196,370 research outputs found
Some Bounds for the Number of Blocks III
Let be a pair of point set and
a set consists of point subsets of which are called
blocks. Let be the maximal cardinality of the intersections between the
distinct two blocks in . The triple is called the
parameter of . Let be the number of the blocks in .
It is shown that inequality
holds for each satisfying , in the paper: Some Bounds for
the Number of Blocks, Europ. J. Combinatorics 22 (2001), 91--94, by R. Noda. If
achieves the upper bound, is called a design. In
the paper, an upper bound and a lower bound, , for of a design are
given. In the present paper we consider the cases when does not achieve the
upper bound or lower bound given above, and get new more strict bounds for
respectively. We apply this bound to the problem of the perfect -codes in
the Johnson scheme, and improve the bound given by Roos in 1983.Comment: Stylistic corrections are made. References are adde
Asymptotically optimal covering designs
A (v,k,t) covering design, or covering, is a family of k-subsets, called
blocks, chosen from a v-set, such that each t-subset is contained in at least
one of the blocks. The number of blocks is the covering's size}, and the
minimum size of such a covering is denoted by C(v,k,t). It is easy to see that
a covering must contain at least (v choose t)/(k choose t) blocks, and in 1985
R\"odl [European J. Combin. 5 (1985), 69-78] proved a long-standing conjecture
of Erd\H{o}s and Hanani [Publ. Math. Debrecen 10 (1963), 10-13] that for fixed
k and t, coverings of size (v choose t)/(k choose t) (1+o(1)) exist (as v \to
\infty).
An earlier paper by the first three authors [J. Combin. Des. 3 (1995),
269-284] gave new methods for constructing good coverings, and gave tables of
upper bounds on C(v,k,t) for small v, k, and t. The present paper shows that
two of those constructions are asymptotically optimal: For fixed k and t, the
size of the coverings constructed matches R\"odl's bound. The paper also makes
the o(1) error bound explicit, and gives some evidence for a much stronger
bound
A Sidon-type condition on set systems
Consider families of -subsets (or blocks) on a ground set of size .
Recall that if all -subsets occur with the same frequency , one
obtains a -design with index . On the other hand, if all
-subsets occur with different frequencies, such a family has been called (by
Sarvate and others) a -adesign. An elementary observation shows that such
families always exist for . Here, we study the smallest possible
maximum frequency .
The exact value of is noted for and an upper bound (best possible
up to a constant multiple) is obtained for using PBD closure. Weaker, yet
still reasonable asymptotic bounds on for higher follow from a
probabilistic argument. Some connections are made with the famous Sidon problem
of additive number theory.Comment: 6 page
Keyword Search on RDF Graphs - A Query Graph Assembly Approach
Keyword search provides ordinary users an easy-to-use interface for querying
RDF data. Given the input keywords, in this paper, we study how to assemble a
query graph that is to represent user's query intention accurately and
efficiently. Based on the input keywords, we first obtain the elementary query
graph building blocks, such as entity/class vertices and predicate edges. Then,
we formally define the query graph assembly (QGA) problem. Unfortunately, we
prove theoretically that QGA is a NP-complete problem. In order to solve that,
we design some heuristic lower bounds and propose a bipartite graph
matching-based best-first search algorithm. The algorithm's time complexity is
, where is the number of the keywords and is a
tunable parameter, i.e., the maximum number of candidate entity/class vertices
and predicate edges allowed to match each keyword. Although QGA is intractable,
both and are small in practice. Furthermore, the algorithm's time
complexity does not depend on the RDF graph size, which guarantees the good
scalability of our system in large RDF graphs. Experiments on DBpedia and
Freebase confirm the superiority of our system on both effectiveness and
efficiency
Some New Bounds For Cover-Free Families Through Biclique Cover
An cover-free family is a family of subsets of a finite set
such that the intersection of any members of the family contains at least
elements that are not in the union of any other members. The minimum
number of elements for which there exists an with blocks is
denoted by .
In this paper, we show that the value of is equal to the
-biclique covering number of the bipartite graph whose vertices
are all - and -subsets of a -element set, where a -subset is
adjacent to an -subset if their intersection is empty. Next, we introduce
some new bounds for . For instance, we show that for
and
where is a constant satisfies the
well-known bound . Also, we
determine the exact value of for some values of . Finally, we
show that whenever there exists a Hadamard matrix of
order 4d
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