2,307 research outputs found
Maximal antichains of minimum size
Let be a natural number, and let be a set . We study the problem to find the smallest possible size of a
maximal family of subsets of such that
contains only sets whose size is in , and for all
, i.e. is an antichain. We present a
general construction of such antichains for sets containing 2, but not 1.
If our construction asymptotically yields the smallest possible size
of such a family, up to an error. We conjecture our construction to be
asymptotically optimal also for , and we prove a weaker bound for
the case . Our asymptotic results are straightforward applications of
the graph removal lemma to an equivalent reformulation of the problem in
extremal graph theory which is interesting in its own right.Comment: fixed faulty argument in Section 2, added reference
Minimizing the regularity of maximal regular antichains of 2- and 3-sets
Let be a natural number. We study the problem to find the
smallest such that there is a family of 2-subsets and
3-subsets of with the following properties: (1)
is an antichain, i.e. no member of is a subset of
any other member of , (2) is maximal, i.e. for every
there is an with or , and (3) is -regular, i.e. every point
is contained in exactly members of . We prove lower
bounds on , and we describe constructions for regular maximal antichains
with small regularity.Comment: 7 pages, updated reference
Reconstructing (super)trees from data sets with missing distances: Not all is lost
The wealth of phylogenetic information accumulated over many decades of biological research, coupled with recent technological advances in molecular sequence generation, present significant opportunities for researchers to investigate relationships across and within the kingdoms of life. However, to make best use of this data wealth, several problems must first be overcome. One key problem is finding effective strategies to deal with missing data. Here, we introduce Lasso, a novel heuristic approach for reconstructing rooted phylogenetic trees from distance matrices with missing values, for datasets where a molecular clock may be assumed. Contrary to other phylogenetic methods on partial datasets, Lasso possesses desirable properties such as its reconstructed trees being both unique and edge-weighted. These properties are achieved by Lasso restricting its leaf set to a large subset of all possible taxa, which in many practical situations is the entire taxa set. Furthermore, the Lasso approach is distance-based, rendering it very fast to run and suitable for datasets of all sizes, including large datasets such as those generated by modern Next Generation Sequencing technologies. To better understand the performance of Lasso, we assessed it by means of artificial and real biological datasets, showing its effectiveness in the presence of missing data. Furthermore, by formulating the supermatrix problem as a particular case of the missing data problem, we assessed Lasso's ability to reconstruct supertrees. We demonstrate that, although not specifically designed for such a purpose, Lasso performs better than or comparably with five leading supertree algorithms on a challenging biological data set. Finally, we make freely available a software implementation of Lasso so that researchers may, for the first time, perform both rooted tree and supertree reconstruction with branch lengths on their own partial datasets
Minimum Weight Flat Antichains of Subsets
Building on classical theorems of Sperner and Kruskal-Katona, we investigate
antichains in the Boolean lattice of all subsets of
, where is flat, meaning that it contains
sets of at most two consecutive sizes, say , where contains only -subsets,
while contains only -subsets. Moreover, we assume
consists of the first -subsets in squashed
(colexicographic) order, while consists of all -subsets
not contained in the subsets in . Given reals , we
say the weight of is
. We characterize the minimum
weight antichains for any given , and we do the
same when in addition is a maximal antichain. We can then derive
asymptotic results on both the minimum size and the minimum Lubell function
On the number of minimal completely separating systems and antichains in a Boolean lattice
An (n)completely separating system C ((n)CSS) is a collection of blocks of [n] = {1,..., n} such that for all distinct a, b ∈ [n] there are blocks A, B ∈C with a ∈ A \ B and b ∈ B \ A. An (n)CSS is minimal if it contains the minimum possible number of blocks for a CSS on [n]. The number of non-isomorphic minimal (n)CSSs is determined for 11 ≤ n ≤ 35. This also provides an enumeration of a natural class of antichains
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