10,637 research outputs found
The covert set-cover problem with application to Network Discovery
We address a version of the set-cover problem where we do not know the sets
initially (and hence referred to as covert) but we can query an element to find
out which sets contain this element as well as query a set to know the
elements. We want to find a small set-cover using a minimal number of such
queries. We present a Monte Carlo randomized algorithm that approximates an
optimal set-cover of size within factor with high probability
using queries where is the input size.
We apply this technique to the network discovery problem that involves
certifying all the edges and non-edges of an unknown -vertices graph based
on layered-graph queries from a minimal number of vertices. By reducing it to
the covert set-cover problem we present an -competitive Monte
Carlo randomized algorithm for the covert version of network discovery problem.
The previously best known algorithm has a competitive ratio of and therefore our result achieves an exponential improvement
The approximate Loebl-Komlos-Sos conjecture and embedding trees in sparse graphs
Loebl, Koml\'os and S\'os conjectured that every -vertex graph with at
least vertices of degree at least contains each tree of order
as a subgraph. We give a sketch of a proof of the approximate version of
this conjecture for large values of .
For our proof, we use a structural decomposition which can be seen as an
analogue of Szemer\'edi's regularity lemma for possibly very sparse graphs.
With this tool, each graph can be decomposed into four parts: a set of vertices
of huge degree, regular pairs (in the sense of the regularity lemma), and two
other objects each exhibiting certain expansion properties. We then exploit the
properties of each of the parts of to embed a given tree .
The purpose of this note is to highlight the key steps of our proof. Details
can be found in [arXiv:1211.3050]
Strictly convex drawings of planar graphs
Every three-connected planar graph with n vertices has a drawing on an O(n^2)
x O(n^2) grid in which all faces are strictly convex polygons. These drawings
are obtained by perturbing (not strictly) convex drawings on O(n) x O(n) grids.
More generally, a strictly convex drawing exists on a grid of size O(W) x
O(n^4/W), for any choice of a parameter W in the range n<W<n^2. Tighter bounds
are obtained when the faces have fewer sides.
In the proof, we derive an explicit lower bound on the number of primitive
vectors in a triangle.Comment: 20 pages, 13 figures. to be published in Documenta Mathematica. The
revision includes numerous small additions, corrections, and improvements, in
particular: - a discussion of the constants in the O-notation, after the
statement of thm.1. - a different set-up and clarification of the case
distinction for Lemma
The approximate Loebl-Koml\'os-S\'os Conjecture II: The rough structure of LKS graphs
This is the second of a series of four papers in which we prove the following
relaxation of the Loebl-Komlos--Sos Conjecture: For every there
exists a number such that for every every -vertex graph
with at least vertices of degree at least
contains each tree of order as a subgraph.
In the first paper of the series, we gave a decomposition of the graph
into several parts of different characteristics; this decomposition might be
viewed as an analogue of a regular partition for sparse graphs. In the present
paper, we find a combinatorial structure inside this decomposition. In the last
two papers, we refine the structure and use it for embedding the tree .Comment: 38 pages, 4 figures; new is Section 5.1.1; accepted to SIDM
- …