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GNRS Conjecture
In theoretical computer science and metric geometry, the GNRS conjecture connects the theory of graph minors, the stretch factor of embeddings, and the approximation ratio of multi-commodity flow problems. It is named after Anupam Gupta, Ilan Newman, Yuri Rabinovich, and Alistair Sinclair, who formulated it in 2004. Formulation One formulation of the conjecture involves embeddings of the shortest path distances of weighted undirected graphs into \ell_1 spaces, real vector spaces in which the distance between two vectors is the sum of their coordinate differences. If an embedding maps all pairs of vertices with distance d to pairs of vectors with distance in the range d,Cd/math> then its stretch factor or distortion is the ratio C/c; an isometry has stretch factor one, and all other embeddings have greater stretch factor. The graphs that have an embedding with at most a given distortion are closed under graph minor operations, operations that delete vertices or edges from a graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science
Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumscribe the theoretical areas precisely. The Association for Computing Machinery, ACM's ACM SIGACT, Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following description: History While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved. Information theory was added to the field with a 1948 mathematical theory of communication by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of Hebbian learning, learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection. Plane graphs can be encoded by combinatorial maps or rotation systems. An equivalence class of topologically equivalent drawings on the sphere, usually with additional assumptions such as the absence of isthmuses, is called a pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximation Algorithms
In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned solution to the optimal one. Approximation algorithms naturally arise in the field of theoretical computer science as a consequence of the widely believed P ≠ NP conjecture. Under this conjecture, a wide class of optimization problems cannot be solved exactly in polynomial time. The field of approximation algorithms, therefore, tries to understand how closely it is possible to approximate optimal solutions to such problems in polynomial time. In an overwhelming majority of the cases, the guarantee of such algorithms is a multiplicative one expressed as an approximation ratio or approximation factor i.e., the optimal solution is always guaranteed to be within a (predetermined) multiplicative factor of the returned solution. However, there are a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The ACM
The ''Journal of the ACM'' 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'' is the monthly journal of the Association for Computing Machinery (ACM). It was established in 1958, with Saul Rosen as its first managing editor. It is sent to all ACM members. Articles are intended for readers with ...'' References External links * Publications established in 1954 Computer science journals Association for Computing Machinery academic journals Bimonthly journals English-language journals {{compu-journal-stub ... [...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. 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, András Frank, Péter Frankl, Zoltán Füredi, András Hajnal, Gyula Katona, László Lovász, László Pyber, Alexander Schrijver, Miklós Simonovits, Vera Sós, Endre Szemerédi, Tamás Szőnyi, Éva Tardos, Gábor Tardos.{{cite web, url=https://www.springer.com/ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIAM Journal On Discrete Mathematics
'' SIAM Journal on Discrete Mathematics'' is a peer-reviewed mathematics journal published quarterly by the Society for Industrial and Applied Mathematics (SIAM). The journal includes articles on pure and applied discrete mathematics. It was established in 1988, along with the ''SIAM Journal on Matrix Analysis and Applications'', to replace the ''SIAM Journal on Algebraic and Discrete Methods''. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.57. According to the ''Journal Citation Reports'', the journal has a 2016 impact factor of 0.755. Although its official ISO abbreviation is ''SIAM J. Discrete Math.'', its publisher and contributors frequently use the shorter abbreviation ''SIDMA''. References External links * Combinatorics journals Publications established in 1988 English-language journals Discrete Mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way ana ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Israel Journal Of Mathematics
'' Israel Journal of Mathematics'' is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem (Magnes Press). Founded in 1963, as a continuation of the ''Bulletin of the Research Council of Israel'' (Section F), the journal publishes articles on all areas of mathematics. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.70, and its 2009 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... was 0.754. External links * Mathematics journals Publications established in 1963 English-language journals Bimonthly journals Hebrew University of Jerusalem {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Cube
