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A key concept in epistemic logic, this problem highlights the importance of common knowledge. Some authors also refer to this as the Two Generals' Paradox, the Two Armies Problem, or the Coordinated Attack Problem. [1] [2] The Two Generals' Problem was the first computer communication problem to be proven to be unsolvable. [3]
Problems 1, 2, 5, 6, [a] 9, 11, 12, 15, and 22 have solutions that have partial acceptance, but there exists some controversy as to whether they resolve the problems. That leaves 8 (the Riemann hypothesis), 13 and 16 [b] unresolved. Problems 4 and 23 are considered as too vague to ever be described as solved; the withdrawn 24 would also be in ...
NP and co-NP together form the first level in the polynomial hierarchy, higher only than P. NP is defined using only deterministic machines. If we permit the verifier to be probabilistic (this, however, is not necessarily a BPP machine [6]), we get the class MA solvable using an Arthur–Merlin protocol with no communication from Arthur to Merlin.
Goldbach’s Conjecture. One of the greatest unsolved mysteries in math is also very easy to write. Goldbach’s Conjecture is, “Every even number (greater than two) is the sum of two primes ...
A problem is hard for a class of problems C if every problem in C can be polynomial-time reduced to . Thus no problem in C is harder than , since an algorithm for allows us to solve any problem in C with at most polynomial slowdown. Of particular importance, the set of problems that are hard for NP is called the set of NP-hard problems.
Therefore, the third readers–writers problem is sometimes proposed, which adds the constraint that no thread shall be allowed to starve; that is, the operation of obtaining a lock on the shared data will always terminate in a bounded amount of time. A solution with fairness for both readers and writers might be as follows:
The problems of finding a Hamiltonian path and a Hamiltonian cycle can be related as follows: In one direction, the Hamiltonian path problem for graph G can be related to the Hamiltonian cycle problem in a graph H obtained from G by adding a new universal vertex x, connecting x to all vertices of G. Thus, finding a Hamiltonian path cannot be ...
Quadratic programming (NP-hard in some cases, P if convex) Subset sum problem [3]: SP13 Variations on the Traveling salesman problem. The problem for graphs is NP-complete if the edge lengths are assumed integers. The problem for points on the plane is NP-complete with the discretized Euclidean metric and rectilinear metric.