According to paper Ising formulations of many NP problems, Vertex Cover problem has the following Ising formulation:
$$\underset{x}{\text{min }} f(x) = a\sum_{(i,j) \in E}(1-x_i)(1-x_j) + b\sum_{i \in V} x_i.$$
I'd like to share some doubts of mine.
Ising formulation has spin variables $\{+1,-1\}$, not logical variable $\{0,1\}$. Shouldn't it be considered a QUBO problem?
By considering this formulation as a QUBO problem, then I should get the Ising formulation by mapping $x_i \rightarrow \frac{1 - \text{Z}_i}{2}$ as follows: $$H = \frac{a}{4}\sum_{(i,j) \in E}(\text{Z}_i + \text{Z}_j + \text{Z}_i\text{Z}_j) - \frac{b}{2}\sum_{i \in V}\text{Z}_i + |V|.$$
However, from qiskit library authors consider the following hamiltonian:
$$H = a\sum_{(i,j) \in E}(1-\text{X}_i)(1-\text{X}_j) +b\sum_{i \in V}\text{Z}_i.$$ which seems different from both previous formulations. Are the two hamiltonians somehow equivalent?