13,701 research outputs found
Some preconditioners for systems of linear inequalities
We show that a combination of two simple preprocessing steps would generally
improve the conditioning of a homogeneous system of linear inequalities. Our
approach is based on a comparison among three different but related notions of
conditioning for linear inequalities
Testing Linear Inequalities of Subgraph Statistics
Property testers are fast randomized algorithms whose task is to distinguish between inputs satisfying some predetermined property ? and those that are far from satisfying it. Since these algorithms operate by inspecting a small randomly selected portion of the input, the most natural property one would like to be able to test is whether the input does not contain certain forbidden small substructures. In the setting of graphs, such a result was obtained by Alon et al., who proved that for any finite family of graphs ?, the property of being induced ?-free (i.e. not containing an induced copy of any F ? ?) is testable.
It is natural to ask if one can go one step further and prove that more elaborate properties involving induced subgraphs are also testable. One such generalization of the result of Alon et al. was formulated by Goldreich and Shinkar who conjectured that for any finite family of graphs ?, and any linear inequality involving the densities of the graphs F ? ? in the input graph, the property of satisfying this inequality can be tested in a certain restricted model of graph property testing. Our main result in this paper disproves this conjecture in the following strong form: some properties of this type are not testable even in the classical (i.e. unrestricted) model of graph property testing.
The proof deviates significantly from prior non-testability results in this area. The main idea is to use a linear inequality relating induced subgraph densities in order to encode the property of being a pseudo-random graph
Linear inequalities for flags in graded posets
The closure of the convex cone generated by all flag -vectors of graded
posets is shown to be polyhedral. In particular, we give the facet inequalities
to the polar cone of all nonnegative chain-enumeration functionals on this
class of posets. These are in one-to-one correspondence with antichains of
intervals on the set of ranks and thus are counted by Catalan numbers.
Furthermore, we prove that the convolution operation introduced by Kalai
assigns extreme rays to pairs of extreme rays in most cases. We describe the
strongest possible inequalities for graded posets of rank at most 5
Minimal Committee Problem for Inconsistent Systems of Linear Inequalities on the Plane
A representation of an arbitrary system of strict linear inequalities in R^n
as a system of points is proposed. The representation is obtained by using a
so-called polarity. Based on this representation an algorithm for constructing
a committee solution of an inconsistent plane system of linear inequalities is
given. A solution of two problems on minimal committee of a plane system is
proposed. The obtained solutions to these problems can be found by means of the
proposed algorithm.Comment: 29 pages, 2 figure
SLIQ: Simple Linear Inequalities for Efficient Contig Scaffolding
Scaffolding is an important subproblem in "de novo" genome assembly in which
mate pair data are used to construct a linear sequence of contigs separated by
gaps. Here we present SLIQ, a set of simple linear inequalities derived from
the geometry of contigs on the line that can be used to predict the relative
positions and orientations of contigs from individual mate pair reads and thus
produce a contig digraph. The SLIQ inequalities can also filter out unreliable
mate pairs and can be used as a preprocessing step for any scaffolding
algorithm. We tested the SLIQ inequalities on five real data sets ranging in
complexity from simple bacterial genomes to complex mammalian genomes and
compared the results to the majority voting procedure used by many other
scaffolding algorithms. SLIQ predicted the relative positions and orientations
of the contigs with high accuracy in all cases and gave more accurate position
predictions than majority voting for complex genomes, in particular the human
genome. Finally, we present a simple scaffolding algorithm that produces linear
scaffolds given a contig digraph. We show that our algorithm is very efficient
compared to other scaffolding algorithms while maintaining high accuracy in
predicting both contig positions and orientations for real data sets.Comment: 16 pages, 6 figures, 7 table
Complexities of 3-manifolds from triangulations, Heegaard splittings, and surgery presentations
We study complexities of 3-manifolds defined from triangulations, Heegaard
splittings, and surgery presentations. We show that these complexities are
related by linear inequalities, by presenting explicit geometric constructions.
We also show that our linear inequalities are asymptotically optimal. Our
results are used in [arXiv:1405.1805] to estimate Cheeger-Gromov
-invariants in terms of geometric group theoretic and knot theoretic
data.Comment: 16 pages, 9 figures. An error in the triangulation argument found by
a referee has been fixed. Constants in Theorems A and B have been improved.
Minor remaining typos have been fixed. To appear in the Quarterly Journal of
Mathematic
Five Guidelines for Partition Analysis with Applications to Lecture Hall-type Theorems
Five simple guidelines are proposed to compute the generating function for
the nonnegative integer solutions of a system of linear inequalities. In
contrast to other approaches, the emphasis is on deriving recurrences. We show
how to use the guidelines strategically to solve some nontrivial enumeration
problems in the theory of partitions and compositions. This includes a
strikingly different approach to lecture hall-type theorems, with new
-series identities arising in the process. For completeness, we prove that
the guidelines suffice to find the generating function for any system of
homogeneous linear inequalities with integer coefficients. The guidelines can
be viewed as a simplification of MacMahon's partition analysis with ideas from
matrix techiniques, Elliott reduction, and ``adding a slice''
Basic solutions of systems with two max-linear inequalities
We give an explicit description of the basic solutions of max-linear systems
with two inequalities.Comment: 16 page
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