentalement que TE produit des décompositions de meilleurs qualité (c-à-d: de plus petites largeurs arborescentes) que les méthodes envisagées dans un contexte distribué. À partir du système structuré construit par TE, nous transformons chaque description locale en une Forme Normale Disjonctive (FND) globalement cohérente.Nous montrons que ce dernier système garantit effectivement un diagnostic distribué correct et complet. En plus, nous exhibons un algorithme capable de vérifier efficacement que tout diagnostic local fait partie d'un diagnostic minimal global, faisant du système structuré de FNDs un système compilé pour le diagnostic distribué.In this thesis, we focus on diagnosing inherently distributed systems such as peer-to-peer, where each peer has access to only a sub-part of the description of an overall system.In addition, due to a too restrictive access control policy, it can be possible that neither peer nor supervisor is able to explain the behaviour of the overall system.The goal of distributed diagnosis is to explain the behaviour of a distributed system by a set of peers (each having a limited local view) as a single diagnosis engine having a global view of the overall system.First, we show that any new system logically equivalent to the initially observed peer-to-peer setting ensures that all diagnosis of a peer may be extended to a global diagnosis (in this case the new system ensures correctness of the distributed diagnosis).Moreover, we prove that if the new system is structured (i.e.it contains a spanning tree for which all peers containing the same variable form a connected graph) then it ensures that any global diagnosis can be found through a set of local diagnoses (in this case the new system ensures the completeness of the distributed diagnoses).For a succinct representation and in order to comply with the privacy policy of the vocabulary of each peer, we present a new algorithm Token Elimination (TE), which decomposes the original peer system to a structured one.We experimentally show that TE produces better quality decompositions (i.e. smaller tree widths) than proposed methods in a distributed context.From the structured system built by TE, we transform each local description into globally consistent DNF.We demonstrate that the latter system is correct and complete for the distributed diagnosis.Finally, we present an algorithm that can effectively check that any local diagnosis is part of a global minimal diagnosis, turning the structured system of DNFs into a compiled system for distributed diagnosis.PARIS11-SCD-Bib. électronique (914719901) / SudocSudocFranceF

Structured Prediction on Dirty Datasets

Many errors cannot be detected or repaired without taking into account the underlying structure and dependencies in the dataset. One way of modeling the structure of the data is graphical models. Graphical models combine probability theory and graph theory in order to address one of the key objectives in designing and fitting probabilistic models, which is to capture dependencies among relevant random variables. Structure representation helps to understand the side effect of the errors or it reveals correct interrelationships between data points. Hence, principled representation of structure in prediction and cleaning tasks of dirty data is essential for the quality of downstream analytical results. Existing structured prediction research considers limited structures and configurations, with little attention to the performance limitations and how well the problem can be solved in more general settings where the structure is complex and rich. In this dissertation, I present the following thesis: By leveraging the underlying dependency and structure in machine learning models, we can effectively detect and clean errors via pragmatic structured predictions techniques. To highlight the main contributions: I investigate prediction algorithms and systems on dirty data with a more realistic structure and dependencies to help deploy this type of learning in more pragmatic settings. Specifically, We introduce a few-shot learning framework for error detection that uses structure-based features of data such as denial constraints violations and Bayesian network as co-occurrence feature. I have studied the problem of recovering the latent ground truth labeling of a structured instance. Then, I consider the problem of mining integrity constraints from data and specifically using the sampling methods for extracting approximate denial constraints. Finally, I have introduced an ML framework that uses solitary and structured data features to solve the problem of record fusion