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Combinatorial Optimization
Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem. In many such problems, such as the ones previously mentioned, exhaustive search is not tractable, and so specialized algorithms that quickly rule out large parts of the search space or approximation algorithms must be resorted to instead. Combinatorial optimization is related to operations research, algorithm theory, and computational complexity theory. It has important applications in several fields, including artificial intelligence, machine learning, auction theory, software engineering, VLSI, applied mathematics and theoretical computer science. Applications Basic applications of combina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimum Spanning Tree
A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. That is, it is a spanning tree whose sum of edge weights is as small as possible. More generally, any edge-weighted undirected graph (not necessarily connected) has a minimum spanning forest, which is a union of the minimum spanning trees for its connected components. There are many use cases for minimum spanning trees. One example is a telecommunications company trying to lay cable in a new neighborhood. If it is constrained to bury the cable only along certain paths (e.g. roads), then there would be a graph containing the points (e.g. houses) connected by those paths. Some of the paths might be more expensive, because they are longer, or require the cable to be buried deeper; these paths would be represented by edges with larger weight ... [...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] |
Spanning Tree
In the mathematical field of graph theory, a spanning tree ''T'' of an undirected graph ''G'' is a subgraph that is a tree which includes all of the vertices of ''G''. In general, a graph may have several spanning trees, but a graph that is not connected will not contain a spanning tree (see about spanning forests below). If all of the edges of ''G'' are also edges of a spanning tree ''T'' of ''G'', then ''G'' is a tree and is identical to ''T'' (that is, a tree has a unique spanning tree and it is itself). Applications Several pathfinding algorithms, including Dijkstra's algorithm and the A* search algorithm, internally build a spanning tree as an intermediate step in solving the problem. In order to minimize the cost of power networks, wiring connections, piping, automatic speech recognition, etc., people often use algorithms that gradually build a spanning tree (or many such trees) as intermediate steps in the process of finding the minimum spanning tree. The Intern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flow Network
In graph theory, a flow network (also known as a transportation network) is a directed graph where each edge has a capacity and each edge receives a flow. The amount of flow on an edge cannot exceed the capacity of the edge. Often in operations research, a directed graph is called a network, the vertices are called nodes and the edges are called arcs. A flow must satisfy the restriction that the amount of flow into a node equals the amount of flow out of it, unless it is a source, which has only outgoing flow, or sink, which has only incoming flow. A flow network can be used to model traffic in a computer network, circulation with demands, fluids in pipes, currents in an electrical circuit, or anything similar in which something travels through a network of nodes. As such, efficient algorithms for solving network flows can also be applied to solve problems that can be reduced to a flow network, including survey design, airline scheduling, image segmentation, and the matching prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shortest-path Tree
In mathematics and computer science, a shortest-path tree rooted at a vertex ''v'' of a connected, undirected graph ''G'' is a spanning tree ''T'' of ''G'', such that the path distance from root ''v'' to any other vertex ''u'' in ''T'' is the shortest path distance from ''v'' to ''u'' in ''G''. In connected graphs where shortest paths are well-defined (i.e. where there are no negative-length cycles), we may construct a shortest-path tree using the following algorithm: # Compute dist(''u''), the shortest-path distance from root ''v'' to vertex ''u'' in ''G'' using Dijkstra's algorithm or Bellman–Ford algorithm. # For all non-root vertices ''u'', we can assign to ''u'' a parent vertex ''pu'' such that ''p''''u'' is connected to ''u'', and that dist(''p''''u'') + edge_dist(''p''''u'',''u'') = dist(''u''). In case multiple choices for ''pu'' exist, choose ''p''''u'' for which there exists a shortest path from ''v'' to ''p''''u'' with as few edges as possible; this tie-breaking rule ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shortest Path
In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The problem of finding the shortest path between two intersections on a road map may be modeled as a special case of the shortest path problem in graphs, where the vertices correspond to intersections and the edges correspond to road segments, each weighted by the length or distance of each segment. Definition The shortest path problem can be defined for graphs whether undirected, directed, or mixed. The definition for undirected graphs states that every edge can be traversed in either direction. Directed graphs require that consecutive vertices be connected by an appropriate directed edge. Two vertices are adjacent when they are both incident to a common edge. A path in an undirected graph is a sequence of vertices P = ( v_1, v_2, \ldots, v_n ) \in V \times V \times \cdots \times V suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements and objective are represented by linear function#As a polynomial function, linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the mathematical optimization, optimization of a linear objective function, subject to linear equality and linear inequality Constraint (mathematics), constraints. Its feasible region is a convex polytope, which is a set defined as the intersection (mathematics), intersection of finitely many Half-space (geometry), half spaces, each of which is defined by a linear inequality. Its objective function is a real number, real-valued affine function, affine (linear) function defined on this polytope. A linear programming algorithm finds a point in the po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial-time Algorithm
In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reservoir
A reservoir (; ) is an enlarged lake behind a dam, usually built to water storage, store fresh water, often doubling for hydroelectric power generation. Reservoirs are created by controlling a watercourse that drains an existing body of water, interrupting a watercourse to form an Bay, embayment within it, excavating, or building any number of retaining walls or levees to enclose any area to store water. Types Dammed valleys Dammed reservoirs are artificial lakes created and controlled by a dam constructed across a valley and rely on the natural topography to provide most of the basin of the reservoir. These reservoirs can either be ''on-stream reservoirs'', which are located on the original streambed of the downstream river and are filled by stream, creeks, rivers or rainwater that surface runoff, runs off the surrounding forested catchments, or ''off-stream reservoirs'', which receive water diversion, diverted water from a nearby stream or aqueduct (water supply), aq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Earth Science
Earth science or geoscience includes all fields of natural science related to the planet Earth. This is a branch of science dealing with the physical, chemical, and biological complex constitutions and synergistic linkages of Earth's four spheres: the biosphere, hydrosphere/ cryosphere, atmosphere, and geosphere (or lithosphere). Earth science can be considered to be a branch of planetary science but with a much older history. Geology Geology is broadly the study of Earth's structure, substance, and processes. Geology is largely the study of the lithosphere, or Earth's surface, including the crust and rocks. It includes the physical characteristics and processes that occur in the lithosphere as well as how they are affected by geothermal energy. It incorporates aspects of chemistry, physics, and biology as elements of geology interact. Historical geology is the application of geology to interpret Earth history and how it has changed over time. Geochemistry studies the che ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supply Chain Optimization
Supply-chain optimization (SCO) aims to ensure the optimal operation of a manufacturing and distribution supply chain. This includes the optimal placement of inventory within the supply chain, minimizing operating costs including manufacturing costs, transportation costs, and distribution costs. Optimization often involves the application of mathematical modelling techniques using computer software. It is often considered to be part of supply chain engineering, although the latter is mainly focused on mathematical modelling approaches, whereas supply chain optimization can also be undertaken using qualitative, management based approaches. Applications Typically, supply-chain managers aim to maximize the profitable operation of their manufacturing and distribution supply chain. This could include measures like maximizing gross margin return on inventory invested ( GMROII) (balancing the cost of inventory at all points in the supply chain with availability to the customer), minimi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
