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Dense Subgraph
In graph theory and computer science, a dense subgraph is a subgraph with many edges per vertex. This is formalized as follows: let be an undirected graph and let be a subgraph of . Then the ''density'' of is defined to be: :d(S) = The densest subgraph problem is that of finding a subgraph of maximum density. The density of the maximally dense subgraph of a graph is sometimes referred to as its subgraph density. In 1984, Andrew V. Goldberg developed a polynomial time algorithm to find the maximum density subgraph using a max flow technique. This has been improved by Gallo, Grigoriadis and Tarjan in 1989 to run in time. A simple LP for finding the optimal solution was given by Charikar in 2000. Subgraph density is asymptotic to the related notion of arboricity and to graph degeneracy. Densest subgraph There are many variations on the densest subgraph problem. One of them is the densest subgraph problem, where the objective is to find the maximum density subgraph on ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dense Subgraph
In graph theory and computer science, a dense subgraph is a subgraph with many edges per vertex. This is formalized as follows: let be an undirected graph and let be a subgraph of . Then the ''density'' of is defined to be: :d(S) = The densest subgraph problem is that of finding a subgraph of maximum density. The density of the maximally dense subgraph of a graph is sometimes referred to as its subgraph density. In 1984, Andrew V. Goldberg developed a polynomial time algorithm to find the maximum density subgraph using a max flow technique. This has been improved by Gallo, Grigoriadis and Tarjan in 1989 to run in time. A simple LP for finding the optimal solution was given by Charikar in 2000. Subgraph density is asymptotic to the related notion of arboricity and to graph degeneracy. Densest subgraph There are many variations on the densest subgraph problem. One of them is the densest subgraph problem, where the objective is to find the maximum density subgraph on ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Time Hypothesis
In computational complexity theory, the exponential time hypothesis is an unproven computational hardness assumption that was formulated by . It states that satisfiability of 3-CNF Boolean formulas cannot be solved more quickly than exponential time in the worst case. The exponential time hypothesis, if true, would imply that P ≠ NP, but it is a stronger statement. It implies that many computational problems are equivalent in complexity, in the sense that if one of them has a subexponential time algorithm then they all do, and that many known algorithms for these problems have optimal or near-optimal time Definition The problem is a version of the Boolean satisfiability problem in which the input to the problem is a Boolean expression in conjunctive normal form (that is, an ''and'' of ''ors'' of variables and their negations) with at most k variables per clause. The goal is to determine whether this expression can be made to be true by some assignment of Boolean values to its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charalampos Tsourakakis
Saint Charalampos ( grc, Ἅγιος Χαράλαμπος) (also variously Charalampas, Charalampus, Charalambos, Haralampus, Haralampos, Haralabos or Haralambos) was an early Christian priest in Magnesia on the Maeander, a city in Asia Minor, in the diocese of the same name. His name means ''glowing with joy'' in Greek. He lived during the reign of Septimius Severus (193–211), when Lucian was Proconsul of Magnesia. According to one source, at the time of his martyrdom in 202, Charalambos was 113 years old. Life and martyrdom Charalambos was Bishop of Magnesia also known as the Assyrian Saint Mar Zayya and spread the Gospel in that region for many years. However, when news of his preaching reached the authorities of the area, the proconsul Lucian and military commander Lucius, the saint was arrested and brought to trial, where he confessed his faith in Christ and refused to offer sacrifice to idols. Despite his advanced age, he was tortured mercilessly. They lacerated his bod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unique Games Conjecture
In computational complexity theory, the unique games conjecture (often referred to as UGC) is a conjecture made by Subhash Khot in 2002. The conjecture postulates that the problem of determining the approximate ''value'' of a certain type of game, known as a ''unique game'', has NP-hard computational complexity. It has broad applications in the theory of hardness of approximation. If the unique games conjecture is true and P ≠ NP, then for many important problems it is not only impossible to get an exact solution in polynomial time (as postulated by the P versus NP problem), but also impossible to get a good polynomial-time approximation. The problems for which such an inapproximability result would hold include constraint satisfaction problems, which crop up in a wide variety of disciplines. The conjecture is unusual in that the academic world seems about evenly divided on whether it is true or not. Formulations The unique games conjecture can be stated in a numbe ... [...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] |
Interval Graph
