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As it is suspected, but unproven, that P≠NP, it is unlikely that any polynomial-time algorithms for NP-hard problems exist. [3] [4] A simple example of an NP-hard problem is the subset sum problem. Informally, if H is NP-hard, then it is at least as difficult to solve as the problems in NP.
The arithmetical hierarchy and polynomial hierarchy classify the degree to which problems are respectively computable and computable in polynomial time. For instance, the level Σ 0 0 = Π 0 0 = Δ 0 0 {\displaystyle \Sigma _{0}^{0}=\Pi _{0}^{0}=\Delta _{0}^{0}} of the arithmetical hierarchy classifies computable, partial functions.
Approximation problems are often known to be NP-hard assuming UGC; such problems are referred to as UG-hard. In particular, assuming UGC there is a semidefinite programming algorithm that achieves optimal approximation guarantees for many important problems.
A dense packing of spheres with a radius ratio of 0.64799 and a density of 0.74786 [22] Many problems in the chemical and physical sciences can be related to packing problems where more than one size of sphere is available.
There are many variations on the densest subgraph problem. One of them is the densest k subgraph problem, where the objective is to find the maximum density subgraph on exactly k vertices. This problem generalizes the clique problem and is thus NP-hard in general graphs.
In computer science, lattice problems are a class of optimization problems related to mathematical objects called lattices.The conjectured intractability of such problems is central to the construction of secure lattice-based cryptosystems: lattice problems are an example of NP-hard problems which have been shown to be average-case hard, providing a test case for the security of cryptographic ...
Packing circles in simple bounded shapes is a common type of problem in recreational mathematics. The influence of the container walls is important, and hexagonal packing is generally not optimal for small numbers of circles. Specific problems of this type that have been studied include: Circle packing in a circle; Circle packing in a square
The hard sphere system exhibits a fluid-solid phase transition between the volume fractions of freezing and melting . The pressure diverges at random close packing η r c p ≈ 0.644 {\displaystyle \eta _{\mathrm {rcp} }\approx 0.644} for the metastable liquid branch and at close packing η c p = 2 π / 6 ≈ 0.74048 {\displaystyle \eta ...