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Szemerédi Regularity Lemma
In extremal graph theory, Szemerédi’s regularity lemma states that a graph can be partitioned into a bounded number of parts so that the edges between parts are regular (in the sense defined below). The lemma shows that certain properties of random graphs can be applied to dense graphs like counting the copies of a given subgraph within graphs. Endre Szemerédi proved the lemma over bipartite graphs for his theorem on arithmetic progressions in 1975 and for general graphs in 1978. Variants of the lemma use different notions of regularity and apply to other mathematical objects like hypergraphs. Statement To state Szemerédi's regularity lemma formally, we must formalize what the edge distribution between parts behaving 'almost randomly' really means. By 'almost random', we're referring to a notion called -regularity. To understand what this means, we first state some definitions. In what follows is a graph with vertex set . Definition 1. Let be disjoint subsets of . The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epsilon Regular Partition
Epsilon (, ; uppercase , lowercase or ; ) is the fifth letter of the Greek alphabet, corresponding phonetically to a mid front unrounded vowel or . In the system of Greek numerals it also has the value five. It was derived from the Phoenician alphabet, Phoenician letter He (letter), He . Letters that arose from epsilon include the Roman E, Ë and Latin epsilon, Ɛ, and Cyrillic Ye (Cyrillic), Е, Ye with grave, È, Yo (Cyrillic), Ё, Ukrainian Ye, Є and E (Cyrillic), Э. The name of the letter was originally (), but it was later changed to ( 'simple e') in the Middle Ages to distinguish the letter from the digraph (orthography), digraph , a former diphthong that had come to be pronounced the same as epsilon. The uppercase form of epsilon is identical to Latin but has its own code point in Unicode: . The lowercase version has two typographical variants, both inherited from history of the Greek alphabet, medieval Greek handwriting. One, the most common in modern typograph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roth's Theorem On Arithmetic Progressions
Roth's theorem on arithmetic progressions is a result in additive combinatorics concerning the existence of arithmetic progressions in subsets of the natural numbers. It was first proven by Klaus Roth in 1953. Roth's theorem is a special case of Szemerédi's theorem for the case k = 3. Statement A subset ''A'' of the natural numbers is said to have positive upper density if :\limsup_\frac > 0. Roth's theorem on arithmetic progressions (infinite version): A subset of the natural numbers with positive upper density contains a arithmetic progression. An alternate, more qualitative, formulation of the theorem is concerned with the maximum size of a Salem–Spencer set which is a subset of = \. Let r_3( be the size of the largest subset of /math> which contains no arithmetic progression. Roth's theorem on arithmetic progressions (finitary version): r_3( = o(N). Improving upper and lower bounds on r_3( is still an open research problem. History The first result in this direc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Computer And System Sciences
The ''Journal of Computer and System Sciences'' (JCSS) is a peer-reviewed scientific journal in the field of computer science. ''JCSS'' is published by Elsevier, and it was started in 1967. Many influential scientific articles have been published in ''JCSS''; these include five papers that have won the Gödel Prize The Gödel Prize is an annual prize for outstanding papers in the area of theoretical computer science, given jointly by the European Association for Theoretical Computer Science (EATCS) and the Association for Computing Machinery Special Inter .... Its managing editor is Michael Segal. Notes References * * External links * Journal homepageScienceDirect accessDBLP information Computer science journals Elsevier academic journals {{compu-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The ACM
The ''Journal of the ACM'' (''JACM'') is a peer-reviewed scientific journal covering computer science in general, especially theoretical aspects. It is an official journal of the Association for Computing Machinery. Its current editor-in-chief is Venkatesan Guruswami. The journal was established in 1954 and "computer scientists universally hold the ''Journal of the ACM'' in high esteem". See also * ''Communications of the ACM ''Communications of the ACM'' (''CACM'') is the monthly journal of the Association for Computing Machinery (ACM). History It was established in 1958, with Saul Rosen as its first managing editor. It is sent to all ACM members. Articles are i ...'' References External links * {{DEFAULTSORT:Journal Of The Acm Academic journals established in 1954 Computer science journals Association for Computing Machinery academic journals Bimonthly journals English-language journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-hard
In computational complexity theory, a computational problem ''H'' is called NP-hard if, for every problem ''L'' which can be solved in non-deterministic polynomial-time, there is a polynomial-time reduction from ''L'' to ''H''. That is, assuming a solution for ''H'' takes 1 unit time, ''H''s solution can be used to solve ''L'' in polynomial time. As a consequence, finding a polynomial time algorithm to solve a single NP-hard problem would give polynomial time algorithms for all the problems in the complexity class NP. As it is suspected, but unproven, that P≠NP, it is unlikely that any polynomial-time algorithms for NP-hard problems exist. A simple example of an NP-hard problem is the subset sum problem. Informally, if ''H'' is NP-hard, then it is at least as difficult to solve as the problems in NP. However, the opposite direction is not true: some problems are undecidable, and therefore even more difficult to solve than all problems in NP, but they are probably not NP- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dense Graph
In mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges (where every pair of vertices is connected by one edge). The opposite, a graph with only a few edges, is a sparse graph. The distinction of what constitutes a dense or sparse graph is ill-defined, and is often represented by 'roughly equal to' statements. Due to this, the way that density is defined often depends on the context of the problem. The graph density of simple graphs is defined to be the ratio of the number of edges with respect to the maximum possible edges. For undirected simple graphs, the graph density is: :D = \frac = \frac For directed, simple graphs, the maximum possible edges is twice that of undirected graphs (as there are two directions to an edge) so the density is: :D = \frac = \frac where is the number of edges and is the number of vertices in the graph. The maximum number of edges for an undirected graph is = \frac2, so the maximal density is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Cut
In a graph, a maximum cut is a cut whose size is at least the size of any other cut. That is, it is a partition of the graph's vertices into two complementary sets and , such that the number of edges between and is as large as possible. Finding such a cut is known as the max-cut problem. The problem can be stated simply as follows. One wants a subset of the vertex set such that the number of edges between and the complementary subset is as large as possible. Equivalently, one wants a bipartite subgraph of the graph with as many edges as possible. There is a more general version of the problem called weighted max-cut, where each edge is associated with a real number, its weight, and the objective is to maximize the total weight of the edges between and its complement rather than the number of the edges. The weighted max-cut problem allowing both positive and negative weights can be trivially transformed into a weighted minimum cut problem by flipping the sign in all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use Conditional (computer programming), conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning). In contrast, a Heuristic (computer science), heuristic is an approach to solving problems without well-defined correct or optimal results.David A. Grossman, Ophir Frieder, ''Information Retrieval: Algorithms and Heuristics'', 2nd edition, 2004, For example, although social media recommender systems are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation. As an e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graphon
In graph theory and statistics, a graphon (also known as a graph limit) is a symmetric measurable function W: ,12\to ,1/math>, that is important in the study of dense graphs. Graphons arise both as a natural notion for the limit of a sequence of dense graphs, and as the fundamental defining objects of exchangeable random graph models. Graphons are tied to dense graphs by the following pair of observations: the random graph models defined by graphons give rise to dense graphs almost surely, and, by the regularity lemma, graphons capture the structure of arbitrary large dense graphs. Statistical formulation A graphon is a symmetric measurable function W: ,12 \to ,1/math>. Usually a graphon is understood as defining an exchangeable random graph model according to the following scheme: # Each vertex j of the graph is assigned an independent random value u_j\sim U ,1/math> # Edge (i,j) is independently included in the graph with probability W(u_i,u_j). A random graph model is an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ca' Foscari University Of Venice
Ca' Foscari University of Venice (), or simply Ca' Foscari, is a public research university and business school in Venice, Italy. Since its foundation in 1868, it has been housed in the Venetian Gothic palace of Ca' Foscari, from which it takes its name. The palace stands on the Grand Canal, between the Rialto and San Marco, in the sestiere of Dorsoduro, while the rest of the University is scattered around the historical centre. In addition to the historical centre of Venice, Ca' Foscari also has campuses in Mestre and Treviso. Ca' Foscari was founded in 1868 after the annexation of the Veneto region in the Kingdom of Italy as the ''Regia Scuola Superiore di Commercio'' (Royal College of Commerce)''.'' As such, it is the second oldest business school in the world, after the Institut Supérieur de Commerce d'Anvers, founded in 1853. Ca' Foscari expanded throughout the 1900s and became a full-fledged university in 1968. It currently has eight departments and almost 21,000 stude ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sparse Graph
In mathematics, a dense graph is a Graph (discrete mathematics), graph in which the number of edges is close to the maximal number of edges (where every pair of Vertex (graph theory), vertices is connected by one edge). The opposite, a graph with only a few edges, is a sparse graph. The distinction of what constitutes a dense or sparse graph is ill-defined, and is often represented by 'roughly equal to' statements. Due to this, the way that density is defined often depends on the context of the problem. The graph density of simple graphs is defined to be the ratio of the number of edges with respect to the maximum possible edges. For undirected simple graphs, the graph density is: :D = \frac = \frac For Directed graph, directed, simple graphs, the maximum possible edges is twice that of undirected graphs (as there are two directions to an edge) so the density is: :D = \frac = \frac where is the number of edges and is the number of vertices in the graph. The maximum number of e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorica
''Combinatorica'' is an international journal of mathematics, publishing papers in the fields of combinatorics and computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, .... It started in 1981, with László Babai and László Lovász as the editors-in-chief with Paul Erdős as honorary editor-in-chief. The current editors-in-chief are Imre Bárány and József Solymosi. The advisory board consists of Ronald Graham, Gyula O. H. Katona, Miklós Simonovits, Vera Sós, and Endre Szemerédi. It is published by the János Bolyai Mathematical Society and Springer Verlag. The following members of the '' Hungarian School of Combinatorics'' have strongly contributed to the journal as authors, or have served as editors: Miklós Ajtai, László Babai, József Beck, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
