207 research outputs found

    Achieving New Upper Bounds for the Hypergraph Duality Problem through Logic

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    The hypergraph duality problem DUAL is defined as follows: given two simple hypergraphs G\mathcal{G} and H\mathcal{H}, decide whether H\mathcal{H} consists precisely of all minimal transversals of G\mathcal{G} (in which case we say that G\mathcal{G} is the dual of H\mathcal{H}). This problem is equivalent to deciding whether two given non-redundant monotone DNFs are dual. It is known that non-DUAL, the complementary problem to DUAL, is in GC(log2n,PTIME)\mathrm{GC}(\log^2 n,\mathrm{PTIME}), where GC(f(n),C)\mathrm{GC}(f(n),\mathcal{C}) denotes the complexity class of all problems that after a nondeterministic guess of O(f(n))O(f(n)) bits can be decided (checked) within complexity class C\mathcal{C}. It was conjectured that non-DUAL is in GC(log2n,LOGSPACE)\mathrm{GC}(\log^2 n,\mathrm{LOGSPACE}). In this paper we prove this conjecture and actually place the non-DUAL problem into the complexity class GC(log2n,TC0)\mathrm{GC}(\log^2 n,\mathrm{TC}^0) which is a subclass of GC(log2n,LOGSPACE)\mathrm{GC}(\log^2 n,\mathrm{LOGSPACE}). We here refer to the logtime-uniform version of TC0\mathrm{TC}^0, which corresponds to FO(COUNT)\mathrm{FO(COUNT)}, i.e., first order logic augmented by counting quantifiers. We achieve the latter bound in two steps. First, based on existing problem decomposition methods, we develop a new nondeterministic algorithm for non-DUAL that requires to guess O(log2n)O(\log^2 n) bits. We then proceed by a logical analysis of this algorithm, allowing us to formulate its deterministic part in FO(COUNT)\mathrm{FO(COUNT)}. From this result, by the well known inclusion TC0LOGSPACE\mathrm{TC}^0\subseteq\mathrm{LOGSPACE}, it follows that DUAL belongs also to DSPACE[log2n]\mathrm{DSPACE}[\log^2 n]. Finally, by exploiting the principles on which the proposed nondeterministic algorithm is based, we devise a deterministic algorithm that, given two hypergraphs G\mathcal{G} and H\mathcal{H}, computes in quadratic logspace a transversal of G\mathcal{G} missing in H\mathcal{H}.Comment: Restructured the presentation in order to be the extended version of a paper that will shortly appear in SIAM Journal on Computin

    Application of hypergraphs in decomposition of discrete systems

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

    An Efficient Architecture for Information Retrieval in P2P Context Using Hypergraph

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    Peer-to-peer (P2P) Data-sharing systems now generate a significant portion of Internet traffic. P2P systems have emerged as an accepted way to share enormous volumes of data. Needs for widely distributed information systems supporting virtual organizations have given rise to a new category of P2P systems called schema-based. In such systems each peer is a database management system in itself, ex-posing its own schema. In such a setting, the main objective is the efficient search across peer databases by processing each incoming query without overly consuming bandwidth. The usability of these systems depends on successful techniques to find and retrieve data; however, efficient and effective routing of content-based queries is an emerging problem in P2P networks. This work was attended as an attempt to motivate the use of mining algorithms in the P2P context may improve the significantly the efficiency of such methods. Our proposed method based respectively on combination of clustering with hypergraphs. We use ECCLAT to build approximate clustering and discovering meaningful clusters with slight overlapping. We use an algorithm MTMINER to extract all minimal transversals of a hypergraph (clusters) for query routing. The set of clusters improves the robustness in queries routing mechanism and scalability in P2P Network. We compare the performance of our method with the baseline one considering the queries routing problem. Our experimental results prove that our proposed methods generate impressive levels of performance and scalability with with respect to important criteria such as response time, precision and recall.Comment: 2o pages, 8 figure

    Efficiently Enumerating Hitting Sets of Hypergraphs Arising in Data Profiling

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    We devise an enumeration method for inclusion-wise minimal hitting sets in hypergraphs. It has delay O(mk* +1 · n2) and uses linear space. Hereby, n is the number of vertices, m the number of hyperedges, and k* the rank of the transversal hypergraph. In particular, on classes of hypergraphs for which the cardinality k* of the largest minimal hitting set is bounded, the delay is polynomial. The algorithm solves the extension problem for minimal hitting sets as a subroutine. We show that the extension problem is W[3]-complete when parameterised by the cardinality of the set which is to be extended. For the subroutine, we give an algorithm that is optimal under the exponential time hypothesis. Despite these lower bounds, we provide empirical evidence showing that the enumeration outperforms the theoretical worst-case guarantee on hypergraphs arising in the profiling of relational databases, namely, in the detection of unique column combinations

    GPU accelerated maximum cardinality matching algorithms for bipartite graphs

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    We design, implement, and evaluate GPU-based algorithms for the maximum cardinality matching problem in bipartite graphs. Such algorithms have a variety of applications in computer science, scientific computing, bioinformatics, and other areas. To the best of our knowledge, ours is the first study which focuses on GPU implementation of the maximum cardinality matching algorithms. We compare the proposed algorithms with serial and multicore implementations from the literature on a large set of real-life problems where in majority of the cases one of our GPU-accelerated algorithms is demonstrated to be faster than both the sequential and multicore implementations.Comment: 14 pages, 5 figure

    Globally Rigid Augmentation of Rigid Graphs

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