What does it mean for a problem to be considered NP-Hard?
user
When a problem is classified as NP-Hard, it means that it is at least as difficult as the hardest problems in the class of problems known as NP (Non-deterministic Polynomial time). NP problems are decision problems that can be solved in polynomial time by a non-deterministic Turing machine, a hypothetical machine that can explore multiple solutions simultaneously.

An NP-Hard problem, on the other hand, is a problem that is at least as hard as the hardest problems in NP. In other words, if an NP-Hard problem could be solved in polynomial time, then all problems in NP could also be solved in polynomial time.

The concept of NP-Hardness is important in computer science and mathematics because it helps to identify problems that are difficult to solve. Many optimization problems, such as the traveling salesman problem and the knapsack problem, are NP-Hard. While it is not currently known whether these problems can be solved efficiently, researchers have developed algorithms that can approximate solutions to within a certain degree of accuracy.
assistant
Explain to me what polynomial time means
user
When a problem can be solved in polynomial time, then given k,C > 0, the steps the algorithm needs to take for an input of size n is at most Cn^k. Though in practice an algorithm with a polynomial running time may not be the most efficient one for different input sizes, we care more about asymptotic behavior when it comes to theory. Besides, it allows us to define an "efficiency" that can be achieved by any machine model.
assistant
What is a non-deterministic Turing machine, and how does it relate to NP problems?
user