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Rödl Nibble
In mathematics, a packing in a hypergraph is a Partition of a set, partition of the set of the hypergraph's edges into a number of disjoint subsets such that no pair of edges in each subset share any vertex. There are two famous algorithms to achieve asymptotically optimal packing in ''k''-uniform hypergraphs. One of them is a random greedy algorithm which was proposed by Joel Spencer. He used a branching process to formally prove the optimal achievable bound under some side conditions. The other algorithm is called the Rödl nibble and was proposed by Vojtěch Rödl et al. They showed that the achievable packing by the Rödl nibble is in some sense close to that of the random greedy algorithm. History The problem of finding the number of such subsets in a ''k''-uniform hypergraph was originally motivated through a conjecture by Paul Erdős and Haim Hanani in 1963. Vojtěch Rödl proved their conjecture asymptotically under certain conditions in 1985. Pippenger and Joel Spencer ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypergraph Packing
In mathematics, a hypergraph is a generalization of a Graph (discrete mathematics), graph in which an graph theory, edge can join any number of vertex (graph theory), vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, a directed hypergraph is a pair (X,E), where X is a set of elements called ''nodes'', ''vertices'', ''points'', or ''elements'' and E is a set of pairs of subsets of X. Each of these pairs (D,C)\in E is called an ''edge'' or ''hyperedge''; the vertex subset D is known as its ''tail'' or ''domain'', and C as its ''head'' or ''codomain''. The order of a hypergraph (X,E) is the number of vertices in X. The size of the hypergraph is the number of edges in E. The order of an edge e=(D,C) in a directed hypergraph is , e, = (, D, ,, C, ): that is, the number of vertices in its tail followed by the number of vertices in its head. The definition above generalizes from a directed graph to a directed hypergraph by defining the h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branching Process
In probability theory, a branching process is a type of mathematical object known as a stochastic process, which consists of collections of random variables indexed by some set, usually natural or non-negative real numbers. The original purpose of branching processes was to serve as a mathematical model of a population in which each individual in generation n produces some random number of individuals in generation n+1, according, in the simplest case, to a fixed probability distribution that does not vary from individual to individual. Branching processes are used to model reproduction; for example, the individuals might correspond to bacteria, each of which generates 0, 1, or 2 offspring with some probability in a single time unit. Branching processes can also be used to model other systems with similar dynamics, e.g., the spread of surnames in genealogy or the propagation of neutrons in a nuclear reactor. A central question in the theory of branchi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Combinatorial Theory
The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applications of combinatorics. ''Series B'' is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as ''JCTA'' and ''JCTB''. The journal was founded in 1966 by Frank Harary and Gian-Carlo Rota.They are acknowledged on the journals' title pages and Web sites. SeEditorial board of JCTA Originally there was only one journal, which was split into two parts in 1971 as the field grew rapidly. In 2020, most of the editorial board of ''JCTA'' resigned to form a new, |
Matching In Hypergraphs
In graph theory, a matching in a hypergraph is a set of hyperedges, in which every two hyperedges are disjoint. It is an extension of the notion of matching in a graph. Definition Recall that a hypergraph is a pair , where is a set of vertices and is a set of subsets of called ''hyperedges''. Each hyperedge may contain one or more vertices. A matching in is a subset of , such that every two hyperedges and in have an empty intersection (have no vertex in common). The matching number of a hypergraph is the largest size of a matching in . It is often denoted by . As an example, let be the set Consider a 3-uniform hypergraph on (a hypergraph in which each hyperedge contains exactly 3 vertices). Let be a 3-uniform hypergraph with 4 hyperedges: : Then admits several matchings of size 2, for example: : : However, in any subset of 3 hyperedges, at least two of them intersect, so there is no matching of size 3. Hence, the matching number of is 2. Interse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steiner System
250px, thumbnail, The Fano plane is a Steiner triple system S(2,3,7). The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line. In combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design, specifically a t-design with λ = 1 and ''t'' = 2 or (recently) ''t'' ≥ 2. A Steiner system with parameters ''t'', ''k'', ''n'', written S(''t'',''k'',''n''), is an ''n''-element set ''S'' together with a set of ''k''-element subsets of ''S'' (called blocks) with the property that each ''t''-element subset of ''S'' is contained in exactly one block. In an alternative notation for block designs, an S(''t'',''k'',''n'') would be a ''t''-(''n'',''k'',1) design. This definition is relatively new. The classical definition of Steiner systems also required that ''k'' = ''t'' + 1. An S(2,3,''n'') was (and still is) called a ''Steiner triple'' (or ''triad'') ''system'', while an S(3,4,''n'') is called a ''Steiner ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere Packing
