49 research outputs found
Exactly Hittable Interval Graphs
Given a set system , where
is a set of elements and is a set of subsets of
, an exact hitting set is a subset of
such that each subset in contains exactly one element in
. We refer to a set system as exactly hittable if it has an exact
hitting set. In this paper, we study interval graphs which have intersection
models that are exactly hittable. We refer to these interval graphs as exactly
hittable interval graphs (EHIG). We present a forbidden structure
characterization for EHIG. We also show that the class of proper interval
graphs is a strict subclass of EHIG. Finally, we give an algorithm that runs in
polynomial time to recognize graphs belonging to the class of EHIG.Comment: 17 pages. arXiv admin note: text overlap with arXiv:1707.0507
Asymptotic Error Free Partitioning over Noisy Boolean Multiaccess Channels
In this paper, we consider the problem of partitioning active users in a
manner that facilitates multi-access without collision. The setting is of a
noisy, synchronous, Boolean, multi-access channel where active users (out
of a total of users) seek to access. A solution to the partition problem
places each of the users in one of groups (or blocks) such that no two
active nodes are in the same block. We consider a simple, but non-trivial and
illustrative case of active users and study the number of steps used
to solve the partition problem. By random coding and a suboptimal decoding
scheme, we show that for any , where and
are positive constants (independent of ), and can be
arbitrary small, the partition problem can be solved with error probability
, for large . Under the same scheme, we also bound from
the other direction, establishing that, for any ,
the error probability for large ; again and
are constants and can be arbitrarily small. These bounds on the number
of steps are lower than the tight achievable lower-bound in terms of for group testing (in which all active users are identified,
rather than just partitioned). Thus, partitioning may prove to be a more
efficient approach for multi-access than group testing.Comment: This paper was submitted in June 2014 to IEEE Transactions on
Information Theory, and is under review no
Algorithmic Graph Theory
The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions
A parametric approach to hereditary classes
The “minimal class approach" consists of studying downwards-closed properties of hereditary graph classes (such as boundedness of a certain parameter within the class) by identifying the minimal obstructions to those properties. In this thesis, we look at various hereditary classes through this lens. In practice, this often amounts to analysing the structure of those classes by characterising boundedness of certain graph parameters within them. However, there is more to it than this: while adopting the minimal class viewpoint, we encounter a variety of interesting notions and problems { some more loosely related to the approach than others. The thesis compiles the author's work in the ensuing research directions