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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] |
Hypergraph Matchings
In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of 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 head or tail of each edge as a set of vertices (C\subseteq X or D\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hall-type Theorems For Hypergraphs
In the mathematical field of graph theory, Hall-type theorems for hypergraphs are several generalizations of Hall's marriage theorem from graphs to hypergraphs. Such theorems were proved by Ofra Kessler, Ron Aharoni, Penny Haxell, Roy Meshulam, and others. Preliminaries Hall's marriage theorem provides a condition guaranteeing that a bipartite graph admits a perfect matching, or - more generally - a matching that saturates all vertices of . The condition involves the number of neighbors of subsets of . Generalizing Hall's theorem to hypergraphs requires a generalization of the concepts of bipartiteness, perfect matching, and neighbors. 1. Bipartiteness: The notion of a bipartiteness can be extended to hypergraphs in many ways (see bipartite hypergraph). Here we define a hypergraph as bipartite if it is ''exactly 2- colorable'', i.e., its vertices can be 2-colored such that each hyperedge contains exactly one yellow vertex. In other words, can be partitioned into two sets ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractional Matching
In graph theory, a fractional matching is a generalization of a matching in which, intuitively, each vertex may be broken into fractions that are matched to different neighbor vertices. Definition Given a graph G=(V,E), a fractional matching in G is a function that assigns, to each edge e\in E, a fraction f(e)\in ,1/math>, such that for every vertex v\in V, the sum of fractions of edges adjacent to v is at most one: \forall v\in V: \sum_f(e)\leq 1 A matching in the traditional sense is a special case of a fractional matching, in which the fraction of every edge is either zero or one: f(e)=1 if e is in the matching, and f(e)=0 if it is not. For this reason, in the context of fractional matchings, usual matchings are sometimes called ''integral matchings''. Size The size of an integral matching is the number of edges in the matching, and the matching number \nu(G) of a graph G is the largest size of a matching in G. Analogously, the ''size'' of a fractional matching is the sum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Programming Duality
The dual of a given linear program (LP) is another LP that is derived from the original (the primal) LP in the following schematic way: * Each variable in the primal LP becomes a constraint in the dual LP; * Each constraint in the primal LP becomes a variable in the dual LP; * The objective direction is inversed – maximum in the primal becomes minimum in the dual and vice versa. The weak duality theorem states that the objective value of the dual LP at any feasible solution is always a bound on the objective of the primal LP at any feasible solution (upper or lower bound, depending on whether it is a maximization or minimization problem). In fact, this bounding property holds for the optimal values of the dual and primal LPs. The strong duality theorem states that, moreover, if the primal has an optimal solution then the dual has an optimal solution too, ''and the two optima are equal''. Pages 81–104. These theorems belong to a larger class of duality theorems in optimizati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Cover
In graph theory, a vertex cover (sometimes node cover) of a graph is a set of vertices that includes at least one endpoint of every edge of the graph. In computer science, the problem of finding a minimum vertex cover is a classical optimization problem. It is NP-hard, so it cannot be solved by a polynomial-time algorithm if P ≠ NP. Moreover, it is hard to approximate – it cannot be approximated up to a factor smaller than 2 if the unique games conjecture is true. On the other hand, it has several simple 2-factor approximations. It is a typical example of an NP-hard optimization problem that has an approximation algorithm. Its decision version, the vertex cover problem, was one of Karp's 21 NP-complete problems and is therefore a classical NP-complete problem in computational complexity theory. Furthermore, the vertex cover problem is fixed-parameter tractable and a central problem in parameterized complexity theory. The minimum vertex cover problem can be formulated ... [...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] |
Hitting Set
A strike is a directed, forceful physical attack with either a part of the human body or with a handheld object (such as a melee weapon), intended to cause blunt or penetrating trauma upon an opponent. There are many different varieties of strikes. A strike with the hand closed into a fist is called a '' punch'', a strike with a fingertip is called a ''jab'', a strike with the leg or foot is called a ''kick'', and a strike with the head is called a ''headbutt''. There are also other variations employed in martial arts and combat sports. "Buffet" or "beat" refer to repeatedly and violently striking an opponent; this is also commonly referred to as a combination, or combo, especially in boxing or fighting video games. Usage Strikes are the key focus of several sports and arts, including boxing, savate, karate, Muay Lao, taekwondo and wing chun. Some martial arts also use the fingertips, wrists, forearms, shoulders, back and hips to strike an opponent as well as the more conven ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transversal (combinatorics)