In graph theory, an interval graph is an undirected graph formed from a set of intervals on the real line, with a vertex for each interval and an edge between vertices whose intervals intersect. It is the intersection graph of the intervals. Interval graphs are chordal graphs and perfect graphs. They can be recognized in linear time, and an optimal graph coloring or maximum clique in these graphs can be found in linear time. The interval graphs include all proper interval graphs, graphs defined in the same way from a set of unit intervals. These graphs have been used to model food webs, and to study scheduling problems in which one must select a subset of tasks to be performed at non-overlapping times. Other applications include assembling contiguous subsequences in DNA mapping, and temporal reasoning. Definition An interval graph is an undirected graph formed from a family of intervals :S_i,\quad i=0,1,2,\dots by creating one vertex for each interval , and connecting two ver ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Applied Mathematics
''Discrete Applied Mathematics'' is a peer-reviewed scientific journal covering algorithmic and applied areas of discrete mathematics. It is published by Elsevier and the editor-in-chief is Endre Boros (Rutgers University). The journal was split off from another Elsevier journal, ''Discrete Mathematics'', in 1979, with that journal's founder Peter Ladislaw Hammer as its founding editor-in-chief. Abstracting and indexing The journal is abstracted and indexing in: According to the ''Journal Citation Reports'', the journal has a 2020 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 ... of 1.139. References External links *{{official website, http://www.journals.elsevier.com/discrete-applied-mathematics/ Combinatorics journals Publications established in 1979 Englis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Split Graph
In graph theory, a branch of mathematics, a split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Split graphs were first studied by , and independently introduced by . A split graph may have more than one partition into a clique and an independent set; for instance, the path is a split graph, the vertices of which can be partitioned in three different ways: #the clique and the independent set #the clique and the independent set #the clique and the independent set Split graphs can be characterized in terms of their forbidden induced subgraphs: a graph is split if and only if no induced subgraph is a cycle on four or five vertices, or a pair of disjoint edges (the complement of a 4-cycle). Relation to other graph families From the definition, split graphs are clearly closed under complementation. Another characterization of split graphs involves complementation: they are chordal graphs the complements of which are also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ..., a tree is an undirected graph in which any two Vertex (graph theory), vertices are connected by ''exactly one'' Path (graph theory), path, or equivalently a Connected graph, connected Cycle (graph theory), acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by ''at most one'' path, or equivalently an acyclic undirected graph, or equivalently a Disjoint union of graphs, disjoint union of trees. A polytreeSee . (or directed tree or oriented treeSee .See . or singly connected networkSee .) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirecte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chordal Graph
In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a ''chord'', which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced cycle in the graph should have exactly three vertices. The chordal graphs may also be characterized as the graphs that have perfect elimination orderings, as the graphs in which each minimal separator is a clique, and as the intersection graphs of subtrees of a tree. They are sometimes also called rigid circuit graphs. or triangulated graphs.. Chordal graphs are a subset of the perfect graphs. They may be recognized in linear time, and several problems that are hard on other classes of graphs such as graph coloring may be solved in polynomial time when the input is chordal. The treewidth of an arbitrary graph may be characterized by the size of the cliques in the chordal graphs that contain it. Perfect elimination and efficient recognit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets U and V may be thought of as a coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denoting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIAM Journal On Computing
The ''SIAM Journal on Computing'' is a scientific journal focusing on the mathematical and formal aspects of computer science. It is published by the Society for Industrial and Applied Mathematics (SIAM). Although its official ISO abbreviation is ''SIAM J. Comput.'', its publisher and contributors frequently use the shorter abbreviation ''SICOMP''. SICOMP typically hosts the special issues of the IEEE Annual Symposium on Foundations of Computer Science (FOCS) and the Annual ACM Symposium on Theory of Computing (STOC), where about 15% of papers published in FOCS and STOC each year are invited to these special issues. For example, Volume 48 contains 11 out of 85 papers published in FOCS 2016. References * External linksSIAM Journal on Computing on |
