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The canonical optimization variant of the above decision problem is usually known as the Maximum-Cut Problem or Max-Cut and is defined as: Given a graph G, find a maximum cut. The optimization variant is known to be NP-Hard. The opposite problem, that of finding a minimum cut is known to be efficiently solvable via the Ford–Fulkerson algorithm.
The satisfiability problem, also called the feasibility problem, is just the problem of finding any feasible solution at all without regard to objective value. This can be regarded as the special case of mathematical optimization where the objective value is the same for every solution, and thus any solution is optimal.
Finding global maxima and minima is the goal of mathematical optimization. If a function is continuous on a closed interval, then by the extreme value theorem, global maxima and minima exist. Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the ...
In an unweighted bipartite graph, the optimization problem is to find a maximum cardinality matching. The problem is solved by the Hopcroft-Karp algorithm in time O ( √ V E ) time, and there are more efficient randomized algorithms , approximation algorithms , and algorithms for special classes of graphs such as bipartite planar graphs , as ...
Low-order polynomials tend to be smooth and high order polynomial curves tend to be "lumpy". To define this more precisely, the maximum number of inflection points possible in a polynomial curve is n-2, where n is the order of the polynomial equation. An inflection point is a location on the curve where it switches from a positive radius to ...
In the undirected edge-disjoint paths problem, we are given an undirected graph G = (V, E) and two vertices s and t, and we have to find the maximum number of edge-disjoint s-t paths in G. Menger's theorem states that the maximum number of edge-disjoint s-t paths in an undirected graph is equal to the minimum number of edges in an s-t cut-set.
Other local search algorithms try to overcome this problem such as stochastic hill climbing, random walks and simulated annealing. Despite the many local maxima in this graph, the global maximum can still be found using simulated annealing. Unfortunately, the applicability of simulated annealing is problem-specific because it relies on finding ...
A graph may have many maximal cliques, of varying sizes; finding the largest of these is the maximum clique problem. Some authors include maximality as part of the definition of a clique, and refer to maximal cliques simply as cliques. Left is a maximal independent set. Middle is a clique, , on the graph complement. Right is a vertex cover on ...