In mathematics, particularly in combinatorics, given a family of sets, here called a collection ''C'', a transversal (also called a cross-section) is a set containing exactly one element from each member of the collection. When the sets of the collection are mutually disjoint, each element of the transversal corresponds to exactly one member of ''C'' (the set it is a member of). If the original sets are not disjoint, there are two possibilities for the definition of a transversal: * One variation is that there is a bijection ''f'' from the transversal to ''C'' such that ''x'' is an element of ''f''(''x'') for each ''x'' in the transversal. In this case, the transversal is also called a system of distinct representatives (SDR). * The other, less commonly used, does not require a one-to-one relation between the elements of the transversal and the sets of ''C''. In this situation, the members of the system of representatives are not necessarily distinct. In computer science, comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Cover In Hypergraphs
In graph theory, a vertex cover in a hypergraph is a set of vertices, such that every hyperedge of the hypergraph contains at least one vertex of that set. It is an extension of the notion of vertex cover in a graph. An equivalent term is a hitting set: given a collection of sets, a set which intersects all sets in the collection in at least one element is called a hitting set. The equivalence can be seen by mapping the sets in the collection onto hyperedges. Another equivalent term, used more in a combinatorial context, is '' transversal''. However, some definitions of transversal require that every hyperedge of the hypergraph contains precisely one vertex from the set. 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 vertex-cover (aka hitting set or transversal) in is set such that, for all hyperedges , it holds that . The vertex-cove ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Cardinality Matching
Maximum cardinality matching is a fundamental problem in graph theory. We are given a graph , and the goal is to find a matching containing as many edges as possible; that is, a maximum cardinality subset of the edges such that each vertex is adjacent to at most one edge of the subset. As each edge will cover exactly two vertices, this problem is equivalent to the task of finding a matching that covers as many vertices as possible. An important special case of the maximum cardinality matching problem is when is a bipartite graph, whose vertices are partitioned between left vertices in and right vertices in , and edges in always connect a left vertex to a right vertex. In this case, the problem can be efficiently solved with simpler algorithms than in the general case. Algorithms for bipartite graphs Flow-based algorithm The simplest way to compute a maximum cardinality matching is to follow the Ford–Fulkerson algorithm. This algorithm solves the more general probl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-dimensional Matching
In the mathematical discipline of graph theory, a 3-dimensional matching is a generalization of bipartite matching (also known as 2-dimensional matching) to 3-partite hypergraphs, which consist of hyperedges each of which contains 3 vertices (instead of edges containing 2 vertices in a usual graph). 3-dimensional matching, often abbreviated as 3DM, is also the name of a well-known computational problem: finding a largest 3-dimensional matching in a given hypergraph. 3DM is one of the first problems that were proved to be NP-hard. Definition Let ''X'', ''Y'', and ''Z'' be finite sets, and let ''T'' be a subset of ''X'' × ''Y'' × ''Z''. That is, ''T'' consists of triples (''x'', ''y'', ''z'') such that ''x'' ∈ ''X'', ''y'' ∈ ''Y'', and ''z'' ∈ ''Z''. Now ''M'' ⊆ ''T'' is a 3-dimensional matching if the following holds: for any two distinct triples (''x''1, ''y''1, ''z''1) ∈ ''M'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Family Of Sets
In set theory and related branches of mathematics, a family (or collection) can mean, depending upon the context, any of the following: set, indexed set, multiset, or class. A collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set family, or a set system. Additionally, a family of sets may be defined as a function from a set I, known as the index set, to F, in which case the sets of the family are indexed by members of I. In some contexts, a family of sets may be allowed to contain repeated copies of any given member, and in other contexts it may form a proper class. A finite family of subsets of a finite set S is also called a '' hypergraph''. The subject of extremal set theory concerns the largest and smallest examples of families of sets satisfying certain restrictions. Examples The set of all subsets of a given set S is called the pow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
