During a class assignment, I was presented with the following question:
Provide an integer program that has an exponential number of branches...(expunged excess)
There was more to the question, but I'm primarily concerned and curious about finding more integer programs than I the ones I found that fit this criterion. A few models I came up with are:
$$\min z = x_{n+1}$$ Subject to: $$2x_1 + 2x_2 + \cdots + 2x_n + x_{n+1} = n$$ $$x_n\in\{0,1\}\forall n$$
when $n$ is odd, and
$$\max z = x_1 + x_2 + \cdots + x_n$$ Subject to: $$x_1 + x_2 + \cdots + x_n \le n-\frac{1}{2},\exists n\in\mathbb{R}^+$$ $$x_n\in\{0,1\}\forall n$$
Both of these are single-constraint knapsack problems.
I mentioned this problem to one of my peers, who said it reminded him of the Klee-Minty Cube. Therefore, I'm wondering/asking if more model types are not a single-constraint knapsack that proves that the branch-and-bound algorithm will take exponential steps, and if there isn't why?
