NP-hardness

The conjecture postulates the NP-hardness of the following promise problem known as label cover with unique constraints.

The strong/weak kinds of NP-hardness are defined analogously.

This is different from much of complexity theory (e.g., NP-hardness), where the term "hard" is meant in the worst-case.

The same results on NP-hardness, inapproximability and approximability apply to both the maximum cut problem and the maximum bipartite subgraph problem.

Results about NP-hardness in theoretical computer science make heuristics the only viable option for a variety of complex optimization problems that need to be routinely solved in real-world applications.

The hardness result of Mulzer and Rote also implies the NP-hardness of finding an approximate solution with relative approximation error at most O(1/n).

Richard M. Karp showed in 1972 that the Hamiltonian cycle problem was NP-complete, which implies the NP-hardness of TSP.

Thus strong NP-completeness or NP-hardness may also be defined as the NP-completeness or NP-hardness of this unary version of the problem.

Therefore, it is possible to test in linear time whether a unique ordering exists, and whether a Hamiltonian path exists, despite the NP-hardness of the Hamiltonian path problem for more general directed graphs .

These two properties of the Moser spindle were used by to show the NP-hardness of testing whether a given graph has a two-dimensional unit distance representation; the proof uses a reduction from 3SAT in which the Moser spindle is used as the central truth-setting gadget in the reduction.

Removing the condition of visiting each city "only once" does not remove the NP-hardness, since it is easily seen that in the planar case there is an optimal tour that visits each city only once (otherwise, by the triangle inequality, a shortcut that skips a repeated visit would not increase the tour length).