Given a set of binary variables $x_{ij} \in X,\ i=0,..,N,\ j=0,..,M$ how do I model an adjacency constraint on $i$'s such that:
- $\sum_i^N\sum_j^Mx_{ij} = \alpha, \;\text{with }\ 0 < \alpha < N$ and
- $i_0,\ldots,i_\alpha$ are adjacent/consecutive numbers.
For example, assuming $\alpha=3$, $\{x_{32}=\ x_{49}=\ x_{57}=1\}$ would be a feasible solution while $\{x_{22}=\ x_{49}=\ x_{57}=1\}$ is not because $i=2$ in $x_{i=2,j=2}$ is not consecutive to $i=4$ in $x_{i=4,j=9}$.
EDIT:
In simpler words given N binary variables $x_{i},x_{i+1},..,x_{i+N}$ exactly $\alpha$ variables can be equal to 1 and they have to be consecutives. For example $\{x_{i} = x_{i+1} = x_{i+2} = 1\}$ is a valid solution while $\{x_{i} = x_{i+1} = x_{i+3} = 1\}$ is not.
