193 research outputs found
The completion of optimal -packings
A 3- packing design consists of an -element set and a
collection of -element subsets of , called {\it blocks}, such that every
-element subset of is contained in at most one block. The packing number
of quadruples denotes the number of blocks in a maximum
- packing design, which is also the maximum number of
codewords in a code of length , constant weight , and minimum Hamming
distance 4. In this paper the undecided 21 packing numbers are shown
to be equal to Johnson bound
where ,
is odd,
Completion and deficiency problems
Given a partial Steiner triple system (STS) of order , what is the order
of the smallest complete STS it can be embedded into? The study of this
question goes back more than 40 years. In this paper we answer it for
relatively sparse STSs, showing that given a partial STS of order with at
most triples, it can always be embedded into a complete
STS of order , which is asymptotically optimal. We also obtain
similar results for completions of Latin squares and other designs.
This suggests a new, natural class of questions, called deficiency problems.
Given a global spanning property and a graph , we define the
deficiency of the graph with respect to the property to be
the smallest positive integer such that the join has property
. To illustrate this concept we consider deficiency versions of
some well-studied properties, such as having a -decomposition,
Hamiltonicity, having a triangle-factor and having a perfect matching in
hypergraphs.
The main goal of this paper is to propose a systematic study of these
problems; thus several future research directions are also given
Set-Codes with Small Intersections and Small Discrepancies
We are concerned with the problem of designing large families of subsets over
a common labeled ground set that have small pairwise intersections and the
property that the maximum discrepancy of the label values within each of the
sets is less than or equal to one. Our results, based on transversal designs,
factorizations of packings and Latin rectangles, show that by jointly
constructing the sets and labeling scheme, one can achieve optimal family sizes
for many parameter choices. Probabilistic arguments akin to those used for
pseudorandom generators lead to significantly suboptimal results when compared
to the proposed combinatorial methods. The design problem considered is
motivated by applications in molecular data storage and theoretical computer
science
Algorithms and complexity analyses for some combinational optimization problems
The main focus of this dissertation is on classical combinatorial optimization problems in two important areas: scheduling and network design.
In the area of scheduling, the main interest is in problems in the master-slave model. In this model, each machine is either a master machine or a slave machine. Each job is associated with a preprocessing task, a slave task and a postprocessing task that must be executed in this order. Each slave task has a dedicated slave machine. All the preprocessing and postprocessing tasks share a single master machine or the same set of master machines. A job may also have an arbitrary release time before which the preprocessing task is not available to be processed. The main objective in this dissertation is to minimize the total completion time or the makespan. Both the complexity and algorithmic issues of these problems are considered. It is shown that the problem of minimizing the total completion time is strongly NP-hard even under severe constraints. Various efficient algorithms are designed to minimize the total completion time under various scenarios.
In the area of network design, the survivable network design problems are studied first. The input for this problem is an undirected graph G = (V, E), a non-negative cost for each edge, and a nonnegative connectivity requirement ruv for every (unordered) pair of vertices &ruv. The goal is to find a minimum-cost subgraph in which each pair of vertices u,v is joined by at least ruv edge (vertex)-disjoint paths. A Polynomial Time Approximation Scheme (PTAS) is designed for the problem when the graph is Euclidean and the connectivity requirement of any point is at most 2. PTASs or Quasi-PTASs are also designed for 2-edge-connectivity problem and biconnectivity problem and their variations in unweighted or weighted planar graphs.
Next, the problem of constructing geometric fault-tolerant spanners with low cost and bounded maximum degree is considered. The first result shows that there is a greedy algorithm which constructs fault-tolerant spanners having asymptotically optimal bounds for both the maximum degree and the total cost at the same time. Then an efficient algorithm is developed which finds fault-tolerant spanners with asymptotically optimal bound for the maximum degree and almost optimal bound for the total cost
Combinatorial structures for anonymous database search
This thesis treats a protocol for anonymous database search (or if one prefer, a protocol for user-private information retrieval), that is based on the use of combinatorial configurations. The protocol is called P2P UPIR. It is proved that the (v,k,1)-balanced incomplete block designs (BIBD) and in particular the finite projective planes are optimal configurations for this protocol. The notion of n-anonymity is applied to the configurations for P2P UPIR protocol and the transversal designs are proved to be n-anonymous configurations for P2P UPIR, with respect to the neighborhood points of the points of the configuration. It is proved that to the configurable tuples one can associate a numerical semigroup. This theorem implies results on existence of combinatorial configurations. The proofs are constructive and can be used as algorithms for finding combinatorial configurations. It is also proved that to the triangle-free configurable tuples one can associate a numerical semigroup. This implies results on existence of triangle-free combinatorial configurations
List colouring hypergraphs and extremal results for acyclic graphs
We study several extremal problems in graphs and hypergraphs. The first one is on list-colouring hypergraphs, which is a generalization of the ordinary colouring of hypergraphs. We discuss two methods for determining the list-chromatic number of hypergraphs. One method uses hypergraph polynomials, which invokes Alon's combinatorial nullstellensatz. This method usually requires computer power to complete the calculations needed for even a modest-sized hypergraph. The other method is elementary, and uses the idea of minimum improper colourings. We apply these methods to various classes of hypergraphs, including the projective planes.
We focus on solving the list-colouring problem for Steiner triple systems (STS). It is not hard using either method to determine that Steiner triple systems of orders 7, 9 and 13 are 3-list-chromatic. For systems of order 15, we show that they are 4-list-colourable, but they are also ``almost'' 3-list-colourable. For all Steiner triple systems, we prove a couple of simple upper bounds on their list-chromatic numbers. Also, unlike ordinary colouring where a 3-chromatic STS exists for each admissible order, we prove using probabilistic methods that for every , every STS of high enough order is not -list-colourable.
The second problem is on embedding nearly-spanning bounded-degree trees in sparse graphs. We determine sufficient conditions based on expansion properties for a sparse graph to embed every nearly-spanning tree of bounded degree. We then apply this to random graphs, addressing a question of Alon, Krivelevich and Sudakov, and determine a probability where the random graph asymptotically almost surely contains every tree of bounded degree. This is nearly optimal in terms of the maximum degree of the trees that we embed.
Finally, we solve a problem that arises from quantum computing, which can be formulated as an extremal question about maximizing the size of a type of acyclic directed graph
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