In graph theory, a partial cube is a graph that is isometric to a subgraph of a hypercube. In other words, a partial cube can be identified with a subgraph of a hypercube in such a way that the distance between any two vertices in the partial cube is the same as the distance between those vertices in the hypercube. Equivalently, a partial cube is a graph whose vertices can be labeled with bit strings of equal length in such a way that the distance between two vertices in the graph is equal to the Hamming distance between their labels. Such a labeling is called a ''Hamming labeling''; it represents an isometric embedding of the partial cube into a hypercube. History was the first to study isometric embeddings of graphs into hypercubes. The graphs that admit such embeddings were characterized by and , and were later named partial cubes. A separate line of research on the same structures, in the terminology of families of sets rather than of hypercube labelings of graphs, was foll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tight Span
In metric geometry, the metric envelope or tight span of a metric space ''M'' is an injective metric space into which ''M'' can be embedded. In some sense it consists of all points "between" the points of ''M'', analogous to the convex hull of a point set in a Euclidean space. The tight span is also sometimes known as the injective envelope or hyperconvex hull of ''M''. It has also been called the injective hull, but should not be confused with the injective hull of a module in algebra, a concept with a similar description relative to the category of ''R''-modules rather than metric spaces. The tight span was first described by , and it was studied and applied by Holsztyński in the 1960s. It was later independently rediscovered by and ; see for this history. The tight span is one of the central constructions of T-theory. Definition The tight span of a metric space can be defined as follows. Let (''X'',''d'') be a metric space, and let ''T''(''X'') be the set of extremal functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johnson–Lindenstrauss Lemma
In mathematics, the Johnson–Lindenstrauss lemma is a result named after William B. Johnson and Joram Lindenstrauss concerning low-distortion embeddings of points from high-dimensional into low-dimensional Euclidean space. The lemma states that a set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are nearly preserved. The map used for the embedding is at least Lipschitz, and can even be taken to be an orthogonal projection. The lemma has applications in compressed sensing, manifold learning, dimensionality reduction, and graph embedding. Much of the data stored and manipulated on computers, including text and images, can be represented as points in a high-dimensional space (see vector space model for the case of text). However, the essential algorithms for working with such data tend to become bogged down very quickly as dimension increases. It is therefore desirable to reduce the dimensi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-outerplanar Graph
In graph theory, a ''k''-outerplanar graph is a planar graph that has a planar embedding in which the vertices belong to at most k concentric layers. The outerplanarity index of a planar graph is the minimum value of k for which it is k-outerplanar. Definition An outerplanar graph (or 1-outerplanar graph) has all of its vertices on the unbounded (outside) face of the graph. A 2-outerplanar graph is a planar graph with the property that, when the vertices on the unbounded face are removed, the remaining vertices all lie on the newly formed unbounded face. And so on. More formally, a graph is k-outerplanar if it has a planar embedding such that, for every vertex, there is an alternating sequence of at most k faces and k vertices of the embedding, starting with the unbounded face and ending with the vertex, in which each consecutive face and vertex are incident to each other. Properties and applications The k-outerplanar graphs have treewidth at most 3k-1. However, some bounded-tree ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique-sum
In graph theory, a branch of mathematics, a clique-sum is a way of combining two graphs by gluing them together at a clique, analogous to the connected sum operation in topology. If two graphs ''G'' and ''H'' each contain cliques of equal size, the clique-sum of ''G'' and ''H'' is formed from their disjoint union by identifying pairs of vertices in these two cliques to form a single shared clique, and then possibly deleting some of the clique edges. A ''k''-clique-sum is a clique-sum in which both cliques have at most ''k'' vertices. One may also form clique-sums and ''k''-clique-sums of more than two graphs, by repeated application of the two-graph clique-sum operation. Different sources disagree on which edges should be removed as part of a clique-sum operation. In some contexts, such as the decomposition of chordal graphs or strangulated graphs, no edges should be removed. In other contexts, such as the SPQR-tree decomposition of graphs into their 3-vertex-connected components, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
