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Can 3SUM be solved in strongly sub-quadratic time, that is, in time O(n 2−ϵ) for some ϵ>0? Can the edit distance between two strings of length n be computed in strongly sub-quadratic time? (This is only possible if the strong exponential time hypothesis is false.) Can X + Y sorting be done in o(n 2 log n) time?
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
That is, assuming a solution for H takes 1 unit time, H ' s solution can be used to solve L in polynomial time. [ 1 ] [ 2 ] 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 .
In computational complexity theory, a complexity class is a set of computational problems "of related resource-based complexity". [1] The two most commonly analyzed resources are time and memory. In general, a complexity class is defined in terms of a type of computational problem, a model of computation, and a bounded resource like time or memory.
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Similarly, a problem hard for a class C is called C-hard, e.g. NP-hard. Normally, it is assumed that the reduction in question does not have higher computational complexity than the class itself. Therefore, it may be said that if a C-complete problem has a "computationally easy" solution, then all problems in "C" have an "easy" solution.
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