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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] |
NP-complete
In computational complexity theory, NP-complete problems are the hardest of the problems to which ''solutions'' can be verified ''quickly''. Somewhat more precisely, a problem is NP-complete when: # It is a decision problem, meaning that for any input to the problem, the output is either "yes" or "no". # When the answer is "yes", this can be demonstrated through the existence of a short (polynomial length) ''solution''. # The correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # The problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. Hence, if we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constant-factor Approximation Algorithms
In computational complexity theory, the class APX (an abbreviation of "approximable") is the set of NP optimization problems that allow polynomial-time approximation algorithms with approximation ratio bounded by a constant (or constant-factor approximation algorithms for short). In simple terms, problems in this class have efficient algorithms that can find an answer within some fixed multiplicative factor of the optimal answer. An approximation algorithm is called an f(n)-approximation algorithm for input size n if it can be proven that the solution that the algorithm finds is at most a multiplicative factor of f(n) times worse than the optimal solution. Here, f(n) is called the ''approximation ratio''. Problems in APX are those with algorithms for which the approximation ratio f(n) is a constant c. The approximation ratio is conventionally stated greater than 1. In the case of minimization problems, f(n) is the found solution's score divided by the optimum solution's score, wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David S
David (; , "beloved one") was a king of ancient Israel and Judah and the Kings of Israel and Judah, third king of the Kingdom of Israel (united monarchy), United Monarchy, according to the Hebrew Bible and Old Testament. The Tel Dan stele, an Canaanite and Aramaic inscriptions, Aramaic-inscribed stone erected by a king of Aram-Damascus in the late 9th/early 8th centuries BCE to commemorate a victory over two enemy kings, contains the phrase (), which is translated as "Davidic line, House of David" by most scholars. The Mesha Stele, erected by King Mesha of Moab in the 9th century BCE, may also refer to the "House of David", although this is disputed. According to Jewish works such as the ''Seder Olam Rabbah'', ''Seder Olam Zutta'', and ''Sefer ha-Qabbalah'' (all written over a thousand years later), David ascended the throne as the king of Judah in 885 BCE. Apart from this, all that is known of David comes from biblical literature, Historicity of the Bible, the historicit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael R
Michael may refer to: People * Michael (given name), a given name * Michael (surname), including a list of people with the surname Michael Given name * Michael (bishop elect), English 13th-century Bishop of Hereford elect * Michael (Khoroshy) (1885–1977), cleric of the Ukrainian Orthodox Church of Canada * Michael Donnellan (fashion designer), Michael Donnellan (1915–1985), Irish-born London fashion designer, often referred to simply as "Michael" * Michael (footballer, born 1982), Brazilian footballer * Michael (footballer, born 1983), Brazilian footballer * Michael (footballer, born 1993), Brazilian footballer * Michael (footballer, born February 1996), Brazilian footballer * Michael (footballer, born March 1996), Brazilian footballer * Michael (footballer, born 1999), Brazilian footballer Rulers Byzantine emperors *Michael I Rangabe (d. 844), married the daughter of Emperor Nikephoros I *Michael II (770–829), called "the Stammerer" and "the Amorian" *Michael III ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gerhard J
Gerhard is a name of Germanic origin and may refer to: Given name * Gerhard (bishop of Passau) (fl. 932–946), German prelate * Gerhard III, Count of Holstein-Rendsburg (1292–1340), German prince, regent of Denmark * Gerhard Barkhorn (1919–1983), German World War II flying ace * Gerhard Berger (born 1959), Austrian racing driver * Gerhard Boldt (1918–1981), German soldier and writer * Gerhard de Beer (born 1994), South African football player * Gerhard Diephuis (1817–1892), Dutch jurist * Gerhard Domagk (1895–1964), German pathologist and bacteriologist and Nobel Laureate * Gerhard Dorn (c.1530–1584), Flemish philosopher, translator, alchemist, physician and bibliophile * Gerhard Ertl (born 1936), German physicist and Nobel Laureate * Gerhard Fieseler (1896–1987), German World War I flying ace * Gerhard Flesch (1909–1948), German Nazi Gestapo and SS officer executed for war crimes * Gerhard Gentzen (1909–1945), German mathematician and logician * Gerhard A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marek Karpinski
Marek Karpinski at the Hausdorff Center for Mathematics, Excellence Cluster is a and known for his research in the theory of s and their applications, , |
Clique Problem
