242 research outputs found
Decomposing 1-Sperner hypergraphs
A hypergraph is Sperner if no hyperedge contains another one. A Sperner
hypergraph is equilizable (resp., threshold) if the characteristic vectors of
its hyperedges are the (minimal) binary solutions to a linear equation (resp.,
inequality) with positive coefficients. These combinatorial notions have many
applications and are motivated by the theory of Boolean functions and integer
programming. We introduce in this paper the class of -Sperner hypergraphs,
defined by the property that for every two hyperedges the smallest of their two
set differences is of size one. We characterize this class of Sperner
hypergraphs by a decomposition theorem and derive several consequences from it.
In particular, we obtain bounds on the size of -Sperner hypergraphs and
their transversal hypergraphs, show that the characteristic vectors of the
hyperedges are linearly independent over the reals, and prove that -Sperner
hypergraphs are both threshold and equilizable. The study of -Sperner
hypergraphs is motivated also by their applications in graph theory, which we
present in a companion paper
Achieving New Upper Bounds for the Hypergraph Duality Problem through Logic
The hypergraph duality problem DUAL is defined as follows: given two simple
hypergraphs and , decide whether
consists precisely of all minimal transversals of (in which case
we say that is the dual of ). This problem is
equivalent to deciding whether two given non-redundant monotone DNFs are dual.
It is known that non-DUAL, the complementary problem to DUAL, is in
, where
denotes the complexity class of all problems that after a nondeterministic
guess of bits can be decided (checked) within complexity class
. It was conjectured that non-DUAL is in . In this paper we prove this conjecture and actually
place the non-DUAL problem into the complexity class which is a subclass of . We here refer to the logtime-uniform version of
, which corresponds to , i.e., first order
logic augmented by counting quantifiers. We achieve the latter bound in two
steps. First, based on existing problem decomposition methods, we develop a new
nondeterministic algorithm for non-DUAL that requires to guess
bits. We then proceed by a logical analysis of this algorithm, allowing us to
formulate its deterministic part in . From this result, by
the well known inclusion , it follows
that DUAL belongs also to . Finally, by exploiting
the principles on which the proposed nondeterministic algorithm is based, we
devise a deterministic algorithm that, given two hypergraphs and
, computes in quadratic logspace a transversal of
missing in .Comment: Restructured the presentation in order to be the extended version of
a paper that will shortly appear in SIAM Journal on Computin
A polynomial regularity lemma for semi-algebraic hypergraphs and its applications in geometry and property testing
Fox, Gromov, Lafforgue, Naor, and Pach proved a regularity lemma for
semi-algebraic -uniform hypergraphs of bounded complexity, showing that for
each the vertex set can be equitably partitioned into a bounded
number of parts (in terms of and the complexity) so that all but an
-fraction of the -tuples of parts are homogeneous. We prove that
the number of parts can be taken to be polynomial in . Our improved
regularity lemma can be applied to geometric problems and to the following
general question on property testing: is it possible to decide, with query
complexity polynomial in the reciprocal of the approximation parameter, whether
a hypergraph has a given hereditary property? We give an affirmative answer for
testing typical hereditary properties for semi-algebraic hypergraphs of bounded
complexity
Antichain cutsets of strongly connected posets
Rival and Zaguia showed that the antichain cutsets of a finite Boolean
lattice are exactly the level sets. We show that a similar characterization of
antichain cutsets holds for any strongly connected poset of locally finite
height. As a corollary, we get such a characterization for semimodular
lattices, supersolvable lattices, Bruhat orders, locally shellable lattices,
and many more. We also consider a generalization to strongly connected
hypergraphs having finite edges.Comment: 12 pages; v2 contains minor fixes for publicatio
Sharp Concentration of Hitting Size for Random Set Systems
Consider the random set system of {1,2,...,n}, where each subset in the power
set is chosen independently with probability p. A set H is said to be a hitting
set if it intersects each chosen set. The second moment method is used to
exhibit the sharp concentration of the minimal size of H for a variety of
values of p.Comment: 11 page
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