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The approximation ratio is defined as the ratio of the computed solution length to the optimal length for a worst-case instance, one that maximizes this ratio. Because the NP-hardness reduction for the k-minimum spanning tree problem preserves the weight of all solutions, it also preserves the hardness of approximation of the problem.
In graph theory, the metric k-center problem or vertex k-center problem is a classical combinatorial optimization problem studied in theoretical computer science that is NP-hard. Given n cities with specified distances, one wants to build k warehouses in different cities and minimize the maximum distance of a city to a warehouse.
A logarithmic chart allows only positive values to be plotted. A square root scale chart cannot show negative values. x: the x-values as a comma-separated list, for dates and time see remark in xType and yType; y or y1, y2, …: the y-values for one or several data series, respectively. For pie charts y2 denotes the radius of the corresponding ...
k = 1 and v = 2 yields a trivial graph of two vertices joined by an edge, k = 3 and v = 10 yields the Petersen graph, k = 7 and v = 50 yields the Hoffman–Singleton graph, discovered by Hoffman and Singleton in the course of this analysis, and; k = 57 and v = 3250 predicts a famous graph that has neither been discovered since 1960, nor has its ...
where g is the degree sum, e is the number of edges in the given graph, v is the number of vertices, and c is the number of connected components. The degree sum of a hypergraph is the sum of the degrees of all the vertices, reducing to 2e for a simple graph, or ke for a k-uniform hypergraph. This formula is symmetric between vertices and edges ...
The above graph represents a state-machine that takes user input as a series of bytes representing ASCII characters and control codes. 48..57 is equivalent to the regular expression [0-9] (i.e. any digit), so only sequences beginning with a digit can be recognised. If 10 (line feed) is encountered, the program is done. 46 is the decimal point ...
A graph is k-edge-connected if and only if the maximum flow from u to v is at least k for any pair (u,v), so k is the least u-v-flow among all (u,v). If n is the number of vertices in the graph, this simple algorithm would perform O ( n 2 ) {\displaystyle O(n^{2})} iterations of the Maximum flow problem, which can be solved in O ( n 3 ...
A variant of the problem asks for a minimum weight k-cut where the output partitions have pre-specified sizes. This problem variant is approximable to within a factor of 3 for any fixed k if one restricts the graph to a metric space, meaning a complete graph that satisfies the triangle inequality. [7]