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.
Best Answer
It's not quite clear to me, but I think you mean there is some $k \in [0, \ldots, N+1-\alpha]$ such that $\sum_{j=0}^M x_{ij} = 1$ for $i = k, k+1, \ldots, k+\alpha - 1$ while $\sum_{j=1}^M x_{ij} = 0$ for all other $i$. You can model this with binary variables $t_i$, $i=0 \ldots N+1-\alpha$, and constraints $$ \eqalign{\sum_{i=0}^{N+1-\alpha} t_i & = 1 \cr \sum_{j=0}^M x_{ij} - \sum_{k=\max(0,i+1-\alpha)}^{\min(i,N+1-\alpha)} t_k &= 0 \text{ for each $ i = 0 \ldots N$}\cr}$$