25 research outputs found
Saturating sets in projective planes and hypergraph covers
Let be an arbitrary finite projective plane of order . A subset
of its points is called saturating if any point outside is collinear
with a pair of points from . Applying probabilistic tools we improve the
upper bound on the smallest possible size of the saturating set to
. The same result is
presented using an algorithmic approach as well, which points out the
connection with the transversal number of uniform multiple intersecting
hypergraphs.Comment: 10 pages, detailed calculations are included compared to V
A Unifying Hierarchy of Valuations with Complements and Substitutes
We introduce a new hierarchy over monotone set functions, that we refer to as
(Maximum over Positive Hypergraphs). Levels of the hierarchy
correspond to the degree of complementarity in a given function. The highest
level of the hierarchy, - (where is the total number of
items) captures all monotone functions. The lowest level, -,
captures all monotone submodular functions, and more generally, the class of
functions known as . Every monotone function that has a positive
hypergraph representation of rank (in the sense defined by Abraham,
Babaioff, Dughmi and Roughgarden [EC 2012]) is in -. Every
monotone function that has supermodular degree (in the sense defined by
Feige and Izsak [ITCS 2013]) is in -. In both cases, the
converse direction does not hold, even in an approximate sense. We present
additional results that demonstrate the expressiveness power of
-.
One can obtain good approximation ratios for some natural optimization
problems, provided that functions are required to lie in low levels of the
hierarchy. We present two such applications. One shows that the
maximum welfare problem can be approximated within a ratio of if all
players hold valuation functions in -. The other is an upper
bound of on the price of anarchy of simultaneous first price auctions.
Being in - can be shown to involve two requirements -- one
is monotonicity and the other is a certain requirement that we refer to as
(Positive Lower Envelope). Removing the monotonicity
requirement, one obtains the hierarchy over all non-negative
set functions (whether monotone or not), which can be fertile ground for
further research
Further results on covering codes with radius R and codimension tR + 1
The length function is the smallest possible length of a -ary linear code with codimension (redundancy) and covering
radius . Let be the smallest size of a -saturating set
in the projective space . There is a one-to-one
correspondence between codes and -saturating -sets in
that implies . In this work, for
, new asymptotic upper bounds on are obtained in the
following form:
The new bounds are essentially better than the known ones. For , a new
construction of -saturating sets in the projective space
, providing sets of small sizes, is proposed. The
codes, obtained by the construction, have minimum distance , i.e. they are almost MDS (AMDS) codes. These codes are taken as the
starting ones in the lift-constructions (so-called "-concatenating
constructions") for covering codes to obtain infinite families of codes with
growing codimension , .Comment: 24 pages. arXiv admin note: text overlap with arXiv:2108.1360
Saturating linear sets of minimal rank
Saturating sets are combinatorial objects in projective spaces over finite
fields that have been intensively investigated in the last three decades. They
are related to the so-called covering problem of codes in the Hamming metric.
In this paper, we consider the recently introduced linear version of such sets,
which is, in turn, related to the covering problem in the rank metric. The main
questions in this context are how small the rank of a saturating linear set can
be and how to construct saturating linear sets of small rank. Recently, Bonini,
Borello, and Byrne provided a lower bound on the rank of saturating linear sets
in a given projective space, which is shown to be tight in some cases. In this
paper, we provide construction of saturating linear sets meeting the lower
bound and we develop a link between the saturating property and the
scatteredness of linear sets. The last part of the paper is devoted to show
some parameters for which the bound is not tight.Comment: 26 page
Matroid Stratifications of Hypergraph Varieties, Their Realization Spaces, and Discrete Conditional Independence Models
We study varieties associated to hypergraphs from the point of view of projective geometry and matroid theory. We describe their decompositions into matroid varieties, which may be reducible and can have arbitrary singularities by the Mnëv–Sturmfels universality theorem. We focus on various families of hypergraph varieties for which we explicitly compute an irredundant irreducible decomposition. Our main findings in this direction are three-fold: (1) we describe minimal matroids of such hypergraphs; (2) we prove that the varieties of these matroids are irreducible and their union is the hypergraph variety; and (3) we show that every such matroid is realizable over real numbers. As corollaries, we give conceptual decompositions of various, previously studied, varieties associated with graphs, hypergraphs, and adjacent minors of generic matrices. In particular, our decomposition strategy gives immediate matroid interpretations of the irreducible components of multiple families of varieties associated to conditional independence models in statistical theory and unravels their symmetric structures which hugely simplifies the computations