In computer science, the clique problem is the computational problem of finding cliques (subsets of vertices, all adjacent to each other, also called complete subgraphs) in a graph. It has several different formulations depending on which cliques, and what information about the cliques, should be found. Common formulations of the clique problem include finding a maximum clique (a clique with the largest possible number of vertices), finding a maximum weight clique in a weighted graph, listing all maximal cliques (cliques that cannot be enlarged), and solving the decision problem of testing whether a graph contains a clique larger than a given size. The clique problem arises in the following real-world setting. Consider a social network, where the graph's vertices represent people, and the graph's edges represent mutual acquaintance. Then a clique represents a subset of people who all know each other, and algorithms for finding cliques can be used to discover these groups ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singleton Set
In mathematics, a singleton (also known as a unit set or one-point set) is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the axiom of regularity guarantees that no set is an element of itself. This implies that a singleton is necessarily distinct from the element it contains, thus 1 and \ are not the same thing, and the empty set is distinct from the set containing only the empty set. A set such as \ is a singleton as it contains a single element (which itself is a set, but not a singleton). A set is a singleton if and only if its cardinality is . In von Neumann's set-theoretic construction of the natural numbers, the number 1 is ''defined'' as the singleton \. In axiomatic set theory, the existence of singletons is a consequence of the axiom of pairing: for any set ''A'', the axiom applied to ''A'' and ''A'' asserts the existence of \, which is the same as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Cover
In the mathematical field of combinatorics, given a collection \mathcal of subsets of a set X, an exact cover is a subcollection \mathcal^ of \mathcal such that each element in X is contained in ''exactly one'' subset in \mathcal^. One says that each element in X ''is covered by'' exactly one subset in \mathcal^. An exact cover is a kind of cover. In other words, \mathcal^ is a partition of X consisting of subsets contained in \mathcal. The exact cover problem to find an exact cover is a kind of constraint satisfaction problem. The elements of \mathcal represent choices and the elements of X represent constraints. It is non-deterministic polynomial time (NP) complete and has a variety of applications, ranging from the optimization of airline flight schedules, cloud computing, and electronic circuit design. An exact cover problem involves the relation ''contains'' between subsets and elements. But an exact cover problem can be represented by any heterogeneous relation between a ... [...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] |
Matching (graph Theory)
In the mathematical discipline of graph theory, a matching or independent edge set in an undirected Graph (discrete mathematics), graph is a set of Edge (graph theory), edges without common vertex (graph theory), vertices. In other words, a subset of the edges is a matching if each vertex appears in at most one edge of that matching. Finding a matching in a bipartite graph can be treated as a Flow network, network flow problem. Definitions Given a Graph (discrete mathematics), graph a matching ''M'' in ''G'' is a set of pairwise non-adjacent edges, none of which are loop (graph theory), loops; that is, no two edges share common vertices. A vertex is matched (or saturated) if it is an endpoint of one of the edges in the matching. Otherwise the vertex is unmatched (or unsaturated). A maximal matching is a matching ''M'' of a graph ''G'' that is not a subset of any other matching. A matching ''M'' of a graph ''G'' is maximal if every edge in ''G'' has a non-empty intersectio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Claw-free Graph
In graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw (graph theory), claw as an induced subgraph. A claw is another name for the complete bipartite graph K_ (that is, a star graph comprising three edges, three leaves, and a central vertex). A claw-free graph is a graph in which no induced subgraph is a claw; i.e., any subset of four vertices has other than only three edges connecting them in this pattern. Equivalently, a claw-free graph is a graph in which the neighborhood (graph theory), neighborhood of any vertex (graph theory), vertex is the complement (graph theory), complement of a triangle-free graph. Claw-free graphs were initially studied as a generalization of line graphs, and gained additional motivation through three key discoveries about them: the fact that all claw-free connected graphs of even order have perfect matchings, the discovery of polynomial time algorithms for finding maximum independent sets in claw-free graphs, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
