Solving the Knapsack problem is pretty easy, it's probably the textbook example for explaining branch-and-bound algorithms. What you probably had in mind is that it's NP-complete, meaning that we don't know any polynomial-time algorithms and most experts think that none can be found, although nobody has been able to prove that (this is the famous P=NP? problem).

Which is why the OP says: > Find counterexamples to each of the following algorithms for the knapsack problem.

Anyway, yes, in the strictest sense of the term, it is unsolved because it's NP-complete, however it's been thoroughly analyzed in many different ways (it's pseudo-P, we have a MITM algo, different flavors of approximations, etc.)

Actually, the knapsack problem allows taking an item more than once. The problem described in the blog is the 0-1 knapsack problem which can be solved in polynomial time.