48 research outputs found
Partially ordered secretaries
The elements of a finite nonempty partially ordered set are exposed at
independent uniform times in to a selector who, at any given time, can
see the structure of the induced partial order on the exposed elements. The
selector's task is to choose online a maximal element. This generalizes the
classical linear order secretary problem, for which it is known that the
selector can succeed with probability and that this is best possible. We
describe a strategy for the general problem that achieves success probability
for an arbitrary partial order.Comment: 5 page
Topics in algorithmic, enumerative and geometric combinatorics
This thesis presents five papers, studying enumerative and
extremal problems on combinatorial structures.
The first paper studies Forman's discrete Morse theory in the case where a group acts on the underlying complex. We generalize the notion of a Morse matching, and obtain a theory that can be used to simplify the description of the G-homotopy type of a simplicial complex. As an application, we determine the S_2xS_{n-2}-homotopy type of the complex of non-connected graphs on n nodes. In the introduction, connections are drawn between the first paper and the evasiveness conjecture for monotone graph properties.
In the second paper, we investigate Hansen polytopes of split graphs. By applying a partitioning
technique, the number of nonempty faces is counted, and in particular we confirm
Kalai's 3^d-conjecture for such polytopes. Furthermore, a characterization of
exactly which Hansen polytopes are also Hanner polytopes is given. We end by
constructing an interesting class of Hansen polytopes having very few faces and
yet not being Hanner.
The third paper studies the problem of packing a pattern as densely as possible into compositions. We are able to find the
packing density for some classes of generalized patterns, including all the three letter patterns.
In the fourth paper, we present combinatorial proofs of the enumeration of derangements with descents in prescribed positions. To this end, we
consider fixed point lambda-coloured permutations, which are easily
enumerated. Several formulae regarding these numbers are given, as
well as a generalisation of Euler's difference tables. We also prove that except in a trivial special case, the event that pi has descents in a set S of positions is positively correlated with the event that pi is a derangement, if pi is chosen uniformly in S_n.
The fifth paper solves a partially ordered generalization of the famous secretary problem. The elements of a finite nonempty partially ordered set are exposed in uniform random order to a selector who, at any given time, can see the relative order of the exposed elements. The selector's task is to choose online a maximal element. We describe a strategy for the general problem that achieves success probability at least 1/e for an arbitrary partial order, thus proving that the linearly ordered set is at least as difficult as any other instance of the problem. To this end, we define a probability measure on the maximal elements of an arbitrary partially ordered set, that may be interesting in its own right
On the Combinatorics of Locally Repairable Codes via Matroid Theory
This paper provides a link between matroid theory and locally repairable
codes (LRCs) that are either linear or more generally almost affine. Using this
link, new results on both LRCs and matroid theory are derived. The parameters
of LRCs are generalized to matroids, and the matroid
analogue of the generalized Singleton bound in [P. Gopalan et al., "On the
locality of codeword symbols," IEEE Trans. Inf. Theory] for linear LRCs is
given for matroids. It is shown that the given bound is not tight for certain
classes of parameters, implying a nonexistence result for the corresponding
locally repairable almost affine codes, that are coined perfect in this paper.
Constructions of classes of matroids with a large span of the parameters
and the corresponding local repair sets are given. Using
these matroid constructions, new LRCs are constructed with prescribed
parameters. The existence results on linear LRCs and the nonexistence results
on almost affine LRCs given in this paper strengthen the nonexistence and
existence results on perfect linear LRCs given in [W. Song et al., "Optimal
locally repairable codes," IEEE J. Sel. Areas Comm.].Comment: 48 pages. Submitted for publication. In this version: The text has
been edited to improve the readability. Parameter d for matroids is now
defined by the use of the rank function instead of the dual matroid. Typos
are corrected. Section III is divided into two parts, and some numberings of
theorems etc. have been change
Enumeration of derangements with descents in prescribed positions
We enumerate derangements with descents in prescribed positions. A generating
function was given by Guo-Niu Han and Guoce Xin in 2007. We give a
combinatorial proof of this result, and derive several explicit formulas. To
this end, we consider fixed point -coloured permutations, which are
easily enumerated. Several formulae regarding these numbers are given, as well
as a generalisation of Euler's difference tables. We also prove that except in
a trivial special case, if a permutation is chosen uniformly among all
permutations on elements, the events that has descents in a set
of positions, and that is a derangement, are positively correlated
Bounds on Binary Locally Repairable Codes Tolerating Multiple Erasures
Recently, locally repairable codes has gained significant interest for their
potential applications in distributed storage systems. However, most
constructions in existence are over fields with size that grows with the number
of servers, which makes the systems computationally expensive and difficult to
maintain. Here, we study linear locally repairable codes over the binary field,
tolerating multiple local erasures. We derive bounds on the minimum distance on
such codes, and give examples of LRCs achieving these bounds. Our main
technical tools come from matroid theory, and as a byproduct of our proofs, we
show that the lattice of cyclic flats of a simple binary matroid is atomic.Comment: 9 pages, 1 figure. Parts of this paper were presented at IZS 2018.
This extended arxiv version includes corrected versions of Theorem 1.4 and
Proposition 6 that appeared in the IZS 2018 proceeding