219 research outputs found
Achieving New Upper Bounds for the Hypergraph Duality Problem through Logic
The hypergraph duality problem DUAL is defined as follows: given two simple
hypergraphs and , decide whether
consists precisely of all minimal transversals of (in which case
we say that is the dual of ). 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
, where
denotes the complexity class of all problems that after a nondeterministic
guess of bits can be decided (checked) within complexity class
. It was conjectured that non-DUAL is in . In this paper we prove this conjecture and actually
place the non-DUAL problem into the complexity class which is a subclass of . We here refer to the logtime-uniform version of
, which corresponds to , 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
bits. We then proceed by a logical analysis of this algorithm, allowing us to
formulate its deterministic part in . From this result, by
the well known inclusion , it follows
that DUAL belongs also to . Finally, by exploiting
the principles on which the proposed nondeterministic algorithm is based, we
devise a deterministic algorithm that, given two hypergraphs and
, computes in quadratic logspace a transversal of
missing in .Comment: Restructured the presentation in order to be the extended version of
a paper that will shortly appear in SIAM Journal on Computin
Incremental Recompilation of Knowledge
Approximating a general formula from above and below by Horn formulas (its
Horn envelope and Horn core, respectively) was proposed by Selman and Kautz
(1991, 1996) as a form of ``knowledge compilation,'' supporting rapid
approximate reasoning; on the negative side, this scheme is static in that it
supports no updates, and has certain complexity drawbacks pointed out by
Kavvadias, Papadimitriou and Sideri (1993). On the other hand, the many
frameworks and schemes proposed in the literature for theory update and
revision are plagued by serious complexity-theoretic impediments, even in the
Horn case, as was pointed out by Eiter and Gottlob (1992), and is further
demonstrated in the present paper. More fundamentally, these schemes are not
inductive, in that they may lose in a single update any positive properties of
the represented sets of formulas (small size, Horn structure, etc.). In this
paper we propose a new scheme, incremental recompilation, which combines Horn
approximation and model-based updates; this scheme is inductive and very
efficient, free of the problems facing its constituents. A set of formulas is
represented by an upper and lower Horn approximation. To update, we replace the
upper Horn formula by the Horn envelope of its minimum-change update, and
similarly the lower one by the Horn core of its update; the key fact which
enables this scheme is that Horn envelopes and cores are easy to compute when
the underlying formula is the result of a minimum-change update of a Horn
formula by a clause. We conjecture that efficient algorithms are possible for
more complex updates.Comment: See http://www.jair.org/ for any accompanying file
Parallel Computation of the Minimal Elements of a Poset
Computing the minimal elements of a partially ordered finite set (poset) is a fundamental problem in combinatorics with numerous applications such as polynomial expression optimization, transversal hypergraph generation and redundant component removal, to name a few. We propose a divide-and-conquer algorithm which is not only cache-oblivious but also can be parallelized free of determinacy races. We have implemented it in Cilk++ targeting multicores. For our test problems of sufficiently large input size our code demonstrates a linear speedup on 32 cores.National Science Foundation (U.S.). (Grant number CNS-0615215)National Science Foundation (U.S.). (Grant number CCF- 0621511
An Efficient Architecture for Information Retrieval in P2P Context Using Hypergraph
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
Controlling Complexity in Spatial Modelling
The present complexity approach is based on two assumptions: A1: measurability of deviations of outcomes with respect to reference values; A2 : extension of A1 to multi-set analysis. Complexity is then defined in terms of multi-set deviation compared to single-set ones; an interpretation is given in terms of information costs; examples show the relevance of the interpretation. As a useful by-product the explicit solution of the quadratic part of the discrete logistic ? one of the examples ? is derived; a set of pij-numbers is introduced, and a workable method for generating them exposed. Extensions are considered, in particular controllability. A further application is then proposed, namely to hypergraph conflict analysis, in particular conflict resolution. Many decisional conflicts at the spatial level can be axiomatised in this form; it is shown how the use of particular structures ? in the mathematical sense of that word ? of the problem allows of reducing greatly the degree of complexity of the problem, and hence the difficulty of finding a solution.Chaos, complexity, conflict, dynamics, hypergraphs, information
Efficiently Enumerating Hitting Sets of Hypergraphs Arising in Data Profiling
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
- …