2,488 research outputs found
Problems on q-Analogs in Coding Theory
The interest in -analogs of codes and designs has been increased in the
last few years as a consequence of their new application in error-correction
for random network coding. There are many interesting theoretical, algebraic,
and combinatorial coding problems concerning these q-analogs which remained
unsolved. The first goal of this paper is to make a short summary of the large
amount of research which was done in the area mainly in the last few years and
to provide most of the relevant references. The second goal of this paper is to
present one hundred open questions and problems for future research, whose
solution will advance the knowledge in this area. The third goal of this paper
is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author
Join-Reachability Problems in Directed Graphs
For a given collection G of directed graphs we define the join-reachability
graph of G, denoted by J(G), as the directed graph that, for any pair of
vertices a and b, contains a path from a to b if and only if such a path exists
in all graphs of G. Our goal is to compute an efficient representation of J(G).
In particular, we consider two versions of this problem. In the explicit
version we wish to construct the smallest join-reachability graph for G. In the
implicit version we wish to build an efficient data structure (in terms of
space and query time) such that we can report fast the set of vertices that
reach a query vertex in all graphs of G. This problem is related to the
well-studied reachability problem and is motivated by emerging applications of
graph-structured databases and graph algorithms. We consider the construction
of join-reachability structures for two graphs and develop techniques that can
be applied to both the explicit and the implicit problem. First we present
optimal and near-optimal structures for paths and trees. Then, based on these
results, we provide efficient structures for planar graphs and general directed
graphs
A Method for Classification of Doubly Resolvable Designs and Its Application
This article presents the principal results of the Ph.D. thesis Investigation and classification of doubly resolvable designs by Stela Zhelezova (Institute of Mathematics and Informatics, BAS), successfully defended at the Specialized Academic Council for Informatics and Mathematical
Modeling on 22 February 2010.The resolvability of combinatorial designs is intensively investigated because of its applications. This research focuses on resolvable designs
with an additional property - they have resolutions which are mutually orthogonal. Such designs are called doubly resolvable. Their specific properties can be used in statistical and cryptographic applications.Therefore the classification of doubly resolvable designs and their sets of mutually orthogonal resolutions might be very important. We develop a method for classification of doubly resolvable designs. Using this method and extending it with some theoretical restrictions we succeed in obtaining a classification of doubly resolvable designs with small parameters. Also we classify 1-parallelisms and 2-parallelisms of PG(5,2) with automorphisms of order 31 and find the first known transitive 2-parallelisms among them. The content of the paper comprises the essentials of the author’s Ph.D. thesis
Exact algorithms for the order picking problem
Order picking is the problem of collecting a set of products in a warehouse
in a minimum amount of time. It is currently a major bottleneck in supply-chain
because of its cost in time and labor force. This article presents two exact
and effective algorithms for this problem. Firstly, a sparse formulation in
mixed-integer programming is strengthened by preprocessing and valid
inequalities. Secondly, a dynamic programming approach generalizing known
algorithms for two or three cross-aisles is proposed and evaluated
experimentally. Performances of these algorithms are reported and compared with
the Traveling Salesman Problem (TSP) solver Concorde
Orthogonal Resolutions and Latin Squares
Resolutions which are orthogonal to at least one other resolution (RORs) and sets of m mutually orthogonal resolutions (m-MORs) of 2-(v, k, λ) designs are considered. A dependence of the number of nonisomorphic RORs and m-MORs of multiple designs on the number of inequivalent sets of v/k − 1 mutually orthogonal latin squares (MOLS) of size m is obtained. ACM Computing Classification System (1998): G.2.1.∗ This work was partially supported by the Bulgarian National Science Fund under Contract
No I01/0003
On the algorithmic complexity of twelve covering and independence parameters of graphs
The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs
Correlation bounds for fields and matroids
Let be a finite connected graph, and let be a spanning tree of
chosen uniformly at random. The work of Kirchhoff on electrical networks can be
used to show that the events and are negatively
correlated for any distinct edges and . What can be said for such
events when the underlying matroid is not necessarily graphic? We use Hodge
theory for matroids to bound the correlation between the events ,
where is a randomly chosen basis of a matroid. As an application, we prove
Mason's conjecture that the number of -element independent sets of a matroid
forms an ultra-log-concave sequence in .Comment: 16 pages. Supersedes arXiv:1804.0307
On multicolor Ramsey numbers of triple system paths of length 3
Let be a 3-uniform hypergraph. The multicolor Ramsey number is the smallest integer such that every coloring of with colors has a monochromatic copy of . Let
be the loose 3-uniform path with 3 edges and
denote the messy 3-uniform path with 3 edges; that is, let and . In this note we
prove and for
sufficiently large. The former result improves on the bound , which was recently established by {\L}uczak and Polcyn.Comment: 18 pages, 3 figure
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