298 research outputs found
Counting packings of generic subsets in finite groups
A packing of subsets in a group is a
sequence such that are
disjoint subsets of . We give a formula for the number of packings if the
group is finite and if the subsets satisfy
a genericity condition. This formula can be seen as a generalization of the
falling factorials which encode the number of packings in the case where all
the sets are singletons
Using homological duality in consecutive pattern avoidance
Using the approach suggested in [arXiv:1002.2761] we present below a
sufficient condition guaranteeing that two collections of patterns of
permutations have the same exponential generating functions for the number of
permutations avoiding elements of these collections as consecutive patterns. In
short, the coincidence of the latter generating functions is guaranteed by a
length-preserving bijection of patterns in these collections which is identical
on the overlappings of pairs of patterns where the overlappings are considered
as unordered sets. Our proof is based on a direct algorithm for the computation
of the inverse generating functions. As an application we present a large class
of patterns where this algorithm is fast and, in particular, allows to obtain a
linear ordinary differential equation with polynomial coefficients satisfied by
the inverse generating function.Comment: 12 pages, 1 figur
Homomorphisms on infinite direct products of groups, rings and monoids
We study properties of a group, abelian group, ring, or monoid which (a)
guarantee that every homomorphism from an infinite direct product
of objects of the same sort onto factors through the direct product of
finitely many ultraproducts of the (possibly after composition with the
natural map or some variant), and/or (b) guarantee that when a
map does so factor (and the index set has reasonable cardinality), the
ultrafilters involved must be principal.
A number of open questions, and topics for further investigation, are noted.Comment: 26 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be
updated more frequently than arXiv copy. Version 2 has minor revisions in
wording etc. from version
Set-Codes with Small Intersections and Small Discrepancies
We are concerned with the problem of designing large families of subsets over
a common labeled ground set that have small pairwise intersections and the
property that the maximum discrepancy of the label values within each of the
sets is less than or equal to one. Our results, based on transversal designs,
factorizations of packings and Latin rectangles, show that by jointly
constructing the sets and labeling scheme, one can achieve optimal family sizes
for many parameter choices. Probabilistic arguments akin to those used for
pseudorandom generators lead to significantly suboptimal results when compared
to the proposed combinatorial methods. The design problem considered is
motivated by applications in molecular data storage and theoretical computer
science
q-analogs of group divisible designs
A well known class of objects in combinatorial design theory are {group
divisible designs}. Here, we introduce the -analogs of group divisible
designs. It turns out that there are interesting connections to scattered
subspaces, -Steiner systems, design packings and -divisible projective
sets.
We give necessary conditions for the existence of -analogs of group
divsible designs, construct an infinite series of examples, and provide further
existence results with the help of a computer search.
One example is a group divisible design over
which is a design packing consisting of blocks
that such every -dimensional subspace in is covered
at most twice.Comment: 18 pages, 3 tables, typos correcte
Approximating the MaxCover Problem with Bounded Frequencies in FPT Time
We study approximation algorithms for several variants of the MaxCover
problem, with the focus on algorithms that run in FPT time. In the MaxCover
problem we are given a set N of elements, a family S of subsets of N, and an
integer K. The goal is to find up to K sets from S that jointly cover (i.e.,
include) as many elements as possible. This problem is well-known to be NP-hard
and, under standard complexity-theoretic assumptions, the best possible
polynomial-time approximation algorithm has approximation ratio (1 - 1/e). We
first consider a variant of MaxCover with bounded element frequencies, i.e., a
variant where there is a constant p such that each element belongs to at most p
sets in S. For this case we show that there is an FPT approximation scheme
(i.e., for each B there is a B-approximation algorithm running in FPT time) for
the problem of maximizing the number of covered elements, and a randomized FPT
approximation scheme for the problem of minimizing the number of elements left
uncovered (we take K to be the parameter). Then, for the case where there is a
constant p such that each element belongs to at least p sets from S, we show
that the standard greedy approximation algorithm achieves approximation ratio
exactly (1-e^{-max(pK/|S|, 1)}). We conclude by considering an unrestricted
variant of MaxCover, and show approximation algorithms that run in exponential
time and combine an exact algorithm with a greedy approximation. Some of our
results improve currently known results for MaxVertexCover
Small doubling, atomic structure and -divisible set families
Let be a set family such that the intersection
of any two members of has size divisible by . The famous
Eventown theorem states that if then , and this bound can be achieved by, e.g., an `atomic'
construction, i.e. splitting the ground set into disjoint pairs and taking
their arbitrary unions. Similarly, splitting the ground set into disjoint sets
of size gives a family with pairwise intersections divisible by
and size . Yet, as was shown by Frankl and Odlyzko,
these families are far from maximal. For infinitely many , they
constructed families as above of size . On the other hand, if the intersection of {\em any number} of
sets in has size divisible by , then it is
easy to show that . In 1983 Frankl
and Odlyzko conjectured that holds
already if one only requires that for some any distinct members
of have an intersection of size divisible by . We
completely resolve this old conjecture in a strong form, showing that
if is chosen
appropriately, and the error term is not needed if (and only if) , and is sufficiently large. Moreover the only extremal
configurations have `atomic' structure as above. Our main tool, which might be
of independent interest, is a structure theorem for set systems with small
'doubling'
- β¦