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions (where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions) or to Non-Euclidean geometry, non-Euclidean spaces such as hyperbolic space. A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible. The proportion of space filled by the spheres is called the ''packing density'' of the arrangement. As the local density of a packing in an infinite space can vary depending on the volume over which it is measured, the problem is usually to maximise the average or asymptotic density, measured over a large enough volume. For equal spheres in three dimensions, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Cover Problem
The set cover problem is a classical question in combinatorics, computer science, operations research, and complexity theory. Given a set of elements (henceforth referred to as the universe, specifying all possible elements under consideration) and a collection, referred to as , of a given subsets whose union equals the universe, the set cover problem is to identify a smallest sub-collection of whose union equals the universe. For example, consider the universe, and the collection of sets In this example, is equal to 4, as there are four subsets that comprise this collection. The union of is equal to . However, we can cover all elements with only two sets: , see picture, but not with only one set. Therefore, the solution to the set cover problem for this and has size 2. More formally, given a universe \mathcal and a family \mathcal of subsets of \mathcal, a set cover is a subfamily \mathcal\subseteq\mathcal of sets whose union is \mathcal. * In the set cover deci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramsey's Theorem
In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. To demonstrate the theorem for two colours (say, blue and red), let and be any two positive integers. Ramsey's theorem states that there exists a least positive integer for which every blue-red edge colouring of the complete graph on vertices contains a blue clique on vertices or a red clique on vertices. (Here signifies an integer that depends on both and .) Ramsey's theorem is a foundational result in combinatorics. The first version of this result was proved by Frank Ramsey. This initiated the combinatorial theory now called Ramsey theory, that seeks regularity amid disorder: general conditions for the existence of substructures with regular properties. In this application it is a question of the existence of ''monochromatic subsets'', that is, subsets of connected edges o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Packing
Set packing is a classical NP-complete problem in computational complexity theory and combinatorics, and was one of Karp's 21 NP-complete problems. Suppose one has a finite set ''S'' and a list of subsets of ''S''. Then, the set packing problem asks if some ''k'' subsets in the list are pairwise disjoint (in other words, no two of them share an element). More formally, given a universe \mathcal and a family \mathcal of subsets of \mathcal, a ''packing'' is a subfamily \mathcal\subseteq\mathcal of sets such that all sets in \mathcal are pairwise disjoint. The size of the packing is , \mathcal, . In the set packing decision problem, the input is a pair (\mathcal,\mathcal) and an integer t; the question is whether there is a set packing of size t or more. In the set packing optimization problem, the input is a pair (\mathcal,\mathcal), and the task is to find a set packing that uses the most sets. The problem is clearly in NP since, given t subsets, we can easily verify that they a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covering Number
In mathematics, a covering number is the number of balls of a given size needed to completely cover a given space, with possible overlaps between the balls. The covering number quantifies the size of a set and can be applied to general metric spaces. Two related concepts are the ''packing number'', the number of disjoint balls that fit in a space, and the ''metric entropy'', the number of points that fit in a space when constrained to lie at some fixed minimum distance apart. Definition Let (''M'', ''d'') be a metric space, let ''K'' be a subset of ''M'', and let ''r'' be a positive real number. Let ''B''''r''(''x'') denote the ball of radius ''r'' centered at ''x''. A subset ''C'' of ''M'' is an ''r-external covering'' of ''K'' if: :K \subseteq \bigcup_ B_r(x). In other words, for every y\in K there exists x\in C such that d(x,y)\leq r. If furthermore ''C'' is a subset of ''K'', then it is an ''r-internal covering''. The external covering number of ''K'', denoted N^_r(K), is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Coloring
In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a Graph (discrete mathematics), graph. The assignment is subject to certain constraints, such as that no two adjacent elements have the same color. Graph coloring is a special case of graph labeling. In its simplest form, it is a way of coloring the Vertex (graph theory), vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an ''edge coloring'' assigns a color to each Edge (graph theory), edges so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each Face (graph theory), face (or region) so that no two faces that share a boundary have the same color. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independent Set (graph Theory)
In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set S of vertices such that for every two vertices in S, there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in S. A set is independent if and only if it is a clique in the graph's complement. The size of an independent set is the number of vertices it contains. Independent sets have also been called "internally stable sets", of which "stable set" is a shortening. A maximal independent set is an independent set that is not a proper subset of any other independent set. A maximum independent set is an independent set of largest possible size for a given graph G. This size is called the independence number of ''G'' and is usually denoted by \alpha(G). The optimization problem of finding such a set is called the maximum independent set problem. It is a strongly NP-hard problem. As ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
