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Cutwidth
In graph theory, the cutwidth of an undirected graph is the smallest integer k with the following property: there is an ordering of the vertices of the graph, such that every cut obtained by partitioning the vertices into earlier and later subsets of the ordering is crossed by at most k edges. That is, if the vertices are numbered v_1,v_2,\dots v_n, then for every \ell=1,2,\dots n-1, the number of edges v_iv_j with i\le\ell and j>\ell is at most k. The cutwidth of a graph has also been called its folding number. Both the vertex ordering that produces the cutwidth, and the problem of computing this ordering and the cutwidth, have been called minimum cut linear arrangement. Relation to other parameters Cutwidth is related to several other width parameters of graphs. In particular, it is always at least as large as the treewidth or pathwidth of the same graph. However, it is at most the pathwidth multiplied by O(\Delta), or the treewidth multiplied by O(\Delta\log n) where \Delta i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pathwidth
In graph theory, a path decomposition of a graph is, informally, a representation of as a "thickened" path graph, and the pathwidth of is a number that measures how much the path was thickened to form . More formally, a path-decomposition is a sequence of subsets of vertices of such that the endpoints of each edge appear in one of the subsets and such that each vertex appears in a contiguous subsequence of the subsets,. and the pathwidth is one less than the size of the largest set in such a decomposition. Pathwidth is also known as interval thickness (one less than the maximum clique size in an interval supergraph of ), vertex separation number, or node searching number. Pathwidth and path-decompositions are closely analogous to treewidth and tree decompositions. They play a key role in the theory of graph minors: the families of graphs that are closed under graph minors and do not include all forests may be characterized as having bounded pathwidth, and the "vortices ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carving Width
In graph theory, the carving width of a graph is a number, defined from the graph, that describes the number of edges separating the clusters in a hierarchical clustering of the graph vertices. Definition and examples The carving width is defined in terms of hierarchical clusterings of the vertices of a given graph, called "carvings". A carving can be described as an unrooted binary tree whose leaves are labeled with the vertices of the given graph. Removing any edge from this tree partitions the tree into two subtrees, and correspondingly partitions the vertices of the tree into two clusters. The vertex clusters, formed in this way, constitute a laminar set family: any two vertex clusters (not just the two complementary clusters formed by removing the same edge) are either disjoint, or one is contained in the other. The width of a carving, defined in this way, is the maximum number of edges that connect two complementary clusters. The carving width of the graph is the minimum wid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Book Embedding
In graph theory, a book embedding is a generalization of planar graph, planar embedding of a Graph (discrete mathematics), graph to embeddings in a ''book'', a collection of half-planes all having the same Line (geometry), line as their boundary. Usually, the vertices of the graph are required to lie on this boundary line, called the ''spine'', and the edges are required to stay within a single half-plane. The book thickness of a graph is the smallest possible number of half-planes for any book embedding of the graph. Book thickness is also called pagenumber, stacknumber or fixed outerthickness. Book embeddings have also been used to define several other graph invariants including the pagewidth and book crossing number. Every graph with vertices has book thickness at most \lceil n/2\rceil, and this formula gives the exact book thickness for complete graphs. The graphs with book thickness one are the outerplanar graphs. The graphs with book thickness at most two are the subhamilt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (graph Theory)
In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices). In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another. From the point of view of graph theory, vertices are treated as featureless and indivisible objects, although they may have additional structure depending on the application from which the graph arises; for instance, a semantic network is a graph in which the vertices represent concepts or classes of objects. The two vertices forming an edge are said to be the endpoints of this edge, and the edge is said to be incident to the vertices. A vertex ''w'' is said to be ... [...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] |
Satish Rao
Satish B. Rao is an American computer scientist who is a professor of computer science at the University of California, Berkeley. Biography Satish Rao received his PhD from the Massachusetts Institute of Technology in 1989 and joined the faculty at the University of California, Berkeley in 1999. Research and awards Rao's research focuses on computational biology, graph partitioning, and single- and multi-commodity flows (maximum flow problem). Rao is an ACM Fellow (2013) and won the Fulkerson Prize with Sanjeev Arora and Umesh Vazirani in 2012 for their work on improving the approximation ratio for graph separators and related problems from O(\log n) to O(\sqrt). Rao teaches discrete mathematics and probability theory at the University of California, Berkeley. Publications Satish Rao has published over 100 publications and is cited frequently. Selected publications * S. Arora, S. Rao, and U. Vazirani. "Expander flows, geometric embeddings and graph partitioning," ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximation Ratio
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 a ... [...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] |
Polynomial-time Approximation Scheme
In computer science (particularly algorithmics), a polynomial-time approximation scheme (PTAS) is a type of approximation algorithm for optimization problems (most often, NP-hard optimization problems). A PTAS is an algorithm which takes an instance of an optimization problem and a parameter and produces a solution that is within a factor of being optimal (or for maximization problems). For example, for the Euclidean traveling salesman problem, a PTAS would produce a tour with length at most , with being the length of the shortest tour. Sanjeev Arora, Polynomial-time Approximation Schemes for Euclidean TSP and other Geometric Problems, Journal of the ACM 45(5) 753–782, 1998. The running time of a PTAS is required to be polynomial in the problem size for every fixed ε, but can be different for different ε. Thus an algorithm running in time or even counts as a PTAS. Variants Deterministic A practical problem with PTAS algorithms is that the exponent of the polynomial co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Programming
An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are linear. Integer programming is NP-complete. In particular, the special case of 0–1 integer linear programming, in which unknowns are binary, and only the restrictions must be satisfied, is one of Karp's 21 NP-complete problems. If some decision variables are not discrete, the problem is known as a mixed-integer programming problem. Canonical and standard form for ILPs In integer linear programming, the ''canonical form'' is distinct from the ''standard form''. An integer linear program in canonical form is expressed thus (note that it is the \mathbf vector which is to be decided): : \begin & \underset && \mathbf^\mathrm \mathbf\\ & \text && A \mathbf \le \mathbf ... [...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] |
