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What is the lowest possible average-case time complexity of Shellsort with a deterministic fixed gap sequence? 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?
Moret, B. M. E.; H. D. Shapiro (1991). Algorithms from P to NP, volume 1 - Design and Efficiency . Redwood City, California: Benjamin/Cummings Publishing Company, Inc. Discusses intractability of problems with algorithms having exponential performance in Chapter 2, "Mathematical techniques for the analysis of algorithms."
List of unsolved problems may refer to several notable conjectures or open problems in various academic fields: Natural sciences, engineering and medicine [ edit ]
[1] The subset sum problem is a special case of the decision and 0-1 problems where each kind of item, the weight equals the value: =. In the field of cryptography, the term knapsack problem is often used to refer specifically to the subset sum problem. The subset sum problem is one of Karp's 21 NP-complete problems. [2]
For example, the decision problem "is the input even?" is formalized as the set of even numbers. A decision problem whose input consists of strings or more complex values is formalized as the set of numbers that, via a specific Gödel numbering , correspond to inputs that satisfy the decision problem's criteria.
The impossibility of the case A = 1 or B = 1 is implied by Catalan's conjecture, proven in 2002 by Preda Mihăilescu. (Notice C cannot be 1, or one of A and B must be 0, which is not permitted.) A potential class of solutions to the equation, namely those with A, B, C also forming a Pythagorean triple, were considered by L. Jesmanowicz in the ...
The variant where variables are required to be 0 or 1, called zero-one linear programming, and several other variants are also NP-complete [2] [3]: MP1 Some problems related to Job-shop scheduling; Knapsack problem, quadratic knapsack problem, and several variants [2] [3]: MP9 Some problems related to Multiprocessor scheduling
The expression problem is a challenging problem in programming languages that concerns the extensibility and modularity of statically typed data abstractions. The goal is to define a data abstraction that is extensible both in its representations and its behaviors, where one can add new representations and new behaviors to the data abstraction, without recompiling existing code, and while ...