52,932 research outputs found
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
Applications of incidence bounds in point covering problems
In the Line Cover problem a set of n points is given and the task is to cover
the points using either the minimum number of lines or at most k lines. In
Curve Cover, a generalization of Line Cover, the task is to cover the points
using curves with d degrees of freedom. Another generalization is the
Hyperplane Cover problem where points in d-dimensional space are to be covered
by hyperplanes. All these problems have kernels of polynomial size, where the
parameter is the minimum number of lines, curves, or hyperplanes needed. First
we give a non-parameterized algorithm for both problems in O*(2^n) (where the
O*(.) notation hides polynomial factors of n) time and polynomial space,
beating a previous exponential-space result. Combining this with incidence
bounds similar to the famous Szemeredi-Trotter bound, we present a Curve Cover
algorithm with running time O*((Ck/log k)^((d-1)k)), where C is some constant.
Our result improves the previous best times O*((k/1.35)^k) for Line Cover
(where d=2), O*(k^(dk)) for general Curve Cover, as well as a few other bounds
for covering points by parabolas or conics. We also present an algorithm for
Hyperplane Cover in R^3 with running time O*((Ck^2/log^(1/5) k)^k), improving
on the previous time of O*((k^2/1.3)^k).Comment: SoCG 201
Covering of Subspaces by Subspaces
Lower and upper bounds on the size of a covering of subspaces in the
Grassmann graph \cG_q(n,r) by subspaces from the Grassmann graph
\cG_q(n,k), , are discussed. The problem is of interest from four
points of view: coding theory, combinatorial designs, -analogs, and
projective geometry. In particular we examine coverings based on lifted maximum
rank distance codes, combined with spreads and a recursive construction. New
constructions are given for with or . We discuss the density
for some of these coverings. Tables for the best known coverings, for and
, are presented. We present some questions concerning
possible constructions of new coverings of smaller size.Comment: arXiv admin note: text overlap with arXiv:0805.352
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