128 research outputs found

    Polynomial Delay Algorithm for Listing Minimal Edge Dominating sets in Graphs

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    The Transversal problem, i.e, the enumeration of all the minimal transversals of a hypergraph in output-polynomial time, i.e, in time polynomial in its size and the cumulated size of all its minimal transversals, is a fifty years old open problem, and up to now there are few examples of hypergraph classes where the problem is solved. A minimal dominating set in a graph is a subset of its vertex set that has a non empty intersection with the closed neighborhood of every vertex. It is proved in [M. M. Kant\'e, V. Limouzy, A. Mary, L. Nourine, On the Enumeration of Minimal Dominating Sets and Related Notions, In Revision 2014] that the enumeration of minimal dominating sets in graphs and the enumeration of minimal transversals in hypergraphs are two equivalent problems. Hoping this equivalence can help to get new insights in the Transversal problem, it is natural to look inside graph classes. It is proved independently and with different techniques in [Golovach et al. - ICALP 2013] and [Kant\'e et al. - ISAAC 2012] that minimal edge dominating sets in graphs (i.e, minimal dominating sets in line graphs) can be enumerated in incremental output-polynomial time. We provide the first polynomial delay and polynomial space algorithm that lists all the minimal edge dominating sets in graphs, answering an open problem of [Golovach et al. - ICALP 2013]. Besides the result, we hope the used techniques that are a mix of a modification of the well-known Berge's algorithm and a strong use of the structure of line graphs, are of great interest and could be used to get new output-polynomial time algorithms.Comment: proofs simplified from previous version, 12 pages, 2 figure

    Computing knock out strategies in metabolic networks

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    Given a metabolic network in terms of its metabolites and reactions, our goal is to efficiently compute the minimal knock out sets of reactions required to block a given behaviour. We describe an algorithm which improves the computation of these knock out sets when the elementary modes (minimal functional subsystems) of the network are given. We also describe an algorithm which computes both the knock out sets and the elementary modes containing the blocked reactions directly from the description of the network and whose worst-case computational complexity is better than the algorithms currently in use for these problems. Computational results are included.Comment: 12 page

    The complexity of acyclic conjunctive queries revisited

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    In this paper, we consider first-order logic over unary functions and study the complexity of the evaluation problem for conjunctive queries described by such kind of formulas. A natural notion of query acyclicity for this language is introduced and we study the complexity of a large number of variants or generalizations of acyclic query problems in that context (Boolean or not Boolean, with or without inequalities, comparisons, etc...). Our main results show that all those problems are \textit{fixed-parameter linear} i.e. they can be evaluated in time f(Q).db.Q(db)f(|Q|).|\textbf{db}|.|Q(\textbf{db})| where Q|Q| is the size of the query QQ, db|\textbf{db}| the database size, Q(db)|Q(\textbf{db})| is the size of the output and ff is some function whose value depends on the specific variant of the query problem (in some cases, ff is the identity function). Our results have two kinds of consequences. First, they can be easily translated in the relational (i.e., classical) setting. Previously known bounds for some query problems are improved and new tractable cases are then exhibited. Among others, as an immediate corollary, we improve a result of \~\cite{PapadimitriouY-99} by showing that any (relational) acyclic conjunctive query with inequalities can be evaluated in time f(Q).db.Q(db)f(|Q|).|\textbf{db}|.|Q(\textbf{db})|. A second consequence of our method is that it provides a very natural descriptive approach to the complexity of well-known algorithmic problems. A number of examples (such as acyclic subgraph problems, multidimensional matching, etc...) are considered for which new insights of their complexity are given.Comment: 30 page

    Application of hypergraphs in decomposition of discrete systems

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    seria: Lecture Notes in Control and Computer Science ; vol. 23

    Proceedings of the 17th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

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    Geometric Inhomogeneous Random Graphs for Algorithm Engineering

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    The design and analysis of graph algorithms is heavily based on the worst case. In practice, however, many algorithms perform much better than the worst case would suggest. Furthermore, various problems can be tackled more efficiently if one assumes the input to be, in a sense, realistic. The field of network science, which studies the structure and emergence of real-world networks, identifies locality and heterogeneity as two frequently occurring properties. A popular model that captures these properties are geometric inhomogeneous random graphs (GIRGs), which is a generalization of hyperbolic random graphs (HRGs). Aside from their importance to network science, GIRGs can be an immensely valuable tool in algorithm engineering. Since they convincingly mimic real-world networks, guarantees about quality and performance of an algorithm on instances of the model can be transferred to real-world applications. They have model parameters to control the amount of heterogeneity and locality, which allows to evaluate those properties in isolation while keeping the rest fixed. Moreover, they can be efficiently generated which allows for experimental analysis. While realistic instances are often rare, generated instances are readily available. Furthermore, the underlying geometry of GIRGs helps to visualize the network, e.g.,~for debugging or to improve understanding of its structure. The aim of this work is to demonstrate the capabilities of geometric inhomogeneous random graphs in algorithm engineering and establish them as routine tools to replace previous models like the Erd\H{o}s-R{\\u27e}nyi model, where each edge exists with equal probability. We utilize geometric inhomogeneous random graphs to design, evaluate, and optimize efficient algorithms for realistic inputs. In detail, we provide the currently fastest sequential generator for GIRGs and HRGs and describe algorithms for maximum flow, directed spanning arborescence, cluster editing, and hitting set. For all four problems, our implementations beat the state-of-the-art on realistic inputs. On top of providing crucial benchmark instances, GIRGs allow us to obtain valuable insights. Most notably, our efficient generator allows us to experimentally show sublinear running time of our flow algorithm, investigate the solution structure of cluster editing, complement our benchmark set of arborescence instances with a density for which there are no real-world networks available, and generate networks with adjustable locality and heterogeneity to reveal the effects of these properties on our algorithms

    Logic-Based Explainability in Machine Learning

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    The last decade witnessed an ever-increasing stream of successes in Machine Learning (ML). These successes offer clear evidence that ML is bound to become pervasive in a wide range of practical uses, including many that directly affect humans. Unfortunately, the operation of the most successful ML models is incomprehensible for human decision makers. As a result, the use of ML models, especially in high-risk and safety-critical settings is not without concern. In recent years, there have been efforts on devising approaches for explaining ML models. Most of these efforts have focused on so-called model-agnostic approaches. However, all model-agnostic and related approaches offer no guarantees of rigor, hence being referred to as non-formal. For example, such non-formal explanations can be consistent with different predictions, which renders them useless in practice. This paper overviews the ongoing research efforts on computing rigorous model-based explanations of ML models; these being referred to as formal explanations. These efforts encompass a variety of topics, that include the actual definitions of explanations, the characterization of the complexity of computing explanations, the currently best logical encodings for reasoning about different ML models, and also how to make explanations interpretable for human decision makers, among others
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